贝叶斯 定理

Have you ever noticed how you can be fuming with anger one second and absolutely calm the next?

您是否注意到过如何一秒钟发怒并在下一秒钟完全冷静？

A bad driver cuts you off on the highway, and you’re raging. A moment later, you notice him pull into the hospital and your anger melts away. “Yeah, maybe he has a patient in the car with him. Or, maybe someone close is dying. I guess he’s not so bad after all.”

坏司机在高速公路上挡住了你，让你发狂。 片刻之后，您发现他被送入医院，您的愤怒消散了。 “是的，也许他有一个病人陪在车上。 或者，也许有人在死。 我猜他毕竟还算不错 。”

An obscure rule from probability theory called Bayes Theorem explains this very well. This 9,000-word blog post is a complete introduction to Bayes Theorem and how to put it to practice. In short, Bayes Theorem is a framework for critical thinking. By the end of this post, you’ll be making better decisions, realise when you’re being unreasonable, and also understand why some people believe in UFOs.

概率论中一个晦涩的规则叫做贝叶斯定理，很好地解释了这一点。 这篇9000字的博客文章全面介绍了贝叶斯定理以及如何将其付诸实践。 简而言之，贝叶斯定理是批判性思维的框架。 在这篇文章的结尾，您将做出更好的决策，意识到自己何时变得不合理，并理解为什么有些人相信UFO。

It’s a hefty promise, and there’s a good chance of failure. Implementing these ideas will take emotional effort, but it’s worth it.

这是一个巨大的承诺，并且很有可能失败。 实施这些想法将需要花大力气，但这是值得的。

Thinking the driver is bad is normal. Bayes Theorem expects the same. The difference between Bayes and us is the intensity with which we believe things. Most times, the seething anger isn’t warranted. This is probably why we feel stupid about all that anger. It melts away so quickly! This is calibration — aligning our emotions to the intensity of the situation — which we’ll cover as well.

认为驾驶员不好是正常的。 贝叶斯定理期望相同。 贝叶斯和我们之间的区别是我们相信事物的强度。 在大多数情况下，这种冒犯性的愤怒是没有必要的。 这可能就是为什么我们对所有这些愤怒感到愚蠢。 它融化的太快了！ 这是校准-使我们的情绪与情况的强度保持一致-我们也将介绍。

There’s no fancy math in this guide. We’re using probability theory, but aren’t going into the derivation, nor are we solving probability problems from school textbooks. These things are tedious without understanding the *why*. Instead, we’ll understand why Bayes Theorem matters and how to apply it.

本指南中没有花哨的数学运算。 我们使用的是概率论，但没有进行推导，也没有从学校教科书中解决概率问题。 这些事情很繁琐，却不了解*原因* 。 相反，我们将了解为什么贝叶斯定理很重要以及如何应用它。

To begin with, let’s play a game. Throughout this game, *notice* how you feel about your decisions. Notice what decisions you’re making, and notice how you find the answer.

首先，让我们玩一个游戏。 在整个游戏中，请*注意*您对自己的决定的感觉。 注意您正在做的决定，并注意如何找到答案。

**The 2, 4, 6 Game****Bayes Explanation****Bayes Theorem****Example: Being Late****Grinding the Gears****The Four Rules for Being a Good Bayesian**Probability is a map of your understanding of the worldUpdate incrementallySeek disconfirming evidenceRemember your priors**Destroying Cognitive Biases****Seeking Disconfirming Evidence for Bayes****The 2, 4, 6 Game Revisited****The Being Late Example Revisited****Getting Stronger****Improve Your Priors****Become a Master Hypothesis Builder****Learn the Math****Putting It All Together in Practice**Hypotheses with frequenciesGrowing the diskSwitching hypotheses**Strong Opinions, Weakly Held?****This Seems Very Different to What I Learned in School****Epilogue: The End is the Beginning****Endnotes**

# 2，4，6游戏 (The 2, 4, 6 Game)

There’s a black box with a formula inside for generating three numbers. Your job is to try and guess this formula. The input box below is connected to the black box. If you give it three numbers, it’s going to tell you whether they follow the formula or not. Separate each number with a comma.

有一个黑匣子，里面有一个公式，用于生成三个数字。 您的工作是尝试猜测此公式。 下面的输入框连接到黑框。 如果给它三个数字，它将告诉您它们是否遵循公式。 用逗号分隔每个数字。

To start you off, (2, 4, 6) follows the pattern. Try it out!

首先，请遵循(2，4，6)的模式。 试试看！

Alternatively, play it here. Write down your answer when you figure it out.

或者， 在这里播放 。 找出答案后写下答案。

# 贝叶斯解释 (Bayes Explanation)

Most people try some sequence of (4, 6, 8), (1, 2, 3)… and end up with either increasing numbers or increasing numbers that are even. Notice how you’re pretty confident in your answer by the time you write it down. You’ve tried a few examples, and **all** of them made sense!

大多数人尝试按(4、6、8)，(1、2、3)的顺序进行排序，最终结果要么增加数字要么增加偶数。 请注意，当您写下答案时，您对答案很有信心。 您已经尝试了一些示例，并且**所有**示例**都有**意义！

But perhaps you didn’t think to try (-1, 2, 10) or (4, 2, 6).

但是也许您没有想到尝试(-1，2，10)或(4，2，6)。

If my comment made your confidence waver, go ahead and try the input box again. See if you can find a pattern that works. The answer is at the bottom of this section, but don’t skip ahead. Every sentence before that is setting up an important idea.

如果我的评论使您信心不足，请继续尝试再次输入框。 看看是否可以找到有效的模式。 答案在本节的底部，但不要跳过。 在此之前的每个句子都在树立一个重要的思想。

# 贝叶斯定理 (Bayes Theorem)

If you’ve heard of Bayes Theorem before, you know this formula:

如果您以前听说过贝叶斯定理，那么您会知道以下公式：

Indeed, that’s all there is to it. I bet you’ve also heard the famous formula: E = mc². That’s all there is to mass-energy equivalence. However, figuring out how to harness nuclear energy is still a hard problem. The formula made it possible, but implementing it still took 40 years.

确实，这就是全部。 我敢打赌，您还听说过著名的公式：E =mc²。 这就是质能等效性的全部。 但是，弄清楚如何利用核能仍然是一个难题。 该公式使之成为可能，但是实施它仍然需要40年。

It’s the same with Bayes Theorem. The formula is exciting because of what it implies. We’re discovering the nuclear energy version of Bayes Theorem.

贝叶斯定理也是如此。 由于其含义，该公式令人兴奋。 我们正在发现贝叶斯定理的核能版本。

Translated to English, the formula goes like this:

翻译成英文，公式如下：

To form accurate beliefs, you always start from the information you already have. You update beliefs. You don’t discard everything you know.

为了形成准确的信念，您总是从已经拥有的信息开始。 您更新信念。 您不会丢弃所有您知道的东西。

The first key component is a hypothesis (H) — the belief we’re talking about.

第一个关键部分是假设(H)-我们正在谈论的信念。

The second key component is the evidence (E) — what data do we have to support/reject the hypothesis?

第二个关键要素是证据(E)-我们必须支持/拒绝该假设的哪些数据？

The third key component is the probability (P) of the above two. This probability is our confidence in the belief.

第三个关键因素是上述两个因素的概率(P)。 这种可能性是我们对信念的信心。

If you’re familiar with probability theory, you learned this in school. If not, don’t worry, there are excellent mathematical introductions to explain it to you. We’ll skip the math and focus on how to use it.

如果您熟悉概率论，那么您是在学校学习的。 如果没有，请不要担心，这里有出色的数学入门向您解释。 我们将跳过数学，而将重点放在如何使用它上。

Our point of interest, and where Bayes truly shines, is where we compare two hypotheses. Instead of uncovering the absolute probabilities, which is hard, this focuses on how much more likely one hypothesis is compared to another. Most reasoning, in our minds, takes this form.

我们的兴趣点以及贝叶斯真正发光的地方是我们比较两个假设的地方。 与其揭示难以确定的绝对概率，不如说是将一种假设与另一种假设进行比较的可能性。 我们认为，大多数推理都采用这种形式。

In this case, the formula looks like:

在这种情况下，公式如下所示：

Posterior odds measure how likely a hypothesis is compared to another one.

后验概率测量一种假设与另一种假设进行比较的可能性。

Prior odds measure how likely it was before we had any new evidence.

事前赔率衡量的是我们获得任何新证据之前的可能性。

Likelihood odds measure how well the evidence explains the current hypothesis compared to the other one. We’ll explore what this means with the help of examples.

可能性几率衡量的是证据相对于另一种假设对当前假设的解释程度。 我们将借助示例探索这意味着什么。

We’ll start with the 2, 4, 6 game to show how qualitatively, math and intuition agree. Then we’ll get into a simpler example where we’re miscalibrated and do the math.

我们将从2、4、6游戏开始，以显示定性，数学和直觉如何吻合。 然后，我们将进入一个更简单的示例，在该示例中我们未正确校准并进行数学计算。

I’m going to choose my path through the 2, 4, 6 game, but I hope yours was similar enough. If not, try doing this on your own!

我将选择参加2、4、6游戏的路线，但我希望你的路途足够相似。 如果没有，请尝试自己做！

I have a hypothesis I want to test, *H-3even* = 3 even numbers in increasing order. It’s implicit here, but the hypothesis I’m testing this against is *H-not-3even*, or that the formula **is not** 3 even numbers in increasing order.

我有一个要测试的假设， *H-3even* = 3个偶数，按递增顺序排列。 它在这里是隐式的，但是我要针对其进行测试的假设是*H-not-3even* ，或者该公式**不是** 3个偶数递增的。

I input (4, 6, 8) and the black box says “Yes.” My confidence in 3 even numbers rises. In Bayesian-speak, my posterior odds have increased, because the likelihood odds have increased. And the likelihood odds have increased, since the probability of (4, 6, 8) saying “Yes” is higher when the formula is *H-3even*.

我输入(4，6，8)，黑框说“是”。 我对3个偶数的信心上升。 用贝叶斯说话，我的后验几率增加了，因为可能性几率增加了。 并且，当公式为*H-3even*时， *(4，6，8* )说“是”的可能性更高，因此可能性可能性也增加了。

You’ll notice how *you feel* every new number that matches your hypothesis makes your belief stronger.

你会发现*你*的*感觉*如何符合你的假设，使你的信念更强的每一个新的号码。

I try (1, 2, 3) next. “Yes.” What? I expected “No!”.

我接下来尝试(1,2,3)。 “是。” 什么？ 我期望“不！”。

Everything tumbles, like it should when you find something that doesn’t follow the pattern. The probability of (1, 2, 3) saying “Yes” is higher with *H-not-3even*, since (1, 2, 3) are not all even. The likelihood odds are in favour of *H-not-3even* now, which means we discard *H-3even*. In this case, one small piece of evidence was enough to completely flip the scales.

一切都在下跌，就像您发现不遵循模式的东西一样。 *H-not-3even*表示(1、2、3)说“是”的可能性更高，因为(1、2、3)都不都是偶数。 现在，似然率支持*H-not-3even* ，这意味着我们丢弃*H-3even* 。 在这种情况下，只有一小部分证据足以完全翻转天平。

Then, which new hypothesis should you try? The clues usually lie in how you disproved the previous hypothesis.

那么，您应该尝试哪种新假设？ 线索通常在于您如何反驳先前的假设。

I tried (1, 2, 3) which said “Yes” when I expected it to say “No.” My new hypothesis thus became “3 increasing numbers”.

我曾尝试(1、2、3)说“是”，但我希望它说“否”。 我的新假设因此变成“ 3个递增的数字”。

Just like in the previous case, (4, 2, 6) saying “Yes” killed this hypothesis. My new hypothesis thus became “3 positive numbers”.

