CRITICAL THINKING - Fundamentals: Bayes' Theorem [HD]

my name is Ian ol Azov I'm a graduate student at the CUNY Graduate Center and today I want to talk to you about Bayes theorem Bayes theorem is a fact about probabilities a version of which was first discovered in the 18th century by Thomas Bayes the theorem is Bayes most famous contribution to the mathematical theory of probability it has a lot of applications and some philosophers even think it's the key to understanding what it means to think rationally in order to understand the theorem though we'll have to understand a little bit about probabilities the probability of a proposition is the chance or likelihood that that proposition is true suppose you know that one student in a class of 20 has the flu but you don't know who if you know that Sally is a student in the class you would say the probability that Sally has the flu is 1 in 20 or 5 percent or point zero 5 we can call this your prior probability that Sally has the flu because it's your probability prior to finding out any new information as a shorthand we'll write P of Sally has the flu equals 0.05 suppose now that there are five girls and 15 boys in the class now you don't know whether the class's flu patient is a boy or a girl but if you were to find out that the patient was a girl your probability that Sally has the flu would go up to one in five or twenty percent or 0.2 on the other hand if you were to find out that the patient was a boy your probability that Sally has the flu would go down to zero because these things are still iffy though remember you don't yet know whether the flu patient is a boy or girl will call these things conditional probabilities your probability that Sally has the flu conditional on the flu patient being a girl is point to your probability that Sally has the flu given that the flu patient is a boy is zero as a shorthand we'll write P of Sally has the flu given that the flu patient is a girl equals point two and P of Sally has the flu given that the flu patient is a boy equals zero the little vertical line tells you that we're talking about conditional probabilities now here's the thing sometimes you don't know what your conditional probabilities should be in other words you know that you might encounter some new evidence in the future but you don't yet know how that evidence should affect the probability you assign to some hypothesis here's where Bayes theorem comes in it gives you a way of figuring out what your conditional probabilities should be so what does Bayes theorem actually say remember our shorthand your probability in some hypothesis let's call it h conditional on some new piece of evidence let's call it e is written P of H given e here's what Bayes theorem tells us P of H given e equals P of e given H times P of H divided by P of e in other words it tells us the three ingredients that go into the probability of a hypothesis conditional on some evidence the probability of the evidence conditional on the hypothesis the prior probability of the hypothesis and the prior probability of the evidence let's look at an example imagine that one morning you don't feel right and you go on WebMD to figure out what's wrong you're browsing around until you find an illness that catches your eye hypothesis so the hypothesis under consideration is that you've come down with hypothesis as you read through the list of symptoms you realize that you have all of them in other words you have all of the symptoms that you would have if you had hypothesis so let's say P of e given H or P of symptoms given hypothesize s equals 0.95 you begin to freak out but then you remember Bayes theorem it tells you that there are two more things you need to know in order to figure out the probability that you have hypothesis the prior probability that you would come down with hypothesis and the prior probability that you would have the symptoms that you actually have with a little more googling you discover that the disease is extremely rare only one in 100,000 people have it so P of hypothesis is point zero zero zero zero one now for the last ingredient what kind of symptoms are they suppose they're very common like a headache and a runny nose lots of people have those google tells you one in a hundred so P of symptoms your prior probability that you would come down with the symptoms you have is point zero one at last you know everything that you need to know in order to figure out the probability that you have hypothesize s given your symptoms Bayes theorem tells you that P of hypothesis given symptoms equals P of symptoms given hypothesize s times P of hypothesis divided by P of symptoms in other words P of hypothesis given symptoms equals 0.001 five or a little less than one in a thousand Bayes theorem is very helpful because in figuring out what to make of some new piece of evidence people often ignore the prior probability of the hypothesis or treat P of H given E as P of e given H this mistake is sometimes known as the base rate fallacy in the case we just looked at P of H given E is very different from P of e given H 1 is less than one tenth of a percent and the other is 95 percent without Bayes theorem you might have gotten a lot more worked up about hypothesis than you need it to be wrapping up then Bayes theorem is a formula that tells you how to calculate conditional probabilities or the probability you should assign to some hypothesis given a piece of evidence even if you forget the formula though try to remember that the conditional probability of H given E is determined by three things the conditional probability of e given H the prior probability of H and the prior probability of e if you leave one of those three things out you don't have a complete picture

Channel: Wireless Philosophy

Views: 338,971

Rating: 4.9009356 out of 5

Keywords: Khan Academy, Philosophy, Wireless Philosophy, Wiphi, video, lecture, course, critical thinking, logic, mathematics, probability, Thomas Bayes, Bayes Theorem, epistemology, bayesian, base rate fallacy, Ian Olasov, CUNY, City University of New York, confirmation theory, statistics

Id: OqmJhPQYRc8

Length: 6min 20sec (380 seconds)

Published: Fri Apr 22 2016

Reddit Comments

[u/liarliarplants4hire]

I'm in medicine and this reminds me of the saying "Uncommon presentations of common diseases is more common than common presentations of uncommon diseases".

