A User's Guide to Bayes' Theorem

What is Bayes' Theorem? How is it used in philosophy, statistics, and beyond? How should it NOT be used? Welcome to the ultimate guide to these questions and more.
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OUTLINE
0:00 Intro and outline
3:10 What is Bayes’ Theorem?
4:50 Belief and credence
14:46 Interpretations of probability
24:52 Epistemic probability
28:32 Bayesian epistemology
30:50 Core normative rules
33:05 Propositional logic background
42:32 Kolmogorov’s axioms
54:25 Dutch Books
57:25 Ratio Formula
1:02:03 Conditionalization principle
1:10:47 Subjective vs. Objective Bayesianism
1:13:50 Bayes’ Theorem: Standard form(s)
1:39:25 Bayes’ Theorem: Odds form
1:53:06 Evidence
2:09:25 Visualizing Bayes’ Theorem
2:49:41 Common mistakes
2:49:53 Base rate fallacy
2:52:28 Evidence for H vs. Making H probable
2:53:43 Total evidence requirement
2:54:58 Fallacy of understated evidence
2:59:36 Confirmation is comparative
3:01:07 Evidential symmetry
3:07:56 Strength asymmetry
3:11:40 Falsifiability as a virtue
3:12:27 Likelihood ratio rigging
3:19:32 Conclusion and Resources
NOTES
(1) The argument I give around the ten-minute mark admittedly doesn't address the view that (i) credences don't exist, and yet (ii) we can still account for the relevant data about our doxastic lives by appeal to beliefs about probabilities. It also doesn't address the view that while credences exist and are distinct from beliefs, beliefs about probabilities suffice to account for the relevant data.
While I have independent reservations for these views, it's worth noting them nonetheless. The belief/credence part of the video was mainly an exercise in warming listeners up to talk of credences so they would be more receptive to the rest of the video. It's a necessary preamble to the main event: Bayes' Theorem. I grant that a proper defense of belief-credence dualism - and a proper defense of the overly-basic motivations I sketched at the ten-minute mark - would need to contend with these alternative proposals!
CORRECTIONS
(1) At 52:55, I meant to say that the SECOND claim entails the FIRST while the FIRST does not entail the SECOND. Oops!
(2) Thankfully, the audio improves at 28:32! Remind me to never record a solo presentation using Zoom…
(3) Here's an important clarification about the roommate/magical marker example given around 2:07:00. In the video, I was not clear about the content of the hypotheses in question and how this affects their likelihoods and priors. Here is how I should have spelled out the example.
​Consider two hypotheses:
H1: My friend wrote on my board
H2: My marker by itself wrote on my board
The data is:
D: "Don't forget to take out the trash!" is written on my board
Now, H1 renders D quite surprising, given that my friend knows all about my diligent habits of taking out the trash, etc. If H1 is true, he would most likely have written something on the board then erased it (since he knows I don't like him touching my board, etc.). And even if he didn't erase it, it would be very odd for him to write this given that he knows I'm super diligent about the trash.
But H2 renders D far, far more surprising. Of all the possible things the marker could conceivably have written on the board by magically floating upwards etc., only an absurdly small fraction are even coherent, let alone English words strung together to compose an intelligible, grammatical, contextually relevant English sentence.
Of course, H1 has higher prior than H2. But the point made in the video stands, since the likelihood of H1 (i.e., P(D|H1)) is very low, but it's still much greater than the likelihood of H2 (i.e., P(D|H2)), and hence data can still be evidence for a hypothesis even though the data is very surprising on that hypothesis.
LINKS
(1) Want the script? Become a patron :)
(2) Videos from @3blue1brown: (i) • Bayes theorem, the geo... , (ii) • The quick proof of Bay... , (iii) • The medical test parad...
(3) Videos from Dr. Ben Page (@thinkingillustrated5281): (i) • How to Think About Pro... , (ii) • How does the Bayesian ...
(4) Page's article on the Bayesian Bar: www.academia.edu/41820082/Int...
(5) Over 100 Arguments for God ANSWERED: • Over 100 Arguments for...
(6) My Springer book: (a) www.amazon.com/Existential-In... (b) link.springer.com/book/10.100...
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[@MajestyofReason]