就像在前面的案例中一样，(4，2，6)说“是”杀死了这个假设。 我的新假设因此变成“ 3个正数”。

I tried (-1, 2, 3), which said “No”! This was all I needed to become reasonably confident in “3 positive numbers”. The more negative numbers I tried, the more confident I got.

我尝试过(-1，2，3)，说“不”！ 这就是我要对“ 3个正数”变得相当有信心的全部。 我尝试的负数越多，我越有信心。

3 positive numbers is, indeed, correct.¹

实际上，3个正数是正确的。¹

Graphically, this is what’s happening with the three hypotheses:

从图形上讲，这是三个假设的结果：

Interlude: Can we prove something to be true with Bayes?No matter how much data you have, you can never say something is true. This is the problem of induction.

插曲：我们可以证明贝叶斯的真实性吗？无论您拥有多少数据，您都无法说出真实的话。这是归纳的问题。

“No amount of observations of white swans can allow the inference that all swans are white, but the observation of a single black swan is sufficient to refute that conclusion.”

“对白天鹅的任何观察都不能得出所有天鹅都是白色的推断，但是观察到一只黑天鹅就足以驳斥这一结论。”

However, after a certain level of confidence, you live your life believing it’s true. Once you start believing is when you must pay close attention to evidence that doesn’t fit.

但是，经过一定程度的自信后，您就会相信自己的生活是真实的。一旦开始相信，就必须密切注意不适当的证据。

Calibration is key. What we’ve just shown is our thinking process, and how Bayes Theorem is mostly aligned with it when we’re thinking well. Bayes Theorem updates beliefs in the same direction our brains do, but what changes is how much each piece of evidence influences us!

校准是关键。 我们刚刚显示的是我们的思维过程，以及在我们进行良好思考时贝叶斯定理如何与之保持一致。 贝叶斯定理以与我们大脑相同的方向更新信念，但是变化的是每条证据对我们的影响有多大！

With this next example, let’s get into the basic math. We’ll revisit the 2, 4, 6 game in a bit.

在下一个示例中，让我们进入基本数学。 我们将再次讨论2、4、6游戏。

# 示例：迟到 (Example: Being Late)

Your colleague is sometimes late to work. They’ve been on time four times and late three times the past week. How many more times would it take you to start believing they’re “always” late?

您的同事有时上班迟到。 他们过去一周来了四次，最后三次都按时到了。 您会开始相信他们“总是”迟到多少次？

In my experience, just a few more times does the trick. But let’s use Bayes to calibrate.

以我的经验，可以完成更多次。 但是，让我们使用贝叶斯进行校准。

Since there’s no good reason to expect tardiness over punctuality, let’s say the prior odds are 1:1². The alternative hypothesis, the one we’re testing against, is “not always being late.” To make this more concrete, let’s say this means they’re late only 10% of the time.³

由于没有充分的理由期待迟到而不是守时，因此，先验赔率为1：1²。 我们正在测试的另一种假设是“并不总是迟到”。 为了更具体一点，可以说这意味着他们只在10％的时间内迟到。³

We’ll use the data we have to calculate the likelihood of being late. We want to contrast the data with us believing that they’re almost always on time, or almost always late. Remember, to figure this out, we imagine believing the first hypothesis, then judge how likely the data is. Then, we imagine believing the second hypothesis and judge how likely the data is.

我们将使用必须的数据来计算迟到的可能性。 我们想与我们对比数据，认为它们几乎总是准时的，或者几乎总是迟到的。 请记住，要弄清楚这一点，我们假设相信第一个假设，然后判断数据的可能性。 然后，我们想象相信第二个假设并判断数据的可能性。

There are several ways to mathematically represent this data, from a binomial function to a beta distribution. However, we’re not getting into that yet. Today is more about an intuitive explanation, one which you’re more likely to use every day. So try to imagine the odds for and against given this data.⁴

从二项式函数到Beta分布，有几种方法可以用数学方式表示此数据。 但是，我们还没有涉及到这一点。 今天的内容更多是关于一种直观的解释，您每天都可能会使用这种解释。 因此，请尝试想象给定此数据的可能性。⁴

It’s a bit surprising to me that for someone who’s supposed to be always late, they’re on-time more times than not. So this evidence is pretty unlikely for our hypothesis, while much more plausible when they’re not always late. I’ll give it likelihood odds of 1:10.⁵

对于我来说应该总是很晚的人，他们准时上班的次数比不上来要多。 因此，对于我们的假设来说，这种证据几乎是不可能的，而当它们并非总是迟到时，则更有说服力。 我给它1:10的可能性。

Now, the posterior odds are 1:10 * 1:1 = 1:10. So, they’re 10 times less likely to be late than not.⁶

现在，后验赔率是1:10 * 1：1 = 1:10。 因此，他们迟到的可能性比不发生的可能性低10倍。⁶

Huh, that’s surprising, isn’t it? Much lower than I imagined.

嗯，这令人惊讶，不是吗？ 比我想象的要低得多。

Let’s take this example further. Say we want to figure out how many more times would they have to be late to get the odds up to 100:1?

让我们进一步讲这个例子。 假设我们想弄清楚他们必须再迟多少次才能使赔率达到100：1？

We’ll do the same updating process again.

我们将再次执行相同的更新过程。

Prior odds = the old posterior odds = 1:10 for being late.

先发赔率=旧的后赔率= 1:10。

Likelihood ratio = ??

可能性比= ??

Posterior odds = 100:1

后验赔率= 100：1

Thus, likelihood ratio = 100:1 / 1:10 = 1000:1. Which means they’ll have to be late for 12 days out of 20 to get that kind of odds.⁷

因此，似然比= 100：1 / 1:10 = 1000：1。 这意味着他们必须在20天之内迟到12天才能获得这种赔率。⁷

These numbers might feel weird. They are. We’re not used to calibrating and in this specific case, we’re comparing two extreme hypotheses, which makes things weirder. Usually, this is a case of “I don’t know what the numbers mean.” We’ll explore this unsettling feeling in a bit.

这些数字可能会令人感到奇怪。 他们是。 我们不习惯进行校准，在这种特定情况下，我们正在比较两个极端假设，这使事情变得很奇怪。 通常，这是“我不知道数字是什么意思”的情况。 我们将稍作探讨。

Neither hypothesis is close to what we’d expect anymore. After all, the colleague is late just 12 out of 20 days!? Why aren’t we considering another hypothesis — say, they’re late 60% of the time? Try this hypothesis out — it should blow the other two out of the park. You can use this calculator I built.

两种假设都不再接近我们的预期。 毕竟，同事在20天之内仅迟到12天！ 我们为什么不考虑另一种假设-比如说，他们迟到了60％？ 尝试一下该假设-它应该将另外两个假设从公园中吹出来。 您可以使用我构建的计算器 。

For now, note that a little math helps us calibrate vastly better for the hypothesis we do have. It puts our emotions and thoughts into perspective.

现在，请注意一点数学可以帮助我们更好地针对我们的假设进行校准。 它使我们的情感和思想成为现实。

Also, notice how we transformed the problem to become easier to grasp. Instead of applying the complicated version requiring pen and paper (what are the probabilities, what are the formulas, how do I find them?), we follow this “old odds times current likelihood equals new odds”.

另外，请注意我们如何将问题转变为更容易掌握的问题。 而不是应用需要笔和纸的复杂版本(概率是多少，公式是什么，如何找到它们？)，我们遵循的是“旧几率乘以当前几率等于新几率”。

When you have this model explaining how the “ideal you” thinks, you get a gears-level understanding! This is epic because you can fiddle with the gears and see how your thinking changes. You can figure out best practices, and become a stronger thinker. This is exactly what we’re going to do next.

当您使用此模型解释“理想的自己”的想法时，您将获得齿轮级的理解 ！ 这是史诗般的，因为您可以摆弄齿轮，看看您的想法如何变化。 您可以找出最佳做法，并成为更强大的思想家。 这正是我们接下来要做的。

Note:The mind doesn’t always work like Bayes Theorem prescribes. There’s lots of things Bayes can’t explain. Don’t try to fit everything you see into Bayes rule. But wherever beliefs are concerned, it’s a good model to use. Find the best beliefs and calibrate them!

注意：头脑并不总是像贝叶斯定理所规定的那样工作。 贝叶斯无法解释很多事情。 不要试图使您看到的所有内容都适合贝叶斯规则。 但是，无论信仰如何，它都是一个很好的模型。 找到最佳信念并进行校准！

# 磨齿轮 (Grinding the Gears)

Let’s start grinding these gears. A meta-goal for this section is to make the gears visible to you. We are inching closer to the nuclear energy version of Bayes Theorem, but so far we’ve just seen how it’s possible. The next step is to learn how to do it ourselves.

让我们开始研磨这些齿轮。 此部分的目标是使齿轮对您可见。 我们离贝叶斯定理(Bayes Theorem)的核能版本越来越近，但是到目前为止，我们仅看到了它的可能性。 下一步是学习如何自己做。

Using Bayes Theorem, try to answer this question:

使用贝叶斯定理，尝试回答以下问题：

**How can two people see the same things and reach different conclusions?**

**两个人如何看待相同的事物并得出不同的结论？**

Let’s first assume they’re Bayesian, and follow the rules. Can they reach different conclusions?

首先让我们假设他们是贝叶斯，然后遵循规则。 他们能否得出不同的结论？

Yes! The current data only gives us the likelihood odds. There’s a second component, the priors, that are yet to be used! In the “being late” example, we thought them up using our background knowledge. This is why they’re called priors: It’s information that we already have.

是! 当前数据仅给我们可能性可能性。 还有另一个组件，先验，还有待使用！ 在“迟到”的例子中，我们使用背景知识来思考它们。 这就是为什么它们被称为先验的原因：这是我们已经拥有的信息。

It’s plausible that two different people have different experiences and different information, which means they can have different priors. Sometimes, these priors can get pretty strong. You, who has seen their colleague be late 3 out of 7 days, will have different priors compared to Bob, who has seen their colleague be late 6 out of 10 days. Bob has seen your colleague be late twice as much!

两个不同的人有不同的经历和不同的信息是合理的，这意味着他们可以有不同的先验条件。 有时，这些先验会变得很强大。 与同事鲍勃迟到六天的鲍勃相比，看到同事迟到7天之三的您的先验时间会有所不同。 鲍勃看到你的同事迟到了两倍！

Seeing your colleague be on time the following day would change your confidence a lot more than Bob. Can you figure out how much?

第二天见到您的同事准时会比鲍勃更能改变您的信心。 你能算出多少吗？

For you = 1:20 (priors, 3 out of 7 days late) * (1:20) = 1:400 = 400x likely to believe their colleague is **not** “always” late.

对于您= 1:20(优先级，晚7天中有3天)*(1:20)= 1：400 = 400倍可能认为他们的同事**没有** “总是”迟到。

For Bob = 300:1 (priors, 6 out of 10 days late) * (1:20) = 15:1 = 15x likely to believe their colleague is “always” late.

对于Bob = 300：1(之前的10天中有6天)*(1:20)= 15：1 = 15倍可能认为自己的同事“总是”迟到了。

But, if both people are following Bayes rule, they’ll both update in the same direction, and the same % amount — the one given by the likelihood ratio. That’s (1:20), in this case. Notice again how the numbers feel weird, like, this shouldn’t be correct.

但是，如果两个人都遵循贝叶斯规则，他们都会朝着相同的方向和相同的百分比金额(由似然比给出的百分比)进行更新。 在这种情况下，就是(1:20)。 再次注意到数字感觉很奇怪，例如，这不正确。

The second case is when people aren’t following Bayes rule at all. Remember, Bayes is a prescription, not a description! Which means it’s how things ought to be, not how they *are*. This is where things get murky. It’s easy to spin evidence against a belief into evidence for one, especially if the hypothesis isn’t clear cut. We’ll get into this soon — just a few more ideas to attack first.