[u/RabidMortal]

If it helps, the way that I was (conceptually) taught Bayes' Theorem was thus:

1) You hear strange noises in the room next door

2) You hypothesize that the noises come from dinosaurs mating

3) You note that "that's exactly what I would hear if there were dinosaurs mating in the next room"

4) BUT then you note that that likelihood of there being dinosaurs in the next room is very LOW...

5)...and the likelihood of hearing (any) strange noises coming from an adjacent room is probably pretty high...

6) ...so in the end, the actual probability of your original hypothesis is very very low.

While it's a non empirical mnemonic, it was a weird image that it stuck with me and I can always reconstruct the formulaic version of Bays theorem from just that example.

[u/absolutezero52]

I'm not a philosophy type, but I figured you all would be a little upset at his definition of probability being "The likelihood or chance of something happening". Otherwise cool video!

[u/lovikavante]

Could you use Bayes theorem to explain the Monty hall problem? First the probability is 1/3, then some new information is given. But which probabilities should go where in the theorem?

[u/JadedIdealist]

If someone wants to expertly make and evaluate likelihood claims, then that person needs to master (not just have a vague acquaintance with) statistics.

IMO, when someone makes a likelihood claim, one should have no qualms asking to be shown a formal likelihood calculation - it's how to cut through the crap.

Edit: aaand as expected this is hugely contentious in this subreddit.

Edit2: ummm.

[u/SheCutOffHerToe]

Arguably the single most important lesson omitted from common American education.

[u/DashingLeech]

I like that they put in the Base Rate Fallacy as an important concept to check. I find it is incredibly common even in popular beliefs, understandings, and observations.

For example, one of the most popular business management books of the past decade is Jim Collins' Good to Great. Most people think of the book, and the research it is based on, as providing the qualities of a company that will help it go from good to great. (It defines what these mean in quantitative terms, of course.) That is, you are interested in the probability that your company will go from good to great if it has these properties. People believe that having these properties gives a high probability of going good to great. That is, it is presented that P(good to great | specific properties) is high.

The problem is that the research actually investigates the probability of the evidence given the hypothesis. It finds the probability of the properties given that a company has gone from good to great, P(specific properties | good to great) and presents only those properties that have high values -- in their case 100%. (To get this they also create vague and subjective properties, which is another problem.) It gets more complicated in that they have a "control" group of companies that didn't go from good to great and made sure those one also didn't have the properties, meaning the result book describes properties for which P(property X | good to great) = 1 AND P(property X | NOT good to great) = 0, based on their cherry picked groups (not randomly selected).

They never investigated P(good to great) which is very small, P(NOT good to great) which is very large, or P(property X) which is completely unknown. So the book is useless to me.

The same base rate error is used in Dan Buettner's "Blue Zones" of unusually long-lived communities of the world, described in a TED Talk entitled "How to live to be 100+". The title claims P(100+ | lifestyle choice X) but, unfortunately, investigates P(lifestyle choice X | 100+), with no mention of P(100+) or P(lifestyle choice X).

I think the same error is true of much of the "social justice" movements and their very bad statistics, particularly with respect to the concept of privilege, and particularly in the use of the Progressive Stack. (Let's ignore that they treat all people by the identity group even if we have direct evidence of the privilege level of the individual.)

What they are actually arguing is that, based on identity properties like gender, skin color, or sexual orientation, they can designate a person's privilege and provide a counteracting policy to exclude members of the higher privileged identity groups and help the members of the lower privileged identity groups.

So, for example, a policy of excluding whites or males will come from the belief that whites and males have power and wealth, and this is determined by noting that those with power and wealth are statistically likely to be white and male. That is, they use the probability that somebody is white given they have wealth and power -- P(white | wealth & power) -- and the probability that they are male -- P(male | wealth & power) -- are higher than for others with wealth and power.

But this is a base rate fallacy, as it says the people with wealth & power tend to be white and male, not the whites and males tend to have wealth & power. That is, their policies are based on the believe that P(wealth & power | white) and P(wealth & power | male) are quite high, compared to others, but they use the inverse probabilities in their reasoning. (Oddly, the conditional information of the actual wealth & power of a given individual seems irrelevant in such policies for some reason.)

I'm finding more and more examples of where people make these serious mistakes and make serious policies based on them. Business practices, lifestyle choices, and social policies are being built on them based on erroneous thinking.

It's still not clear to me why this base rate fallacy is so common a psychological error. Most people understand that all crows are birds, but only a few birds are crows, but when you switch from absolute essentialist statements to probabilities, the concept gets mixed up so easily.

[u/MaximalAggregate]

How does a person determine prior probabilities of H and E?

[u/That2009WeirdEmoKid]

Anyone who wants a more in-depth explanation should check this out:

http://www.yudkowsky.net/rational/bayes

It's the first explanation that actually made it click in my head