CORRECTIONS: (1) At 52:55, I meant to say that the SECOND claim entails the FIRST while the FIRST does not entail the SECOND. Oops! (2) Thankfully, the audio improves at 28:32! Remind me to never record a solo presentation using Zoom… (3) Here's an important clarification about the roommate/magical marker example given around 2:07:00. In the video, I was not exactly clear about the content of the hypotheses in question and how this affects their likelihoods and priors. Here is how I should have spelled out the example in the clearest way possible. ​Consider two hypotheses: H1: My friend wrote on my board H2: My marker by itself wrote on my board The data is: D: "Don't forget to take out the trash!" is written on my board Now, H1 renders D quite surprising, given that my friend knows all about my diligent habits of taking out the trash, etc. If H1 is true, he would most likely have written something on the board then erased it (since he knows I don't like him touching my board, etc.). And even if he didn't erase it, it would be very odd for him to write *this* given that he knows I'm super diligent about the trash. But H2 renders D _far, far_ more surprising. Of all the possible things the marker could conceivably have written on the board by magically floating upwards etc., only an absurdly small fraction are even coherent, let alone English words strung together to compose an intelligible, grammatical, contextually relevant English sentence. Of course, H1 has higher prior than H2. But the point made in the video stands, since the likelihood of H1 (i.e., P(D|H1)) is very low, but it's still much greater than the likelihood of H2 (i.e., P(D|H2)), and hence data can still be evidence for a hypothesis even though the data is very surprising on that hypothesis.

[@ReverendDr.Thomas]

Keep-up the good work, Joseph! 😇

[@princechukwuemeka2639]

I really needed this. Thank you so much! Do you have any recommendations for someone who wants to learn set theory and closely related topics in a way that is related to philosophy and logic? Video and book recommendations are welcome!

[@MajestyofReason]

@@princechukwuemeka2639 Yes! An excellent, recent, concise, cheap introductory book is in the Cambridge Elements series and is entitled "Set Theory".

[@princechukwuemeka2639]

@@MajestyofReason Okay. Thanks. I'll check it out

[@kaiserquasar3178]

I think I noticed a slight error around 2:07:00. In the example you gave, it's not the likelihood that's very small, but the prior probability of the 2 hypotheses. P(the reminder being written | the marker magically writing it)=1, because, I mean, think about it. It's a tautology. If a marker writes something, that thing is written, necessarily. The prior probability, or P(the marker magically wrote everything) is, in fact, tiny. It's also the case that P(the whiteboard has stuff written on it | your roommate wrote it)=1, necessarily. What's also small, but not as small as the first prior, is the prior probability of your roommate (I impulsively typed "friend" instead of roommate, xd) having written everything. I'm half-expecting it to be the case that you did this on purpose, just so we'd have something to think about, but we'll see.

[@ShinyBaboon]

Joe released a 3 hour video about Bayes Theorem? Woohoo! Sounds like I know what I'm doing with my Friday night!

[@SuckonDeezonuts]

😂 nerd.

[@TheAnalyticChristian]

This may be my favorite video you've ever made. It was a perfect intro to a topic that many people (me included) find difficult. I will be applying what I've learned in my own future videos. Thanks Joe!

[@MajestyofReason]

♥♥♥

[@christthinker6345]

Literally just read **Bayes' Rule: A Tutorial Introduction to Bayesian Analysis** and was scoping out more information on Bayes. We’re on similar wavelengths, Joe!

[@lizjackson111]

Love this video! Amazing job. From the description: "The argument I give around the ten-minute mark admittedly doesn't address the view that (i) credences don't exist, and yet (ii) we can still account for the relevant data about our doxastic lives by appeal to beliefs about probabilities." True. That said, I actually think much of what you say later could still be true even if credences reduce to beliefs about probabilities, or credences don't exist and probability-beliefs play the relevant roles. Probabilism, conditionalization, (etc) would just be normative constraints on our probability-beliefs. (However: those "belief-first" or "belief-only" views are false.)

[@MajestyofReason]

They are, indeed, false, as this one philosopher -- have you heard of her? -- named Dr. Liz Jackson has convincingly shown!