第二种情况是人们根本不遵循贝叶斯规则。 记住，贝叶斯是处方，不是描述！ 这意味着它是如何的事情应该是，他们没有怎么样*了* 。 这就是事情变得晦暗的地方。 将一种信念的证据转化为一种信念的证据很容易，尤其是在假设不明确的情况下。 我们将尽快进行讨论-首先要提出的其他一些想法。

# 成为好贝叶斯的四个规则 (The Four Rules for Being a Good Bayesian)

Bayes Theorem has been around for 200 years, and in this time, people have experimented with the gears to come up with best practices. Thanks to these people, we’ll explore four ideas driven from Bayes Theorem. Mastering these is key to becoming a better thinker.

贝叶斯定理已经存在了200年，在这个时候，人们已经尝试了一些齿轮来提出最佳实践。 多亏了这些人，我们将探讨贝叶斯定理带来的四个想法。 掌握这些是成为更好的思想家的关键。

## 概率是您对世界的了解的地图 (Probability is a map of your understanding of the world)

“Ignorance

exists in the map, not in the territory.”“无知

存在于地图中，而不存在于领土内。”

In this lens, the world is probabilistic. Everything has a probability of happening based on *your* knowledge of the world. The probability is intrinsic to you, not the objects or situations.

从这个角度来看，世界是概率性的。 根据*您*对世界*的*了解，一切都有发生的可能性。 概率是您固有的，而不是物体或情况。

For example, attacking in a gunfight might seem implausible to you, but mafia thugs would disagree. Your knowledge of the world determines your probabilities. These probabilities are your confidence in a belief.

例如，用枪战进行攻击对您来说似乎难以置信，但黑手党暴徒会不同意。 您对世界的了解决定了您的概率。 这些概率是您对信念的信心。

Consider this seemingly weird example:

考虑以下看似奇怪的示例：

You meet a mathematician and she tells you she has two children. You ask if at least** one** of them is a boy. She says yes. You then decide the probability of both of them being boys **must be 1/3**.

您遇到一位数学家，她告诉您她有两个孩子。 您问**其中**至少有**一个**是男孩。 她说是的。 然后，您确定他们俩都是男孩的概率**必须为1/3** 。

You meet another mathematician. She tells you she has two children. You ask if the **elder** one is a boy. She says yes. You then decide the probability of both of them being boys **must be 1/2**.

您遇到了另一位数学家。 她告诉你她有两个孩子。 您问**长者**是否是男孩。 她说是的。 然后，您确定他们俩都是男孩的概率**必须为1/2** 。

You meet another mathematician. She tells you she has two children. You ask if the **younger** one is a boy. She says yes. You then decide the probability of both of them being boys **must be 1/2**.

您遇到了另一位数学家。 她告诉你她有两个孩子。 您问**小**一点的是男孩。 她说是的。 然后，您确定他们俩都是男孩的概率**必须为1/2** 。

If at least one of them is a boy, then either the elder or the younger one is a boy. So what gives? Why are the probabilities different? If the children had an inherent property of being boys, the answer should always be the same.

如果其中至少有一个是男孩，则年龄较大或较小的是男孩。 那有什么呢？ 为什么概率不同？ 如果孩子们天生就是男孩，答案应该总是相同的。

The probabilities are different because each question tells you something different about the world. Each question carries different amounts of information.

概率是不同的，因为每个问题都告诉您有关世界的一些不同信息。 每个问题带有不同数量的信息。

To resolve this paradox, we can draw it out. In the first case,

为了解决这个矛盾，我们可以解决它。 在第一种情况下，

Notice how the question slices the possibilities differently than below. The first question removes only one possibility, GG, while the other two questions remove two possibilities each.⁸

请注意，问题如何对可能性进行了不同于下面的剖析。 第一个问题仅消除了一个可能性GG，而其他两个问题均消除了两个可能性。⁸

This same idea applies to figuring out your priors. It’s okay for different people to arrive at different odds, because the probability is a function of the observer, not a property of the objects! Probabilities represent your beliefs and information. The sun has no probability; it’s you who believes it will rise every day. And how much you believe can scale with the number of days you’ve been alive, or the number of days humanity has survived and recorded the sun, or, if you believe the scientists and their methods — a few billion years. Of course, if you’re living near the North or South pole, your priors are bound to be different. Just because the sun exists doesn’t mean it rises every day.

同样的想法适用于弄清您的先验。 可以由不同的人得出不同的赔率，因为概率是观察者的函数，而不是物体的属性！ 概率代表您的信念和信息。 太阳没有几率。 是您谁相信它会每天上升。 您相信多少可以与您存活的天数，人类赖以生存并记录太阳的天数成正比，或者，如果您相信科学家及其方法，则可以与数十亿年成正比。 当然，如果您居住在北极或南极附近，您的先验必然会有所不同。 仅仅因为太阳存在并不意味着它每天都在升起。

Probability lies in the eyes of the beholder.

概率在情人眼中。

## 逐步更新 (Update incrementally)

*Every extra piece of evidence changes your beliefs.*

*每多一份证据都会改变您的信念。*

In the 2, 4, 6 game, your belief becomes stronger as you discovered more evidence in favour of your hypothesis. The game is brutal: One failure and it’s game over for the hypothesis. This is a property of the hypothesis.

在2、4、6游戏中，当您发现更多支持该假设的证据时，您的信念就会增强。 这场比赛是残酷的：一次失败，这就是假设的终结。 这是假设的性质。

Real life hypotheses are usually not like that. Things are fuzzier. Hence, your increments can tumble up and down depending on the evidence you find. In practice, this means not taking the newest evidence at face value. Always keep it in context with what you already know.

现实生活中的假设通常并非如此。 事情变得更加模糊。 因此，您的增量可能会上下波动，具体取决于找到的证据。 实际上，这意味着不以最新的面额证据为依据。 始终将其与您已经知道的内容保持关联。

Like in the being late example, not updating incrementally would look something like: They’re late today, so I believe “they’re always late.” Notice how often you do this.

就像在最近的示例中一样，不进行增量更新看起来像：他们今天很晚，所以我相信“他们总是很晚”。 请注意您执行此操作的频率。

Of course, we’re not perfect at doing it wrong, either. If they’re on time instead, I’ll believe *everyone is supposed to be on time, no big deal,* instead of, say, *they’re always on time*. This is called motivated reasoning.

当然，我们也不是完美无误的。 如果他们是准时的，我相信*每个人都应该准时，没什么大不了的，*而不是说*他们总是准时的* 。 这称为动机推理。

“Evidence should not determine beliefs, but update them”

“证据不应该决定信念，而应该更新信念”

In the context of our examples, this seems very clear cut. Of course, we’ll update incrementally. It would be dumb to discard all the information we’ve already acquired.

在我们的示例上下文中，这似乎很明确。 当然，我们将逐步更新。 丢弃我们已经获得的所有信息将是愚蠢的。

Here’s a more realistic example.

这是一个更现实的例子。

When I was 14, I heard how working out with weights can stunt your growth. I believed: “Working out is evil and every big guy takes steroids.”

我14岁那年，我听说如何进行举重锻炼会阻碍您的成长。 我相信：“锻炼是邪恶的，每个大个子都服用类固醇。”

When I was 15, I heard how you can work out using just your body weight. That doesn’t stunt growth. I believed: “Bodyweight workouts are amazing to get buff.”

我15岁那年，我听说您如何只用体重就能锻炼身体。 这不会阻碍增长。 我相信：“举重锻炼真是令人振奋。”

A few weeks later, with no gains, I was dejected. I believed: “None of this works. It’s all a sham for corporations to make money.” I didn’t know who exactly was making money from my bodyweight workouts, but then again, I wasn’t a critical thinker.

几周后，我一无所获，感到沮丧。 我相信：“这些都不起作用。 对于公司来说，这完全是一种虚假。” 我不知道到底是谁从我的体重锻炼中赚钱，但话又说回来，我不是一个批判性的思想家。

Since then, I’ve done my own research. I understand how it’s a complex problem that depends not just on working out, not just on how long you work out, but also on what you’re eating.

从那时起，我进行了自己的研究。 我知道这是一个复杂的问题，不仅取决于锻炼，不仅取决于锻炼时间，还取决于您吃的东西。

I began updating incrementally. I took each piece of evidence in context with everything I had seen so far. Not everyone does steroids. Strength training works for at least some people. Intermittent fasting and running works for others. There’s no way, given all the diverse and contradicting evidence, that I can have close to 90% confidence in any belief about working out.

我开始逐步更新。 到目前为止，我都结合了我所看到的所有证据。 并非所有人都使用类固醇。 力量训练至少对某些人有效。 间歇性禁食和奔跑对他人有效。 鉴于所有各种各样且相互矛盾的证据，我无法对任何关于锻炼的信念拥有将近90％的信心。

A sign of not updating incrementally is flip-flopping between beliefs. You’re taking the evidence you’ve seen today as the complete truth.

不逐步更新的迹象是信念之间的相互影响。 您正在获取今天已经视为完整事实的证据。

Another failure mode is rehearsing your arguments. When one argument keeps coming back to your mind, again and again, that’s a case of failing to update incrementally. You’re pitting the same evidence, which you’ve used already, to bloat up confidence.

另一种失败模式是演习您的论点 。 当一个论点一次又一次地浮现在脑海中时，那就是无法增量更新的情况。 您正在使用已经使用过的相同证据来增强信心。

A crude example of this is repeating to yourself, “it’s going to be alright, it’s going to be alright” when facing a new risky situation. Sometimes, it’s necessary to calm your nerves. Other times, like the stock market bubbles, it’s you reinforcing a wrong belief when evidence points against it.

在面对新的风险情况时，一个重复的例子很粗鲁，“一切都会好起来，一切都会好起来”。 有时，有必要使您的神经平静。 在另一些时候，就像股市泡沫一样，当证据表明反对时，就是在强化错误的信念。

## 寻求确凿的证据 (Seek disconfirming evidence)

*In a world where your pet theory is not true, would you expect to see something different?*

*在您的宠物理论不正确的世界中，您是否希望看到不同的东西？*

In the 2, 4, 6 game, if you don’t test anything that you expect to be false, you can only ever find evidence in favour of your hypothesis. You can only become more confident in your ideas.

在2、4、6游戏中，如果您不检验任何期望为假的东西，则只能找到支持您的假设的证据。 您只能对自己的想法更有信心。

Disconfirming evidence is anything you don’t expect to fit in your hypothesis. In the 2, 4, 6 game, these are situations where you expect the black box to say “no”. So, for the hypothesis “3 even increasing numbers”, (1, 2, 3) and (1, -1, 2) are both possible disconfirming evidence.

不一致的证据是您不希望与假设相符的任何内容。 在2、4、6游戏中，这些情况是您希望黑匣子说“不”的情况。 因此，对于“ 3个偶数递增”的假设，(1、2、3)和(1，-1、2)都是可能的证据。

(1, -1, 2) shows you that your hypothesis, 3 even positive numbers might still be correct, while (1, 2, 3) shows you that it’s indeed wrong. You expected a no for (1, 2, 3) but got a yes.⁹

(1，-1，2)向您证明您的假设，甚至3个正数可能仍然是正确的，而(1,2,3)向您证明这确实是错误的。 您期望(1、2、3)否，但得到肯定。⁹

In the being late example, disconfirming evidence for the “always late” hypothesis is looking for times when your colleague came on time. We tend to forget that, looking only at the times they were late. This leads to new problems.

在最近的例子中，为“总是迟到”的假设提供证据的证据是在寻找同事准时到来的时间。 我们只会忘记他们迟到的时间而忘记了这一点。 这导致新的问题。

Until you find disconfirming evidence and test your hypothesis, your beliefs are a castle of glass. And castles of glass are fragile. Indeed, the role of disconfirming evidence is to destroy castles or to make them stronger in the process.