[@danielzhang7506]

I love this channel, since it doesn’t push an agenda either way on theism or atheism and the comment section seems to be genuinely respectful towards both sides, unlike certain other channels which seem to demonise theists.

[@ILoveLuhaidan]

The work you put into your videos is unfathomable

[@logos8312]

This is a great start! If you end up with time to either add a part 2 or splice some things into this vid, here are some suggestions: 1. You mentioned why one might be an objective Bayesian but not why they might be a subjective Bayesian. A short discussion about Van Fraasen's (sp?) square thought experiment and the Principle of Indifference would be helpful, also a discussion about which levels of a hypothesis the Principle of Indifference could be applied to. 2. I've now reached a point where I'm jaded about the use of Bayes in philosophy, however, I make an exception of people are willing to express things via the Law of Total Probability rather than these abstract ratios which hide the fallacies mention in this video. Let's consider something like the FTA (I'd link to my blog post about it but sometimes youtube hides comments with links). P(LPU) = P(LPU | R) x P(R) + P(LPU | NR) x P(NR) This splits P(LPU) (Life Permitting Universe) into Random or Nonrandom causes. P(LPU) = [P(LPU | R and D) x P(D | R) + P(LPU | R and ~D) x P(~D | R)] x P(R) + P(LPU | NR) x P(NR) This splits P(LPU | R) into D and ~D, i.e. randomness following a statistical distribution or non statistically distributed randomness. P(LPU) = [[P(LPU | R and D and U] x P(U | D and R) + P(LPU | R and D and ~U) x P(~U | D and R)] x P(D) + ... This splits P(LPU | R and D) into a Unform Distribution (Principle of Indifference on constants) vs ~U (some non Uniform distribution on constants) So basically if you take what "N" means in the FTA, it's actually: R and D and U (universe is random, obeys a statistical distribution, with uniformly distributed, independent constants. Now what happens if this particularized idea of "N" vanishes for FTA? You still have everything else. P(LPU) = [P(LPU | R and D and ~U) x P(~U | D and R) + P(LPU | R and ~D) x P(~D | R)] x P(R) + P(LPU | NR) x P(NR) Note that FTA proponents could try to shrink this further by saying that P(~U) ought to be very small, but that's not so. Van Fraasen arguments, etc. may show that even if we have reason to favor P(U), there's no reason the preference ought to be overwhelming. Even if it's 70/30, that still leaves P(~U) at .3, and so if P(LPU | R and D and ~U) isn't super low (maybe it's .7), then you could get a reasonable probability like .21 out of all the P(LPU | R) space, especially relevant given that we don't know what P(LPU | R) is in a vacuum (it might be .4 for all we know and so this option yields over half the probability space). 3. The last issue I have with this philosophical use of Bayes is how to numerically evaluate non quantifiable propositions. Cancer tests are one thing, the probability that God wants morally relevant creatures, given that God exists, is quite another. I've been told so many times that one ought to think this probability is high, but I haven't been given any "objective" reason to think that it's high, important given the supposed "objectivity" of the prior. Without a systematic way to quantify propositions, free from disagreement, this project is DOA. Note that applied Bayes in the real world lacks this problem because different priors get iterated sequences of new information which should converge them to a true prior. But something like the FTA doesn't give iterated sequences of information to update priors and resolve disagreements. What you see is what you get, so you have to be 100% correct on the first try, or deal with an infinite recursion problem (the probability that my probability that my probability ... that my probability is correct is...)

[@MajestyofReason]

Excellent additions!