在您找到不确定的证据并检验您的假设之前，您的信念简直是杯水车薪。 玻璃城堡很脆弱。 的确，对证据提出异议的作用是摧毁城堡或使城堡更坚固。

The brain’s natural reaction is to protect this castle of glass. The brain views disconfirming evidence as attacks, and instead of being grateful, we tend to shut out the new evidence. If you dispel disconfirming evidence as rubbish, then *of course* you’ll never change your mind. You’re always updating in favour.

大脑的自然React是保护这座玻璃城堡。 大脑认为将无法证明的证据视为攻击，而不是心存感激，我们倾向于将新证据拒之门外。 如果您消除令人怀疑的证据是垃圾，那么您*当然*不会改变主意。 您总是会不断更新。

You could input a million numbers that follow your pattern in the 2, 4, 6 game, and things would look good, but you’d never realise how your hypothesis is a subset of another (all positive numbers) — until one day your assumption fails you. There’s nothing Bayes can do about this.¹⁰

您可以按照2、4、6游戏中的模式输入一百万个数字，事情看起来会很好，但是您永远都不会意识到自己的假设是另一个(全部为正数)的子集-直到一天您的假设让你失望。 贝叶斯对此无能为力。¹⁰

Thus, always seek disconfirming evidence. To do so, you can look for data that doesn’t fit your hypothesis, like we did above. The other way is to look for hypotheses that can better explain the data you already have. In Bayes-speak, you’re looking for hypotheses with better prior odds. Not all hypotheses are created equal. Some can explain the data better than others.

因此，请始终寻求令人怀疑的证据。 为此，您可以像上面那样查找不符合您的假设的数据。 另一种方法是寻找可以更好地解释您已经拥有的数据的假设。 用贝叶斯说话，您正在寻找先验概率更高的假设。 并非所有假设都是一样的。 有些人可以比其他人更好地解释数据。

Julia Galef has a good example of this, tracking jealousy between friends. Just because Karen complains doesn’t mean she’s jealous. Perhaps any reasonable person would do the same.

Julia·加莱夫(Julia Galef)就是一个很好的例子， 跟踪朋友之间的嫉妒 。 仅仅因为凯伦(Karen)抱怨并不意味着她嫉妒。 也许任何有理智的人都会这样做。

## 记住你的先验 (Remember your priors)

*Every moment you’ve lived is information you can use to figure out the next step.*

*您生活的每一刻都是可以用来确定下一步的信息。*

Last week, I woke up to a painful left eye. Even blinking hurt. It was swollen, and on closer investigation, I found something like a pimple inside my eyelid. I freaked out. “Holy crap, am I going to need surgery now? Will my eyesight be okay?!”

上周，我醒来时痛苦的左眼。 甚至眨眼受伤。 肿胀了，在仔细检查后，我发现眼睑内有粉刺。 我吓坏了。 “天哪，我现在需要手术吗？ 我的视力还好吗？！”

But, what if it’s just a pimple? It sure looks like one, but it’s the first one that hurts. Maybe it isn’t that dangerous.¹¹

但是，如果只是粉刺怎么办？ 它肯定看起来像一个，但它是第一个受伤的人。 也许不是那么危险。¹¹

I was panicking in the bathroom, and then I noticed I was panicking. That was a cue for me to explore why. I did a quick mental calibration.

我在浴室里惊慌失措，然后发现自己在惊慌失措。 这是我探索原因的线索 。 我做了一个快速的心理校准。

“I believe eye surgery to be 100x less likely than no eye surgery”. Most people I know haven’t gotten an eye surgery done, except LASIK, which doesn’t count. In fact, even when I scratched my eye as a kid, it recovered without surgery.

“我相信眼外科手术的可能性比没有眼外科手术的可能性低100倍”。 我认识的大多数人都没有做过眼科手术，除了LASIK手术(不算数)。 实际上，即使我还是个小孩子时，我的眼睛也无需手术即可恢复。

“The pimple has 5x the odds of eye surgery than no pimple”. I’m not a doctor, but this sounds reasonable to me.¹²

“丘疹的眼部手术几率是无丘疹的5倍”。 我不是医生，但这对我来说听起来很合理。¹²

Thus, in effect, odds for getting surgery are (1:100) * (5:1) = 1:20 . I’m 20x likely to not have eye surgery. Huh. Not too bad.

因此，实际上，进行手术的几率是(1：100)*(5：1)= 1:20。 我有20倍可能没有进行眼科手术。 嗯 还不错

This happened quickly. These odds I calculated are the posterior odds! The prior odds were (1:100) and the likelihood odds (5:1). If I had discarded my priors, and only considered the new information I got that day, I’d be panicking at the (5:1) odds for surgery.

这很快发生了。 我计算出的这些赔率是后验赔率！ 先验赔率是(1：100)，可能性赔率(5：1)。 如果我放弃了先验知识，只考虑了当天获得的新信息，那我将为手术的赔率(5：1)感到恐慌。

A few google searches, once I was off the toilet, helped confirm my estimates. I was still overestimating the danger. Styes are common and aren’t dangerous.

我上完厕所后，进行了一些Google搜索，帮助确认了我的估计。 我仍在高估危险。 麦粒肿很常见，并不危险。

These posterior odds have now become my new priors. Next time, I’d be pissed, not panicking.

这些后验赔率现在已经成为我的新先验。 下次，我会生气，不要惊慌。

This story has an interesting insight: When we’re facing something new, our priors are usually miscalibrated. Taking the time to think through it can help put things in perspective. Remember your priors.

这个故事有一个有趣的见解：当我们面对新事物时，我们的先验通常会被错误地校准。 花时间去思考它可以帮助您将事情看得更清楚。 记住你的先验。

In some circles, this is also called Base Rate Neglect. You want to start a startup, and you think you’ll be successful because you’re awesome, you’re smart, and you’re hard-working. What most people fail to realise is that so were most other startup founders! Yet, 90% of startups fail. Remember your priors.

在某些圈子中，这也称为基本速率忽略。 您想启动一家初创企业，并且认为自己会成功，因为您超赞，聪明，并且努力工作。 大多数人没有意识到的是，大多数其他创业公司创始人也是如此！ 但是，有90％的初创企业失败了。 记住你的先验。

On the flip side, don’t get too stuck on your priors. When new data comes in, remember to update incrementally. Being “set in your ways” is a criticism, not a compliment.

另一方面，不要太拘泥于先验。 当有新数据输入时，请记住以增量方式进行更新。 “以自己的方式行事”是一种批评，而不是一种夸奖。

These four ideas together form the basis of good Bayesian thinking. The formula itself can’t think for you. It can just do its best with all the inputs you hunt down for it.

这四个思想共同构成了良好的贝叶斯思想的基础。 公式本身无法为您考虑。 它可以为您寻找的所有输入尽力而为。

Even then, there are lots of tricks our mind falls for, famously called “cognitive biases”. Almost all of these are about incorrectly applying Bayes Rule, fudging with the inputs, or misunderstanding the outputs.

即使那样，我们的头脑还是有很多花招，被称为“认知偏差”。 几乎所有这些都与错误地应用贝叶斯规则，弄乱输入或误解输出有关。

# 破坏认知偏见 (Destroying Cognitive Biases)

Since Bayes Theorem is a basic framework for critical thinking, you can try it against some well known cognitive biases. These are times when we fail to think well. The strength of this framework depends on how well it can explain the biases.

由于贝叶斯定理是批判性思维的基本框架，因此您可以针对某些众所周知的认知偏差进行尝试。 这些时候我们想得不好。 该框架的优势取决于它能很好地解释这些偏见。

Availability Bias: The tendency to overestimate the likelihood of events with greater “availability” in memory

可用性偏差 ：倾向于高估内存中具有更大“可用性”的事件的可能性

= Using the evidence most readily available to you

=使用最容易获得的证据

= Discarding your priors

=丢弃先验

= Not updating incrementally

=不增量更新

Halo effect: The tendency for a person’s positive or negative traits to “spill over” from one personality area to another in others’ perceptions of them

晕轮效应 ：一个人的积极或消极特质从一个人格领域“溢出”到另一个人的感知中的趋势

= Using evidence for a different hypothesis as proxy for another

=将不同假设的证据用作另一个假设的替代

= Updating the wrong hypothesis

=更新错误的假设

= Answering an easier question

=回答一个简单的问题

Dunning Kruger effect: The tendency for unskilled individuals to overestimate their own ability and the tendency for experts to underestimate their own ability

邓宁·克鲁格效应(Dunning Kruger effect) ：非熟练人员倾向于高估自己的能力，专家倾向于低估自己的能力的趋势

= Not seeking disconfirming evidence

=不寻求不确凿的证据

Confirmation Bias: The tendency to search for, interpret, focus on and remember information in a way that confirms one’s preconceptions

确认偏见 ：倾向于以确认自己的先入之见的方式搜索，解释，关注和记住信息的倾向

= Not seeking disconfirming evidence

=不寻求不确凿的证据

Base Rate neglect: The tendency to ignore general information and focus on information only pertaining to the specific case, even when the general information is more important

基本费率忽略 ：倾向于忽略一般信息，而只关注与特定情况有关的信息，即使一般信息更为重要

= Forgetting your priors

=忘记先验

This is very cool, but it is all post-facto. I know the bias, so now I know which part I misused. Can I figure this out a priori? Before I’ve made the mistake? That would be huge. Indeed, that’s something we’ll tackle a few sections below with the brain compass.

这很酷，但是事后都是如此。 我知道偏见，所以现在我知道我滥用了哪一部分。 我可以先验一下吗？ 在我犯错之前？ 那将是巨大的。 确实，这是我们将在下面用大脑指南针解决的几个部分。

And of course, there are some which we have no idea how to explain:

当然，有些我们不知道如何解释：

Self-serving bias: The tendency to claim more responsibility for successes than failures

自私自利的偏见 ：倾向于主张成功胜于失败

Outgroup homogenity: Individuals see members of their own group as being relatively more varied than members of other groups.

群体外的同质性 ：个人认为自己的群体成员比其他群体的成员相对变化更大。

Cheerleader effect: The tendency for people to appear more attractive in a group than in isolation.

啦啦队效应 ：人们在群体中显得比在孤立时更具吸引力的趋势。

In summary, almost every social bias, ever. You can try out a few more from the list on Wikipedia.

总之，几乎所有的社会偏见都是如此。 您可以从Wikipedia上尝试更多。

# 寻求证实贝叶斯的证据 (Seeking Disconfirming Evidence for Bayes)

Applying the same ideas to Bayes Theorem, what can it not explain? Where do we expect Bayes to not work? That’s an opportunity for a better model to take its place.

将相同的想法应用于贝叶斯定理，它不能解释什么？ 我们期望贝叶斯在哪里不起作用？ 这是一个更好的模型取代它的机会。

We explored this in the cognitive biases section. Bayes can’t explain every bias, which means, at minimum, Bayes Theorem is not a complete model for how to think well.

我们在认知偏见部分对此进行了探讨。 贝叶斯无法解释所有偏见，这意味着，至少，贝叶斯定理不是如何思考的完整模型。

The biggest gripe against Bayes is in scientific research. The Frequentists claim that the priors are subjective — too personal to drive at any objective truth. You need to see things happen and assign probabilities based on how frequently they occur. The probabilities are a property of the object, not of your beliefs.

对贝叶斯最大的困扰是科学研究。 惯常论者声称先验是主观的-太过个性化而无法追求任何客观真理。 您需要查看发生的事情并根据发生的频率分配概率。 概率是对象的属性，而不是您的信念。

This argument is tangential to what we’re talking about. I don’t care about objective truths today; I want to focus on why we believe what we believe and how can we change *our* beliefs. Changing the entire world’s beliefs is the next step — too far away, too intractable, when I haven’t gotten a grasp of my own beliefs. I mention this argument anyway to keep us grounded.

这个论点与我们在说的是相切的。 我今天不在乎客观真理。 我想集中讨论为什么我们相信我们所相信的东西，以及如何改变我们的信念。 下一步是改变整个世界的信念-如果我还没有掌握自己的信念，那就太遥不可及了。 无论如何，我提到这个论点是为了让我们保持扎根。

For personal beliefs, it might seem (and will in the following sections) that I’m pulling numbers out of my ass and calling them the likelihood odds. That’s a fair question. However, arguing over the likelihood and prior odds is easier than arguing over statements in English. You can discuss which factors increase and decrease the likelihood odds, and have a fruitful discussion about why people believe what they believe. Contrast this with “You’re wrong! No, you’re wrong! Because of X, Y, Z. Ha! That’s wrong too!”

为了个人的信念，我似乎(并且在以下各节中)会从屁股中取出数字，并称其为可能性几率。 这是一个公平的问题。 但是，争论可能性和先验赔率比争辩英语陈述要容易。 您可以讨论哪些因素会增加或减少可能性的可能性，并可以就人们为什么相信自己的信仰进行富有成果的讨论。 将此与“您错了！ 不你错了！ 因为X，Y，Z。哈！ 也是错的！”

The bigger gripe for us is that it’s hard to figure out all hypotheses — which means it’s very hard to know for sure your belief is right or wrong. This is a problem anyway. It’s important to note the relativity of wrong here — some things are more right, and more wrong than others.

对我们而言，更大的麻烦是很难弄清所有假设-这意味着很难确定您的信念是对还是错。 无论如何这是一个问题。 重要的是要注意这里错误的相对性 -有些事情比其他事情更正确，更多错误。

“When people thought the earth was flat, they were wrong. When people thought the earth was spherical, they were wrong. But if you think that thinking the earth is spherical is just as wrong as thinking the earth is flat, then your view is wronger than both of them put together”

“当人们认为地球平坦时，他们错了。 当人们认为地球是球形时，他们是错的。 但是，如果您认为认为地球是球形与认为地球是平面一样错误，那么您的观点就比他们两个人提出的观点要错误。”

— Isaac Asimov, Relativity of Wrong¹³

—艾萨克·阿西莫夫(Isaac Asimov)， 相对论错误 ¹³

Not being able to figure out the absolute truth is no reason to not move in the right direction.

无法弄清绝对真理，没有理由不朝正确的方向前进。

For us, we ought to use whatever we can learn, whether it comes from Bayesianism or frequentism. When you can count how many times an event has occurred to calibrate your priors, do. “Us vs. Them” has no place in Bayesian thinking.

对于我们来说，无论是贝叶斯主义还是频繁主义，我们都应该使用我们能学到的任何东西。 当您可以计算事件发生的次数来校准先验条件时，请执行。 “我们与他们”在贝叶斯思想中没有地位。

# 重温2、4、6游戏 (The 2, 4, 6 Game Revisited)

Tip:This section, and the following one, explore a lot of nuance. If it gets too heavy, skip to the Getting Stronger or learning to put it to practice section!

提示：本节以及以下部分探讨了许多细微差别。 如果太重，请跳到“变得更强”或学习将其练习！

There’s one thing I left out during the 2, 4, 6 game analysis: the math.

在2、4、6游戏分析中，我忽略了一件事：数学。

It’s qualitatively different from the being late example since just a single piece of evidence was enough to completely destroy the hypothesis. Can you guess why this is the case?

从本质上来说，它与后来的例子不同，因为仅凭一条证据就可以完全摧毁该假设。 您能猜出为什么会这样吗？

It’s a quality of the hypothesis. Some hypotheses are more specific and certain than others, and thus evidence increases and decreases a proportional amount.

这是假设的性质。 一些假设比其他假设更为具体和确定，因此，证据会按比例增加和减少。

Let’s say you’re testing out the hypothesis: “The Earth is flat”. What you see every day raises your confidence in this hypothesis. The Earth is flat as far as you can see. The sky is up, the ground is down, and the sun moves from one edge to another. But, as soon as you see a video of Earth rotating from space, this hypothesis dies.

假设您正在检验以下假设：“地球是平坦的”。 您每天看到的内容会增强您对该假设的信心。 据您所见，地球是平坦的。 天空向上，地面向下，太阳从一侧移到另一侧。 但是，一旦您看到地球从太空旋转的视频，这个假设就会消失。

This is synonymous with us trying out (2, 4, 6), (4, 6, 8), (8, 10, 12) and gaining confidence, while crashing and burning with (1, 2, 3).

这与我们尝试(2、4、6)，(4、6、8)，(8、10、12)并获得信心，同时以(1、2、3)崩溃和燃烧一样，是同义词。

We are at the extremities of probabilistic reasoning, where instead of being uncertain, our hypotheses have 100% certainty. No room for error. If we can’t tolerate errors and uncertainties, we’re back in the world of logic. The spectrum from 0 to 1 dissolves into just two binary options: 0 or 1.

我们处于概率推理的极端，在那里我们的假设具有100％的确定性，而不是不确定的。 没有错误的余地。 如果我们不能容忍错误和不确定性，那么我们就回到了逻辑世界。 从0到1的频谱仅分解为两个二进制选项：0或1。

This “no room for error” is inherent in the hypothesis.

这种“没有错误的余地”是假设中固有的。

“The Earth is flat”, “The formula is 3 even increasing numbers”, “He is always late”.

“地球是平坦的”，“公式是3个偶数递增的数字”，“他总是迟到”。

Sometimes, our language hides our implicit assumptions. Uncovering those, I can rewrite the above three like:

有时，我们的语言隐藏了我们的隐含假设。 发现这些内容后，我可以将以上三个内容重写为：

“The Earth is flat 100% of the time.”

“地球100％的时间都是平坦的。”

“The formula is 3 even increasing numbers 100% of the time.”

“公式是3甚至在100％的时间内递增数字。”

“He is late more than 95% of the time.” This specific hypothesis is a bit tricky. When we *say* “always” late, we don’t mean always. Taken literally, being on time even once would crush this hypothesis. This is ambiguity we’ve got to be careful about when talking in language instead of math.

“他迟到超过95％的时间。” 这个特定的假设有些棘手。 当我们*说* “总是”迟到时，我们并不意味着总是。 从字面上看，准时甚至一次都会粉碎这个假设。 这是含糊不清的，我们在用语言而不是数学说话时必须要小心。

So, we’ve uncovered one quality of these hypotheses: They have a frequency attached to them. But, what about hypotheses like “I’ll be a billionaire” or “Trump will be president next election” or “I’ll need surgery for my eye”?

So, we've uncovered one quality of these hypotheses: They have a frequency attached to them. But, what about hypotheses like “I'll be a billionaire” or “Trump will be president next election” or “I'll need surgery for my eye”?

There’s no frequency here. It doesn’t make sense to say I’ll be a billionaire 100% of the time. You don’t get extra lives or parallel universes to test this hypothesis out. It happens once, and then you’re done. All we have is our confidence in the hypothesis, which we update when we find new data, according to Bayes rule.

There's no frequency here. It doesn't make sense to say I'll be a billionaire 100% of the time. You don't get extra lives or parallel universes to test this hypothesis out. It happens once, and then you're done. All we have is our confidence in the hypothesis, which we update when we find new data, according to Bayes rule.

In summary, there are two kinds of hypothesis: those based on a frequency of something happening, and those without. Frequency makes sense when you can repeat an experiment. Both kinds have confidence levels, which we update using Bayes rule.

In summary, there are two kinds of hypothesis: those based on a frequency of something happening, and those without. Frequency makes sense when you can repeat an experiment. Both kinds have confidence levels, which we update using Bayes rule.

Coming back to the 2,4,6 game, let’s now demonstrate all the details involved in an update. Our hypothesis is: The formula is 3 even increasing numbers 100% of the time. We’re comparing against “the formula is not 3 even increasing numbers (100% of the time).”

Coming back to the 2,4,6 game, let's now demonstrate all the details involved in an update. Our hypothesis is: The formula is 3 even increasing numbers 100% of the time. We're comparing against “the formula is not 3 even increasing numbers (100% of the time).”

With (4, 6, 8), I’d say likelihood odds are 2:1 for 3 even increasing numbers. I end up with posterior odds of 2:1.

With (4, 6, 8), I'd say likelihood odds are 2:1 for 3 even increasing numbers. I end up with posterior odds of 2:1.

With (10, 12, 14), the same story. Posterior odds now become 4:1. I’m getting pretty confident in this castle of glass.

With (10, 12, 14), the same story. Posterior odds now become 4:1. I'm getting pretty confident in this castle of glass.

A few more pieces of confirming evidence, like (12, 14, 16), (6, 8, 10) and now I’m up to 16:1.

A few more pieces of confirming evidence, like (12, 14, 16), (6, 8, 10) and now I'm up to 16:1.

With (1, 2, 3), the likelihood odds are 0, since it’s not 3 even increasing numbers. Posterior odds become 0, too.

With (1, 2, 3), the likelihood odds are 0, since it's not 3 even increasing numbers. Posterior odds become 0, too.

Whenever the hypothesis has a 100% frequency, a single counterexample is enough to destroy all confidence. It’s the land of no uncertainty. The extreme case for Bayes Theorem.

Whenever the hypothesis has a 100% frequency, a single counterexample is enough to destroy all confidence. It's the land of no uncertainty. The extreme case for Bayes Theorem.

There should probably never be a hypothesis in which you have 100% confidence.

There should probably never be a hypothesis in which you have 100% confidence.

Notice how I made even this hypothesis uncertain by adding the “probably”.

Notice how I made even this hypothesis uncertain by adding the “probably”.

# The Being Late Example Revisited (The Being Late Example Revisited)

We have one final property left to uncover. This is the weirdness we faced while doing the math. But before that, let’s explore the new frequency property we just learned.

We have one final property left to uncover. This is the weirdness we faced while doing the math. But before that, let's explore the new frequency property we just learned.

This cool graph should help prime your intuition. There are five lines, each of which is a different hypothesis: from late 0% of the time to late 100% of the time. In every case, the data we get is the person being late four days in a row, then on time the 5th day. Priors update every day. I used the accurate math calculations for this graph, to show how exactly the calibration works with specific hypotheses.

This cool graph should help prime your intuition. There are five lines, each of which is a different hypothesis: from late 0% of the time to late 100% of the time. In every case, the data we get is the person being late four days in a row, then on time the 5th day. Priors update every day. I used the accurate math calculations for this graph, to show how exactly the calibration works with specific hypotheses.

Compare this being late example to the stye in my eye example. What were the possible hypotheses for the stye in my eye? Either I need surgery or I don’t. It’s a hypothesis without frequency.¹⁴ Being late, on the other hand, has thousands of hypotheses with varying frequency: My colleague is late 5% of the time, or maybe 10% of the time, or maybe 50% of the time, or maybe 80% of the time, or maybe 95% of the time.

Compare this being late example to the stye in my eye example. What were the possible hypotheses for the stye in my eye? Either I need surgery or I don't. It's a hypothesis without frequency.¹⁴ Being late, on the other hand, has thousands of hypotheses with varying frequency: My colleague is late 5% of the time, or maybe 10% of the time, or maybe 50% of the time, or maybe 80% of the time, or maybe 95% of the time.

You’re not just comparing two hypotheses — late 95% of the time vs. late 5% of the time, but the entire universe of hypotheses. The (2, 4, 6) game also had lots of possible hypotheses, but we didn’t face this problem because of two reasons. First, there was no easy way to parametrize the hypothesis. We couldn’t just increase or decrease a number to create a new hypothesis. Second, all the hypotheses had 100% frequency and would die quickly, with just one counterexample. New hypotheses would uncover themselves thanks to the disconfirming evidence.