[@314god-pispeaksjesusislord]

Interesting, can we move away from the God question for a moment. Everything you and Joe discuss is now being considered in the use of AI as a "court". That is to say that chatGPT, for example may be used by both parties and then by the judge. Okay, now let's take a case like RUBY RIDGE (I just watched Joe Rogan discussing it, so that is the hat I pulled this rabbit from). Obviously the government claimed to have acted with 100% confidence as did the accused. At the core of the conflict is the issue of ENTRAPMENT (you can find the elements at the justice website, 645 Elements of Entrapment, Joe didn't refer to that BTW, which he should have). As in all crimes, it's necessary to prove not only the criminal act but also criminal intent or MENS REA. Now, humans are going to apply experience and intuition to gauge intent, but I AI will have to rely on Bayes and will be tasked with either it's own novel approach or perhaps a statistical measure of past decisions, STARE DECISIS, made by humans PRESUMING it's been programmed to have human values. What database of values will it qualify? US cases only, or of all known history and then quantify from that data set? Now, I'm not debating whether AI should or shouldn't be used, it is being used, but instead pointing out that this discussion of philosophy is an example of the difficulty with proving the point you guys are making, because AI cannot NOT apply some type of systematic logical construct, while humans on the other hand are capable of evaluating the application of concepts like mercy and the ability to repent. Bayes is applied by courts to these issues in likelihood of recidivism. This does in fact bring in the God question, although it's not necessary to prove God's existence, because the belief of the judge as well as the belief of the convict affect the outcome. Moreover it affects the society which has other pressures on the system like overcrowding and funding. This is precisely why the "founding fathers" said the US system can only be maintained by a Christian people. Now, the post Christian nation demands a different solution which is where AI is being introduced. Do you like that system? It's being applied in China. Thomas Hobbes pointed out in THE LEVIATHAN that belief in God who makes moral judgements is NECESSARY to ensure that promises will be kept without absolute state surveillance, and that is why China is imposing absolute state surveillance. So belief in God is the necessary and most fundamental choice for a free society, consequently it then evident that it's immoral not only to be an atheist or agnostic but even more so to teach others to be so. Now, you can see how this is also necessary for the change proven necessary By Peter Singer in his 1972 essay on FAMINE, AFFLUENCE AND MORALITY. Even if AI feeds the world and absolutely ends crime, it does nothing to make men actually moral. IF evolution by natural selection is true, it selects for belief in God as the greatest good, whether God actually exists or not, so it's immoral to choose to disbelieve regardless of the evidence for the non existence of God, and you can use Bayes to arrive at an extremely high confidence of that consequence.

[@calebp6114]

A lovely reason to procrastinate my history degree- grateful as always Joe!

[@MsJavaWolf]

Cool video, I already knew Bayes' Theorem from high school and my CS degree, but the video made some things more intuitive, I also didn't know about the odds form.

[@JohnnyHofmann]

Awesome video, Joe! Very very helpful.

[@dougjordan9385]

I did not know that this could be applied to Philosophy but I have seen it applied to history by Richard Carrier. As another viewer, I have an engineering background so thanks for the proofs. Great presentation.

[@MiladTabasy]

Ontology, metaphysics, epistemology, religion etc. Why is your domain of philosophical investigation incredibly vast? Thanks for the feeling of responsibility and conscientiousness for making these long videos. I wish you a great future. How about your own theories? tell us about them.

[@STAR0SS]

I have a strong prior that this video is great, I'll update you after I watch it.

[@christianidealism7868]

Good video Joe, this was fairly unbiased since even within the bayesian literature there is disagreement on things like intrinsic probabilities and the structure of probabilities. For example, the Orthodox view of bayesianism says that basic probabilities are the unconditional probabilities of complete worlds, while explanationism claims that basic probabilities are the atomic hypotheses conditional on potential direct explanations. This means that there is disagreement about if all probabilities are conditional or not. So when investigating any sort of evidence, There is always going to be conditional constraints on such and such evidence. For example in my research on the problem of evil I have discovered that theism makes it's predictions based on axiology. If theism is the proposition that an all-good, all-powerful, all-knowing being created the world, then this proposition only makes predictions about what the world looks like in conjunction with a particular axiology (theory of the good). Crudely, the explanatory diagram is: {Theism, Naturalism, Other theories of ultimate reality} --> Empirical facts about the world

[@queencabbage3689]

Dude, you need like 50 times the subscribers you've got; this was magnificent.

[@MajestyofReason]

I’m at least happy it helped you🙂❤️

[@batesthommie2660]

Statistician here! I absolutely love this take!

[@MajestyofReason]

Yesssss, I think you'll appreciate how the statistics and probability relate to philosophical questions about evidence and rationality :)

[@joelturnbull4038]

Wow! Thank you so much for this, Joe! It was much needed, I think. This is perhaps the best-explained video of yours that I have seen. I probably could not have understood this video just a few years ago, but thanks to you, Capturing Christianity, TAC, Liz Jackson and others, I feel so much more comfortable with philosophy than I did before. One question: how does Bayes’ theorem intersect with arguments to the best explanation and theoretical virtues? Are they closely related, or are they different things altogether? I’m still a bit confused about that.

[@MajestyofReason]

Glad my video could help you!! They are closely related. The Bayesian, in fact, will generally argue that comparing theoretical virtues is best formalized by using Bayes’ theorem. In short, when comparing theoretical virtues, there are two main virtues: explanatory power and simplicity. (This is why you’ll hear Oppy say things like “…whichever theory best manages the trade off between maximizing explanatory power and minimizing theoretical commitments…”. So when we compare theories or hypotheses, we compare them along these dimensions. But notice that this is basically the odds form of Bayes’ theorem! The likelihood ratio corresponds to explanatory power. The hypothesis which better explains the data is the one that better predicts it, ie, that renders it less surprising or more expected than it is on the other hypothesis. This is the likelihood ratio. The hypothesis which is simpler is (generally) the one with the higher prior probability, and hence comparing simplicity corresponds to comparing the ratio of the priors. We can also fit in other theoretical virtues into this analysis. The theory which fits worse with our background knowledge has a lower prior. The theory which is less coherent - which has parts in tension with one another - has a lower prior. And so on 🙂

[@joelturnbull4038]

@@MajestyofReason thanks - that helps

[@kaiserquasar3178]

I swear 75% of the best channels out there have less than 100k subs. Insane.

[@MajestyofReason]

❤❤❤

[@NontraditionalCatholic]

I've been waiting for this! I have a BS in Engineering, so I took a decent amount of higher level maths (up through Calc 4), and so I know just enough to know how much I don't know. But I can also tell when other people don't know haha, and I have been super skeptical of the way that some apologist types have been using Baye's theorum for some time now. Looking forward to watching this one - and likely re-watching, a few times!

[@MajestyofReason]

Very happy to help, and always great to see you around!🙂

[@analyticallysound2716]

I just dropped out of university level calc 3 based probability course because it was so difficult. Probability is never explained well. This video does very well though in explaining.

[@NontraditionalCatholic]

@@analyticallysound2716 I'm 49 minutes in and following so far, but I still have plenty of time to get confused haha

[@analyticallysound2716]

@@NontraditionalCatholic Oh I'm talking about a probability course which covers combinatorics, permutations, conditional probability, axioms of probability, random variables, limit theorems, markov chains, etc. I don't find conditional probability very difficult either.

[@gleon1602]

I finally finished this video. I finally understand Bayes Theorem

[@jenst.]

This is amazing. I have been studying Bayesian logic for 3 years now and this is the best summary+explanation for Bayesian Reasoning I have come across so far. It would have been so much easier if I had started with this instead of Math books and Philosphy Papers. If you ever do a re-work/edit of these videos I suggest to make 3 additional slides on: 1) The Handling of several pieces of evidence at once compared to consecutive updates. Although this does not change the outcome, so to speak, it does make a difference in how you present an argument or rather how easy it can be comprehended and thus scrutinized). 2) The common misconception that a high probability for E given H does not entail a low probability of E given not-H. This rarely happens with rough estimations of the Odds version, but easily happens when assessing some form of data based probabilities directly. 3) The use of BT to show what type of Evidence would indeed tip the scale. I think that this can also help to understand when an argument makes a stong case compared to an agnostic conclusion that can easily sway to one side or the other. And my final comment relates to the principle of indifference. I think that you could put more emphasis on the notion that its not valid to start with 1:1 if you DO have information that tips the prior to one way or the other.

[@jenst.]

Sorry. Some double negation on 2). I meant that the two probabilities of the likelihood ratio are independent, but that there is a common misconception that they arent. (Probably because people look at the prior ratio where this is the case).