You're not just comparing two hypotheses — late 95% of the time vs. late 5% of the time, but the entire universe of hypotheses. The (2, 4, 6) game also had lots of possible hypotheses, but we didn't face this problem because of two reasons. First, there was no easy way to parametrize the hypothesis. We couldn't just increase or decrease a number to create a new hypothesis. Second, all the hypotheses had 100% frequency and would die quickly, with just one counterexample. New hypotheses would uncover themselves thanks to the disconfirming evidence.

Here, the hypotheses are on a spectrum. In such a case, sometimes you’re more interested in figuring out what the frequency with the highest confidence is. For the math nerds, and for those who will continue learning about the math: this problem is called parameter estimation. In effect, you’re short-circuiting comparing two hypotheses by comparing all of them together and settling on the one with the highest confidence.

Here, the hypotheses are on a spectrum. In such a case, sometimes you're more interested in figuring out what the frequency with the highest confidence is. For the math nerds, and for those who will continue learning about the math: this problem is called parameter estimation. In effect, you're short-circuiting comparing two hypotheses by comparing all of them together and settling on the one with the highest confidence.

A good rule of thumb here is to choose the mean. If you’ve seen someone be late 12 out of 20 days, the hypothesis with the highest confidence level would turn out to be the one with the same frequency: 12/20 or 60% of the time. This is why I claimed earlier that the 60% hypothesis would blow the 95% hypothesis out of the park. This was a new hypothesis being uncovered thanks to the data.

A good rule of thumb here is to choose the mean. If you've seen someone be late 12 out of 20 days, the hypothesis with the highest confidence level would turn out to be the one with the same frequency: 12/20 or 60% of the time. This is why I claimed earlier that the 60% hypothesis would blow the 95% hypothesis out of the park. This was a new hypothesis being uncovered thanks to the data.

Indeed, if you used the calculator or did it yourself, likelihood odds in favour of being late 60% of the time is 67,585 to 1. Similarly, for being late 5% of the time, likelihood odds in favour of 60% of the time is 8,807,771,533 to 1. Remember, 95% of the time had 130,000:1 odds vs. late 5% of the time. If you divide the two above, you get the same odds.

Indeed, if you used the calculator or did it yourself, likelihood odds in favour of being late 60% of the time is 67,585 to 1. Similarly, for being late 5% of the time, likelihood odds in favour of 60% of the time is 8,807,771,533 to 1. Remember, 95% of the time had 130,000:1 odds vs. late 5% of the time. If you divide the two above, you get the same odds.

This also answers the numbers looking weird feeling we had earlier. We were comparing 95% to the 5% hypothesis, and 95% won by a huge margin. But at the same time, a hypothesis we didn’t know about, the 60% blew both of them out of the park. It felt weird because 95% seemed like the best, when the data clearly pointed in a different direction.

This also answers the numbers looking weird feeling we had earlier. We were comparing 95% to the 5% hypothesis, and 95% won by a huge margin. But at the same time, a hypothesis we didn't know about, the 60% blew both of them out of the park. It felt weird because 95% seemed like the best, when the data clearly pointed in a different direction.

This should give you a good idea of how learning math can be worthwhile. Once you’ve ingrained the habit of thinking like a Bayesian, the next step is calibrating better with Math. But there are two more things you can do before that, which are up next.

This should give you a good idea of how learning math can be worthwhile. Once you've ingrained the habit of thinking like a Bayesian, the next step is calibrating better with Math. But there are two more things you can do before that, which are up next.

# Getting Stronger (Getting Stronger)

Following Bayesian thinking is hard. It’s counterintuitive to reason about things you’re used to feeling out. In most day-to-day low-risk situations, you probably don’t even need it. But, the way you prepare yourself for the few times a day you do need it is by practicing on the low-risk situations.

Following Bayesian thinking is hard. It's counterintuitive to reason about things you're used to feeling out. In most day-to-day low-risk situations, you probably don't even need it. But, the way you prepare yourself for the few times a day you do need it is by practicing on the low-risk situations.

This section looks at what you can do, and the next section focuses on practice.

This section looks at what you can do, and the next section focuses on practice.

# Improve Your Priors (Improve Your Priors)

*Experience is an exercise in improving your priors.*

*Experience is an exercise in improving your priors.*

Read widely; learn about things you don’t know exist. If you’re trying to improve your thinking in a specific field, go broad within that field. Figure out what all can happen and how frequently it happens.

Read widely; learn about things you don't know exist. If you're trying to improve your thinking in a specific field, go broad within that field. Figure out what all can happen and how frequently it happens.

Sometimes, numbers can help a lot. For example, this cheatsheet for engineers is very famous, and I visit it often — whenever I’m building new systems or reasoning about timing bugs. It gives me a more realistic idea of how slow or fast things can be, precisely because my priors are shit! A very experienced engineer could tell without looking, precisely because they have better priors.

Sometimes, numbers can help a lot. For example, this cheatsheet for engineers is very famous, and I visit it often — whenever I'm building new systems or reasoning about timing bugs. It gives me a more realistic idea of how slow or fast things can be, precisely because my priors are shit! A very experienced engineer could tell without looking, precisely because they have better priors.

Someone heard this same tip and went on to create better systems. In Bayesian-speak, they improved their priors. I’d go further: Experience is an exercise in improving your priors.

Someone heard this same tip and went on to create better systems. In Bayesian-speak, they improved their priors. I'd go further: Experience is an exercise in improving your priors.

# Become a Master Hypothesis Builder (Become a Master Hypothesis Builder)

*You can’t test the idea if you don’t have the idea in the first place.*

*You can't test the idea if you don't have the idea in the first place.*

Using Bayes Theorem depends a lot on the hypothesis you’re testing. If you’ve ever noticed yourself trying the same thing again and again, without progress, your hypothesis-generating machine has stalled. You can’t come up with an alternative explanation, so you end up trying the same thing again.

Using Bayes Theorem depends a lot on the hypothesis you're testing. If you've ever noticed yourself trying the same thing again and again, without progress, your hypothesis-generating machine has stalled. You can't come up with an alternative explanation, so you end up trying the same thing again.

This can get painful.

This can get painful.

The alternative is to become a master hypothesis builder. An idea machine. Train yourself to think of alternating viewpoints, different ways to explain the same thing, and different things you can try to fix them.

The alternative is to become a master hypothesis builder. An idea machine. Train yourself to think of alternating viewpoints, different ways to explain the same thing, and different things you can try to fix them.

*Y*our beliefs can only be as accurate as your best hypothesis.

*Y* our beliefs can only be as accurate as your best hypothesis.

The more ideas you can surface, the more ideas you can try, and the better your chances of winning! Like we saw in the being late example, the 60% hypothesis may never come to you, and if you’re stuck with only the 95% and 5% hypothesis, you’re bound to be wrong.

The more ideas you can surface, the more ideas you can try, and the better your chances of winning! Like we saw in the being late example, the 60% hypothesis may never come to you, and if you're stuck with only the 95% and 5% hypothesis, you're bound to be wrong.

However, that’s not all. Some hypotheses are better than others. As we’ve seen before, not all hypotheses are created equal. At the minimum, your hypothesis needs to be falsifiable. It needs to be clear cut. It needs to be specific enough. The hypothesis “anything can happen” is true, but it’s not very useful. The inputs determine the quality of your output.

However, that's not all. Some hypotheses are better than others. As we've seen before, not all hypotheses are created equal. At the minimum, your hypothesis needs to be falsifiable. It needs to be clear cut. It needs to be specific enough. The hypothesis “anything can happen” is true, but it's not very useful. The inputs determine the quality of your output.

A good way to check the quality of a hypothesis is to decide beforehand what will convince you that your hypothesis is wrong. If the answer is nothing, that’s a bad sign. Remember, all our beliefs and convictions are hypotheses — and challenging our beliefs is a good place to try this out. What would convince you that climate change is real?

A good way to check the quality of a hypothesis is to decide beforehand what will convince you that your hypothesis is wrong. If the answer is nothing, that's a bad sign. Remember, all our beliefs and convictions are hypotheses — and challenging our beliefs is a good place to try this out. What would convince you that climate change is real ?

“The remarkable thing about small children is that you can tell them the most ridiculous things and they will accept it all with wide open eyes, open mouth, and it never occurs to them to question you. They will believe anything you tell them. Adults learn to make mental allowance for the reliability of the source when told something hard to believe.”

“The remarkable thing about small children is that you can tell them the most ridiculous things and they will accept it all with wide open eyes, open mouth, and it never occurs to them to question you. They will believe anything you tell them. Adults learn to make mental allowance for the reliability of the source when told something hard to believe.”

— E.T Jaynes in Probability Theory,¹³ pg 99.

— ET Jaynes in Probability Theory ,¹³ pg 99.

A part of growing up is learning how to come up with hypotheses. Children learn from their mistakes. They soon learn the truth about Santa Claus, the neighbourhood monster, and the tooth fairy. They learn other people aren’t always the most reliable sources. This is step one of becoming a master hypothesis builder.

A part of growing up is learning how to come up with hypotheses. Children learn from their mistakes. They soon learn the truth about Santa Claus, the neighbourhood monster, and the tooth fairy. They learn other people aren't always the most reliable sources. This is step one of becoming a master hypothesis builder.

Here’s another example: In the 2, 4, 6 game, why didn’t you think of a hypothesis like: “It’s either (-2, -2, -2), or 3 positive numbers”?

Here's another example: In the 2, 4, 6 game, why didn't you think of a hypothesis like: “It's either (-2, -2, -2), or 3 positive numbers”?

# Learn the Math (Learn the Math)

Math helps you calibrate better. Since our intuitions usually flip us up, it’s good to get grounded in the Math. I like to defer this step to the end, after you’ve put Bayesian thinking to practice.

Math helps you calibrate better. Since our intuitions usually flip us up, it's good to get grounded in the Math. I like to defer this step to the end, after you've put Bayesian thinking to practice.

# Putting It All Together in Practice (Putting It All Together in Practice)

I have one final analogy for you, tying everything we’ve learned together. This is why you’ve spent so long reading this monster blog post.

I have one final analogy for you, tying everything we've learned together. This is why you've spent so long reading this monster blog post.

Like tradition so far, here’s a story to demonstrate the idea. This is Richard Feynman working through seemingly counterintuitive theorems:

Like tradition so far, here's a story to demonstrate the idea. This is Richard Feynman working through seemingly counterintuitive theorems:

I had a scheme, which I still use today when somebody is explaining something that I’m trying to understand: I keep making up examples. For instance, the mathematicians would come in with a terrific theorem, and they’re all excited. As they’re telling me the conditions of the theorem, I construct something which fits all the conditions. You know, you have a set (one ball) — disjoint (two balls). Then the balls turn colors, grow hairs, or whatever, in my head as they put more conditions on. Finally they state the theorem, which is some dumb thing about the ball which isn’t true for my hairy green ball thing, so I say, “False!”.

I had a scheme, which I still use today when somebody is explaining something that I'm trying to understand: I keep making up examples. For instance, the mathematicians would come in with a terrific theorem, and they're all excited. As they're telling me the conditions of the theorem, I construct something which fits all the conditions. You know, you have a set (one ball) — disjoint (two balls). Then the balls turn colors, grow hairs, or whatever, in my head as they put more conditions on. Finally they state the theorem, which is some dumb thing about the ball which isn't true for my hairy green ball thing, so I say, “False!”.

If it’s true, they get all excited, and I let them go on for a while. Then I point out my counterexample.

If it's true, they get all excited, and I let them go on for a while. Then I point out my counterexample.

“Oh. We forgot to tell you that it’s Class 2 Hausdorff homomorphic.”

“Oh. We forgot to tell you that it's Class 2 Hausdorff homomorphic.”

“Well, then,” I say, “It’s trivial! It’s trivial!”

“Well, then,” I say, “It's trivial! It's trivial!”

I guessed right most of the time because although the mathematicians thought their topology theorems were counterintuitive, they weren’t really as difficult as they looked. You can get used to the funny properties of this [..] and do a pretty good job of guessing how it will turn out.

I guessed right most of the time because although the mathematicians thought their topology theorems were counterintuitive, they weren't really as difficult as they looked. You can get used to the funny properties of this [..] and do a pretty good job of guessing how it will turn out.

We can build a similar model for our reasoning!

We can build a similar model for our reasoning!

Everything we’ve talked about reduces down to two exciting use cases for Bayes Theorem: figuring out the hypothesis with the highest confidence so far and then calibrating well.

Everything we've talked about reduces down to two exciting use cases for Bayes Theorem: figuring out the hypothesis with the highest confidence so far and then calibrating well.

Imagine you are a compass. A brain compass.

Imagine you are a compass. A brain compass.

You’re at the center.

You're at the center.

Around you is all the possible data — things you know, expect, as well as things you know that you don’t know. It’s the disk of priors.

Around you is all the possible data — things you know, expect, as well as things you know that you don't know. It's the disk of priors.

A hypothesis cuts this disk of priors into two parts — what you expect to happen, and what you don’t. The more specific the hypothesis is, the smaller the “truthy” section of the disk is.

A hypothesis cuts this disk of priors into two parts — what you expect to happen, and what you don't. The more specific the hypothesis is, the smaller the “truthy” section of the disk is.

The intensity of the colour is your confidence level in the hypothesis. The brighter the color, the higher the confidence.

The intensity of the colour is your confidence level in the hypothesis. The brighter the color, the higher the confidence.

But wait — you usually have several hypotheses. I like to think of the hypothesis as covering some part of the territory. Thus, I call the hypothesis *a hypothesis cover*. There are several hypothesis covers, each with their own intensity, all placed around the disk.

But wait — you usually have several hypotheses. I like to think of the hypothesis as covering some part of the territory. Thus, I call the hypothesis *a hypothesis cover* . There are several hypothesis covers, each with their own intensity, all placed around the disk.

Next is our evidence. Incoming data is balls falling on this brain compass. Imagine yourself taking this data in through your senses. Your senses package whatever you know and see into these balls of data.

Next is our evidence. Incoming data is balls falling on this brain compass. Imagine yourself taking this data in through your senses. Your senses package whatever you know and see into these balls of data.

Every ball becomes part of the disk of priors. It’s real-world data merging with what we think is possible.

Every ball becomes part of the disk of priors. It's real-world data merging with what we think is possible.

If it falls through a hypothesis cover to make it to the territory, the intensity of the hypothesis cover increases. This is us updating incrementally.

If it falls through a hypothesis cover to make it to the territory, the intensity of the hypothesis cover increases. This is us updating incrementally.

If we see all the balls falling into the hypothesis cover, it’s time to seek disconfirming evidence. Make sure some balls have the opportunity to fall outside the hypothesis cover!

If we see all the balls falling into the hypothesis cover, it's time to seek disconfirming evidence. Make sure some balls have the opportunity to fall outside the hypothesis cover!

If every ball of data falls inside your hypothesis cover anyway, maybe your hypothesis is too general to be useful. “Anything is possible” will accept all the balls, but it’s not very useful. If you imagine the brain compass here, you’ll see that the hypothesis cover covers the entire territory!

If every ball of data falls inside your hypothesis cover anyway, maybe your hypothesis is too general to be useful. “Anything is possible” will accept all the balls, but it's not very useful. If you imagine the brain compass here, you'll see that the hypothesis cover covers the entire territory!

This mental picture helps me remember how to use Bayes well. It visually shows me the difference between my priors and how much the hypothesis can explain. It’s a tool for thought, where the better I get at visualising, the better I get at using Bayes Theorem.

This mental picture helps me remember how to use Bayes well. It visually shows me the difference between my priors and how much the hypothesis can explain. It's a tool for thought, where the better I get at visualising, the better I get at using Bayes Theorem.

To make things more concrete, let's walk through one example using the brain compass. Say, the hypothesis is “I’ll become a millionaire”.

To make things more concrete, let's walk through one example using the brain compass. Say, the hypothesis is “I'll become a millionaire”.

In my mind, I immediately see a hypothesis cover saying “Neil = Millionaire”. This then *triggers* a search for priors. We usually start by thinking the hypothesis cover is the entire disk. Don’t stop there. Figure out the entire disk. Don’t discard your priors!

In my mind, I immediately see a hypothesis cover saying “Neil = Millionaire”. This then *triggers* a search for priors. We usually start by thinking the hypothesis cover is the entire disk. Don't stop there. Figure out the entire disk. Don't discard your priors!

The hypothesis cover is “Neil = Millionaire”, and the rest of the disk is “Neil != Millionaire”. I then realise that incoming data can’t really fit into this. I can’t test whether I’m a millionaire or not in the future using just this, since I have 0 data.¹⁵

The hypothesis cover is “Neil = Millionaire”, and the rest of the disk is “Neil != Millionaire”. I then realise that incoming data can't really fit into this. I can't test whether I'm a millionaire or not in the future using just this, since I have 0 data.¹⁵

So, I change the domain. I start looking for data I can use today that reliably predicts millionaires. First, the disk of priors becomes the number of people who are millionaires vs. those who aren’t. Hypothesis cover falls on the millionaires.

So, I change the domain. I start looking for data I can use today that reliably predicts millionaires. First, the disk of priors becomes the number of people who are millionaires vs. those who aren't. Hypothesis cover falls on the millionaires.

The incoming data is the properties of these people. Who is hardworking, lucky, etc. Balls fall both in the hypothesis and outside. The ratio falling inside is the likelihood ratio, and the hypothesis color darkens if the ratio inside is higher.

The incoming data is the properties of these people. Who is hardworking, lucky, etc. Balls fall both in the hypothesis and outside. The ratio falling inside is the likelihood ratio, and the hypothesis color darkens if the ratio inside is higher.

You’ll notice it’s hard to reason about this now, while without the brain compass, you could just use whatever ambiguous criteria you wanted to: “I’m smart, hard-working, lucky, so I have very good chances of becoming a millionaire.” Now, you can *see* all the other smart, hard-working, and lucky people who aren’t millionaires.

You'll notice it's hard to reason about this now, while without the brain compass, you could just use whatever ambiguous criteria you wanted to: “I'm smart, hard-working, lucky, so I have very good chances of becoming a millionaire.” Now, you can *see* all the other smart, hard-working, and lucky people who aren't millionaires.

The Brain Compass stops you from answering an easier question. Further, building the disk of priors forces you to think of not just the hypothesis cover, but what lies outside it. This also forces you to think of how big the cover should be. How much of the territory does it cover?

The Brain Compass stops you from answering an easier question . Further, building the disk of priors forces you to think of not just the hypothesis cover, but what lies outside it. This also forces you to think of how big the cover should be. How much of the territory does it cover?

A good way to realise you’re messing up is when you can’t visualise the complete Brain Compass for your hypothesis.

A good way to realise you're messing up is when you can't visualise the complete Brain Compass for your hypothesis.

There are trade-offs, though. This brain compass visualisation isn’t as accurate as the math. But it’s excellent for everything that comes before the math. I use it a lot more than I use the math, since with most decisions, I can live without the perfect calibration.

There are trade-offs, though. This brain compass visualisation isn't as accurate as the math. But it's excellent for everything that comes before the math. I use it a lot more than I use the math, since with most decisions, I can live without the perfect calibration.

It’s more about building a tool for thought that warns you *before* you fall for the pesky cognitive biases. As you uncover new biases, you can incorporate them into your brain compass.

It's more about building a tool for thought that warns you *before* you fall for the pesky cognitive biases. As you uncover new biases, you can incorporate them into your brain compass.

For example, you can visualise confirmation bias as looking for balls that only fall inside the hypothesis cover, and your brain throwing away the balls falling on the rest of the disk of priors. Perhaps in this case you can’t even see the disk of priors.

For example, you can visualise confirmation bias as looking for balls that only fall inside the hypothesis cover, and your brain throwing away the balls falling on the rest of the disk of priors. Perhaps in this case you can't even see the disk of priors.

Slowly, you’ll start seeing the *shape* of biases in your brain.

Slowly, you'll start seeing the *shape* of biases in your brain.

## Hypotheses with frequencies (Hypotheses with frequencies)

It’s a bit hard to demonstrate, but things get a bit weird when dealing with hypotheses having frequencies. Say the hypothesis is my colleague is late 75% of the time.

It's a bit hard to demonstrate, but things get a bit weird when dealing with hypotheses having frequencies. Say the hypothesis is my colleague is late 75% of the time.

When visualising this, some data points that fall outside your hypothesis cover will increase your confidence! I find it best not to try and calibrate using the brain compass in this case. I still use it to get to the point where I’ve collected some data, figured out my hypothesis, and now need to find confidence levels.

When visualising this, some data points that fall outside your hypothesis cover will increase your confidence! I find it best not to try and calibrate using the brain compass in this case. I still use it to get to the point where I've collected some data, figured out my hypothesis, and now need to find confidence levels.

## Growing the disk (Growing the disk)

Incoming data plays a second crucial role: Growing your disk of priors.

Incoming data plays a second crucial role: Growing your disk of priors.

Increasing the size of the disk grows your mind. If everything you’ve seen gives aliens on earth a 100:1 chance of existing, perhaps notice the balls falling outside your disk. Grow the size to include those outside your model.

Increasing the size of the disk grows your mind. If everything you've seen gives aliens on earth a 100:1 chance of existing, perhaps notice the balls falling outside your disk. Grow the size to include those outside your model.

Surprise is something you didn’t predict, or something you didn’t even know existed! Mathematically they’re the same, but qualitatively, one’s revising your hypothesis, the other is broadening your disk of priors to what else is possible.

Surprise is something you didn't predict, or something you didn't even know existed! Mathematically they're the same, but qualitatively, one's revising your hypothesis, the other is broadening your disk of priors to what else is possible.

## Switching hypotheses (Switching hypotheses)

It might be tempting to think of the hypothesis cover as a slider — you move it to wherever most of the balls are falling. But that’s not what actually happens. Looking at the brain compass from the top hides how the hypothesis covers are positioned.

It might be tempting to think of the hypothesis cover as a slider — you move it to wherever most of the balls are falling. But that's not what actually happens. Looking at the brain compass from the top hides how the hypothesis covers are positioned.

Here’s a side view:

Here's a side view:

When data falls outside your hypothesis cover, your hypothesis begins fading. At the same time, wherever this data is falling, it brightens a hypothesis cover in that area. You notice every hypothesis that gets bright enough. This is similar to how, in the 2, 4, 6 game, whatever data destroyed our hypothesis gave us a hint to our next hypothesis.

When data falls outside your hypothesis cover, your hypothesis begins fading. At the same time, wherever this data is falling, it brightens a hypothesis cover in that area. You notice every hypothesis that gets bright enough. This is similar to how, in the 2, 4, 6 game, whatever data destroyed our hypothesis gave us a hint to our next hypothesis.

At the same time, our beliefs are always the ones with the highest confidence. Taken together, this is how all 3 hypotheses in the 2, 4, 6 game look like:

At the same time, our beliefs are always the ones with the highest confidence. Taken together, this is how all 3 hypotheses in the 2, 4, 6 game look like:

# Strong Opinions, Weakly Held? (Strong Opinions, Weakly Held?)

As with every idea that makes it into pop culture, it gets simplified and re-interpreted to mean whatever the arguer is saying. I’d like to reframe this saying via the lens of the brain compass.

As with every idea that makes it into pop culture, it gets simplified and re-interpreted to mean whatever the arguer is saying. I'd like to reframe this saying via the lens of the brain compass.

An opinion is a hypothesis cover. A strong opinion is a narrow cover. The narrower the territory, the stronger the hypothesis.

An opinion is a hypothesis cover. A strong opinion is a narrow cover. The narrower the territory, the stronger the hypothesis.