[@manavkhatarkar9983]

Where did you learn all of this from 🙂? It's so in-depth... providing mathematical proofs and whatnot 🙂🙂. ❤❤ Learning a lot from you 🙏🏻🙏🏻

[@Jblues87]

Nice video. What paper of Draper's is Real Theology drawing from in the video on understated evidence?

[@shanesullivan460]

This, right here? This is a treasure.

[@newglof9558]

super cool video

[@yuko4678]

Hi i’m new to this channel and found this very interesting i was wondering if you could do a video or have a video on immaterial and material things in metaphysics

[@kyle2591]

LETS GOOOOOOOOOO! Thanks bro!!

[@boringturtle]

What's the difference between the classical and frequency definitions of probability?

[@lolroflmaoization]

i wonder what you think of the views presented in the paper "Credence-and Chance-Without Numbers (and with the Euclidean Property) " By Tim Maudlin.

[@vinegar10able]

It reminds me of David Papineau's book, "Philosophical Devices"

[@tymmiara5967]

That disease and test question was a first question when I was interviewed at Oxford for physics studies, haha. And then they asked what happens if she tested twice.

[@Breakdowns04]

Hi, Joe! I really appreciate your work! I have a question about the section on propositional logic. I heard that a statement is vacuously true when the antecedent is true and the consequence is false. If that is the case, how can a logical entailment be a contradiction? Wouldn’t the entailment render the statement false?

[@MajestyofReason]

A conditional statement of the form ‘if P, then Q’ is false in exactly the following circumstances: P is true and Q is false Otherwise, the conditional as a whole is true. The conditional is said to be vacuously true when the antecedent, P, is false. So, for instance, this conditional is vacuously true: (1) if Joe does not exist, then Arsenal will win the premier league. And so is this: (2) if Joe does not exist, then Arsenal will NOT win the premier league.

[@Breakdowns04]

@@MajestyofReason oh, shoot! I got my truth values mixed up. Thank you! :)

[@DeepDrinks]

Hey Joe, have you read Humes's Abject Failure? He uses Bayes Theorem to argue against Humes philosophy on Miracles. I have heard him criticised for this, I was wondering your thoughts? Looking forward to having you on Deep Drinks soon.

[@MajestyofReason]

Unfortunately I haven’t!

[@MajestyofReason]

This is also relevant: kzbin.info/www/bejne/ipaVn59ngMmqrbM

[@logans.butler285]

Hey Master Joe (I see you as a teacher at this point), what do you think of Loke's use of the Bayes Theorem in his book Investigating the Resurrection of Jesus Christ? His use of the theorem seems crucial for his case.

[@MajestyofReason]

I haven’t read his book on the resurrection, so I can’t comment!

[@dr.shousa]

Interestingly, I just gave an invited lecture at a philsci conference on how philosophy misunderstands (subjective) Bayesian epistemology. I think this videos is a good intro to Bayesian epistemology, as used in philosophy, but is very different than how it's discussed/used in statistics and decision theory. Both fields could benefit from interactions for sure, but as someone who has studied all three fields, I'd like to see philosophers pick up a book on Bayes in stats/decision and not act like everything stopped at 1954. For example, a subjective Bayesian (who actually doesn't hold to the 5 norms you talk about, as they can be derived from more basic axioms) with constraints is not an objective Bayesian. The idea of objective Bayesianism sort of died when people realized that it's (mathematically and conceptually) unattainable and being subjective had better decision making properties (ie you have less risk being wrong).

[@MajestyofReason]

There are definitely different ways to characterize the subjective/objective Bayesian divide! I’ve read several authors characterize it differently (and incompatibly…), such as Titelbaum and Huemer articulating it differently. I ultimately went with one prominent way that lots of philosophers have articulated it🙂 And you’re right that there needs to be more interdisciplinary work here. Was your lecture recorded? I’d love to watch!

[@dr.shousa]

@@MajestyofReason I've found both Titelbaum and Huemer to be quite lacking. The earlier stuff (Howson, Earman, etc) was much better, but I think people have stopped reading the primary material (Ramsey, Savage, Lindley, de Finetti, etc) and since parroted the secondary literature (Howson etc), which is a bit unfortunate. Fortunately, for me, not recorded haha. I think the goal is to write a paper based on the lecture(s), so hopefully I'll have some time in the future for it.