Weakly held means updating incrementally in the face of counter-evidence. Huh, that sounds a lot like something we know, doesn’t it?

Weakly held means updating incrementally in the face of counter-evidence. Huh, that sounds a lot like something we know, doesn't it?

A proper reframing would be “Strong opinions, Bayesianly held.” But I guess that’s not catchy enough.

A proper reframing would be “Strong opinions, Bayesianly held.” But I guess that's not catchy enough.

There are lots more examples we can reframe with the brain compass, but I’ll stop here.

There are lots more examples we can reframe with the brain compass, but I'll stop here.

# This Seems Very Different to What I Learned in School (This Seems Very Different to What I Learned in School)

Go to any online introduction to Bayes Theorem, and here’s what you’ll see:

Go to any online introduction to Bayes Theorem, and here's what you'll see:

“You think being late to work has a 50% chance of happening. You know it depends on your alarm clock not going off in the morning. Your alarm clock didn’t go off. What’s the chance of being late to work now?”

“You think being late to work has a 50% chance of happening. You know it depends on your alarm clock not going off in the morning. Your alarm clock didn't go off. What's the chance of being late to work now?”

Except, that’s not how we think.¹⁶ The funny thing is, when I first started writing this post two months ago, this was indeed the introduction. This framing is very useful for scientific hypothesis testing, but not if you want a quick and intuitive grasp for everyday life. You don’t have time to get your scientist's notebook out and compute every time you want to use Bayes.

Except, that's not how we think.¹⁶ The funny thing is, when I first started writing this post two months ago, this was indeed the introduction. This framing is very useful for scientific hypothesis testing, but not if you want a quick and intuitive grasp for everyday life. You don't have time to get your scientist's notebook out and compute every time you want to use Bayes.

The way you achieve the intuitive grasp is by reframing. You transform the problem so your brain can grasp it quickly. For example, CPUs can do lots of calculations one by one very well. But when it comes to graphics, not only do we need to do lots of calculations, we need to do them in parallel so that we can see the entire screen render at the same time.¹⁷ Thus, we break the problem down and feed it to the GPUs in a way that’s easy to process. Parallel processing algorithms follow the same idea!

The way you achieve the intuitive grasp is by reframing. You transform the problem so your brain can grasp it quickly. For example, CPUs can do lots of calculations one by one very well. But when it comes to graphics, not only do we need to do lots of calculations, we need to do them in parallel so that we can see the entire screen render at the same time.¹⁷ Thus, we break the problem down and feed it to the GPUs in a way that's easy to process. Parallel processing algorithms follow the same idea!

We broke up the parts of Bayes theorem, the hypothesis, evidence, and the priors — and reordered them in the way we intuitively think. See the footnote above for more details.¹⁶

We broke up the parts of Bayes theorem, the hypothesis, evidence, and the priors — and reordered them in the way we intuitively think. See the footnote above for more details.¹⁶

# Epilogue: The End is the Beginning (Epilogue: The End is the Beginning)

To summarize, there are lots of subtle ideas we tackled. We started with the (2, 4, 6) game, realised how our thinking process was roughly Bayesian, and how we’re miscalibrated. We then figured out there are two kinds of hypotheses: those with frequency and those without.

To summarize, there are lots of subtle ideas we tackled. We started with the (2, 4, 6) game, realised how our thinking process was roughly Bayesian, and how we're miscalibrated. We then figured out there are two kinds of hypotheses: those with frequency and those without.

Every hypothesis has a confidence level, which we update using Bayes theorem. Any hypothesis with a 100% frequency or confidence faces one-hit KOs. When one hypothesis dies, another one takes its place. We always believe something, and we’d do well to believe the most probable thing given all the data we’ve seen so far, and our priors!

Every hypothesis has a confidence level, which we update using Bayes theorem. Any hypothesis with a 100% frequency or confidence faces one-hit KOs. When one hypothesis dies, another one takes its place. We always believe something, and we'd do well to believe the most probable thing given all the data we've seen so far, and our priors!

How we disprove our hypothesis is a hint for the next hypothesis. This is systematized by always seeking disconfirming evidence. Remember we can never prove something, only disprove. There’s always some uncertainty in our beliefs. But of course, this doesn’t mean we’ll go about believing the hypothesis that’s 1000x less likely. Some things are more wrong than others.

How we disprove our hypothesis is a hint for the next hypothesis. This is systematized by always seeking disconfirming evidence. Remember we can never prove something, only disprove. There's always some uncertainty in our beliefs. But of course, this doesn't mean we'll go about believing the hypothesis that's 1000x less likely. Some things are more wrong than others.

Bayes Theorem, and by extension critical thinking, is about finding the most probable beliefs and then calibrating.

Bayes Theorem, and by extension critical thinking, is about finding the most probable beliefs and then calibrating.

The next phase is figuring out better tools to more accurately map the territory.

The next phase is figuring out better tools to more accurately map the territory .

Learn the math. Bayesian Statistics the fun way¹³ is an excellent book, and I can’t recommend it enough! Better tools help model reality better. A good next step from the book is the Beta Distribution, which helps model the entire space of frequencies for the being late example.

Learn the math. Bayesian Statistics the fun way ¹³ is an excellent book, and I can't recommend it enough! Better tools help model reality better. A good next step from the book is the Beta Distribution, which helps model the entire space of frequencies for the being late example.

Learn why any model of thinking that is consistent and qualitatively in line with common sense must end up at Bayes Theorem. This is a lecture from ET Jaynes, on building a system of plausible reasoning. You can see all the other lectures too.

Learn why any model of thinking that is consistent and qualitatively in line with common sense must end up at Bayes Theorem. This is a lecture from ET Jaynes , on building a system of plausible reasoning. You can see all the other lectures too.

Notice why you believe what you believe. Notice how you reach conclusions. Notice when you

*jump*to conclusions. Notice how that errors out for you. Cultivate The Art of Noticing.Notice why you believe what you believe. Notice how you reach conclusions. Notice when you

*jump*to conclusions. Notice how that errors out for you. Cultivate The Art of Noticing .- Bayes Theorem won’t be useful if you forget to apply it when you need it the most. Not surprisingly, when you need it the most is when it’s most likely to slip out of your mind. Bayes Theorem won't be useful if you forget to apply it when you need it the most. Not surprisingly, when you need it the most is when it's most likely to slip out of your mind.
It’s also worthwhile to look at some cool applications of Bayes Theorem — it’s been used widely, from detecting cancer, email spam, to environmental damage. Check out the link on environmental damage — every paper has the formula somewhere at the top, but the part you’re interested in are the diagrams. This is a hypothesis with frequency, and we’re using beta distributions to model damage, which also leads to the rule of thumb in the being late revisited example.

It's also worthwhile to look at some cool applications of Bayes Theorem — it's been used widely, from detecting cancer, email spam , to environmental damage . Check out the link on environmental damage — every paper has the formula somewhere at the top, but the part you're interested in are the diagrams. This is a hypothesis with frequency, and we're using beta distributions to model damage, which also leads to the rule of thumb in the being late revisited example.

Learn about the Bayesian Model of the Brain (a.k.a Predictive Coding) in Surfing Uncertainty. Scott Alexander also has an excellent book review.

Learn about the Bayesian Model of the Brain (aka Predictive Coding) in Surfing Uncertainty . Scott Alexander also has an excellent book review.

Oh, and finally, for the 2, 4, 6 game, I know I said three positive numbers is correct, but it isn’t. Despite all our efforts, sometimes we fail to arrive at the right answers. But of course, as Isaac Asimov said, three positive numbers is less wrong than three even increasing numbers.

Oh, and finally, for the 2, 4, 6 game, I know I said three positive numbers is correct, but it isn't. Despite all our efforts, sometimes we fail to arrive at the right answers. But of course, as Isaac Asimov said, three positive numbers is less wrong than three even increasing numbers.

And in the end, that’s what critical thinking is: becoming less wrong.

And in the end, that's what critical thinking is: becoming less wrong.

# Endnotes (Endnotes)

- Correct because I already know the formula. In a real-life uncertainty situation, you’ll never be sure! Your confidence levels will keep inching towards 100%, but never touch 100%. Correct because I already know the formula. In a real-life uncertainty situation, you'll never be sure! Your confidence levels will keep inching towards 100%, but never touch 100%.
- We’ll see a bit later where the 1:1 comes from, and how you can change it. We'll see a bit later where the 1:1 comes from, and how you can change it.
- We go into what these words mean in the following section. This is also why I have “always” in quotes. We go into what these words mean in the following section. This is also why I have “always” in quotes.
- Of course, the math is more accurate! When you have decisions to make with a lot riding on them, it makes sense to take the extra time and figure out what you can. This intuitive calculation is the first step that instills the habit. Of course, the math is more accurate! When you have decisions to make with a lot riding on them, it makes sense to take the extra time and figure out what you can. This intuitive calculation is the first step that instills the habit.
- Mathematically, 1:10 is close to being late 90% of the time, and 1:20 is close to being late 95% of the time. Mathematically, 1:10 is close to being late 90% of the time, and 1:20 is close to being late 95% of the time.
If you’re familiar with the probability form of Bayes Theorem, this might seem weird. Check this explanation to understand the difference between odds and probability multiplication.

If you're familiar with the probability form of Bayes Theorem, this might seem weird. Check this explanation to understand the difference between odds and probability multiplication.

The calculation here is based on the binomial distribution. I know, I said no fancy math. Here’s a calculator I built to do this yourself. I’m assuming 90% late for the main hypothesis, 10% for the alternate, and 12 successes, 8 failures. The odds should come out to 6,000:1 — I rounded that down to 1000:1.

The calculation here is based on the binomial distribution. I know, I said no fancy math. Here's a calculator I built to do this yourself. I'm assuming 90% late for the main hypothesis, 10% for the alternate, and 12 successes, 8 failures. The odds should come out to 6,000:1 — I rounded that down to 1000:1.

Here’s an in-depth explanation.

Here's an in-depth explanation .

- (1, -1, 2) is disconfirming evidence, too! Just that in this case, it failed to disconfirm your hypothesis. (You expected No, and got No). (1, -1, 2) is disconfirming evidence, too! Just that in this case, it failed to disconfirm your hypothesis. (You expected No, and got No).
- In this case, where the brain just can’t reject the results, the natural step is to blame someone else. At least the castle didn’t come down because of you. In this case, where the brain just can't reject the results, the natural step is to blame someone else. At least the castle didn't come down because of you.
- This was me trying to calibrate my sense of danger. This was me trying to calibrate my sense of danger.
- This was me figuring out likelihood odds. This was me figuring out likelihood odds.
- Affiliate link. 会员链接。
- There’s a way to formulate this problem as a hypothesis with frequency, but that becomes a different problem — and a different hypothesis. There's a way to formulate this problem as a hypothesis with frequency, but that becomes a different problem — and a different hypothesis.
- This is the difference between hypotheses with frequency and without! This is the difference between hypotheses with frequency and without!
- Did you figure out the difference between this formulation and how we usually think? This one’s starting with the priors! On the other hand, we start with the incoming data (the alarm clock didn’t go off), which raises the hypothesis — “Am I going to be late today?”, then adjust confidence according to our priors (50% chance of being late). Did you figure out the difference between this formulation and how we usually think? This one's starting with the priors! On the other hand, we start with the incoming data (the alarm clock didn't go off), which raises the hypothesis — “Am I going to be late today?”, then adjust confidence according to our priors (50% chance of being late).
- If you’ve ever seen a photo load slowly, part by part, that’s the CPU processing it layer by layer. Or a slow internet connection slowing the CPU down. If you've ever seen a photo load slowly, part by part, that's the CPU processing it layer by layer. Or a slow internet connection slowing the CPU down.

翻译自: https://medium.com/better-programming/bayes-theorem-a-framework-for-critical-thinking-1ba54f96dc6a

贝叶斯 定理