[@MajestyofReason]

@@dr.shousa well, if the paper gets published, please send it to me! If that happens, I could even invite you on my channel for a chat on it🙂

[@cynicviper]

BAYESED

[@Sui_Generis0]

Got 8.33...% in trivia, does that mean I passed and get to skip this lecture? All jokes aside, excited to watch this video with all the applications it has within certain philosophical questions

[@jmike2039]

Joe you need to team up with scott clifton and both star in the new soap opera, Our Doxastic Lives 😅

[@Shrubbist]

It's always grape to see a post by Majesty of Raisin.

[@kaiserquasar3178]

I have a question with regards to your showing that if E is evidence for H, then ~E is evidence against H. Your first step after supposing that's the case was to say that this means P(E | H)/P(E | ~H)>1, but is this entailed by the definition the video had us become accustomed to (E is evidence for H if P(H | E)>P(H)? Maybe, but in that case, I guess explaining that would have been nice. It's intuitive that E is evidence for H if it's more expected on H than ~H, but it seems also true that E is evidence for H just in case P(H | E)>P(H). So which is it? Or is it both?

[@MajestyofReason]

It’s both🙂 Moreover, P(H|E)>P(H) iff P(E|H)>P(E|~H).

[@manavkhatarkar9983]

Btw Joe, in the breast cancer problem, does the 8% probability of her having cancer given that she tested positive, mean that the test is not good? 😅

[@MiladTabasy]

Please make a video about permissivism .

[@chrisaddington7379]

If brest cancer scenario had been, a group of 100 screened, then group told 1 person (not known to group) had tested positive then the chance for any one is 8%, But NOT when that 1 person IS informed, changes to 90% chance

[@Mrfunny663vnb83]

This thumbnail is giving me 3blue1brown vibes and I loved it :)

[@teenagesatanworship]

Thank you so much Joe, your videos are great and this one was especially great! Gonna be watching a few more times to let it all sink in.

[@DigitalGnosis]

Great video Joe. Would love to see you do something to address the heterodox use of Bayes theorem by philosophers of religion, particularly Theists/apologists, trying to create an illusion of scientific sophistication to make views intimidating and difficult to engage with.

[@gabbiewolf1121]

0:43 My answer to the quiz (I'm cheating because I already know Bayes Rule x3): 0.08333 with the 3s repeating

[@bencnnw]

Damn 43 seconds and I'm here 0_0

[@vinegar10able]

I always thought librarians were far more common than farmers. I've met librarians, but I've never met a farmer.

[@TheOtherCaleb]

Why are you so smart, Joe?

[@zombieinjeans]

Bayes' theroem is super useful as a tool, but it utterly fails as an epistemology. We don't know things based on probability or have an order of possibility for our theories. Knowledge is explanatory, and we only ever have our best explanation. In the extremely rare case we have two good explanations (ie: Newtonian Gravity and Relativity), we must devise a crucial experiment to distinguish them. That's why falsifiability is such an important component to a good explanation. Highly recommend the work of David Deutsch, and Brett Hall's KZbin channel on epistemology.

[@kamilgregor]

tl;dr extraordinary claims require extraordinary evidence

[@MajestyofReason]

I mention that in the video! :)

[@Oskar1000]

2:07:15 This is just anti supernaturalist bias.

[@zsoltnagy5654]

3:01:07: *Evidential symmetry:* *If E is evidence for H, then ¬E is evidence against H.* More precisely: E is evidence for H, if and only if ¬E is evidence against H. So spread the word, that *"Absence of evidence E for H is evidence ¬E of absence of H or against H!"* Just sayin.

[@chrisaddington7379]

Hi, you are wrong re breast cancer- if 90% correct result then 90% chance breast cancer. Statistical result of 1% pop'n having cancer is irrelevant if had a positive screening. REGARDLESS OF 'MATHEMATICAL' DEBATES,, IF EVEN THE SLIGHTEST CONCERN FOLLOW UP WITH FURTHER TESTS. sincerely Chris Addington