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Visualizing Bayes’ theorem

I recently came up with what I think is an intuitive way to explain Bayes’ Theorem. I searched in google for a while and could not find any article that explains it in this particular way.

Of course there’s the wikipedia page, that long article by Yudkowsky, and a bunch of other explanations and tutorials. But none of them have any pictures. So without further ado, and with all the chutzpah I can gather, here goes my explanation.

Probabilities

One of the easiest ways to understand probabilities is to think of them in terms of Venn Diagrams. You basically have a Universe with all the possible outcomes (of an experiment for instance), and you are interested in some subset of them, namely some event. Say we are studying cancer, so we observe people and see whether they have cancer or not. If we take as our Universe all people participating in our study, then there are two possible outcomes for any particular individual, either he has cancer or not. We can then split our universe in two events: the event “people with cancer” (designated as A), and “people with no cancer” (or ~A). We could build a diagram like this:

venn-aSo what is the probability that a randomly chosen person has cancer? It is just the number of elements in A divided by the number of elements of U (the Universe). We denote the number of elements of A as |A|, and read it the cardinality of A. And define the probability of A, P(A), as

eq01

Since A can have at most the same number of elements as U, the probability P(A) can be at most one.

Good so far? Okay, let’s add another event. Let’s say there is a new screening test that is supposed to measure something. That test will be “positive” for some people, and “negative” for some other people. If we take the event B to mean “people for which the test is positive”. We can create another diagram:

venn-bSo what is the probability that the test will be “positive” for a randomly selected person? It would be the number of elements of B (cardinality of B, or |B|) divided by the number of elements of U, we call this P(B), the probability of event B occurring.

eq02

Note that so far, we have treated the two events in isolation. What happens if we put them together?

venn-lastWe can compute the probability of both events occurring (AB is a shorthand for A∩B) in the same way.

eq04

But this is where it starts to get interesting. What can we read from the diagram above?

We are dealing with an entire Universe (all people), the event A (people with cancer), and the event B (people for whom the test is positive). There is also an overlap now, namely the event AB which we can read as “people with cancer and with a positive test result”. There is also the event B – AB or “people without cancer and with a positive test result”, and the event A – AB or “people with cancer and with a negative test result”.

Now, the question we’d like answered is “given that the test is positive for a randomly selected individual, what is the probability that said individual has cancer?”. In terms of our Venn diagram, that translates to “given that we are in region B, what is the probability that we are in region AB?” or stated another way “if we make  region B our new Universe, what is the probability of A?”.  The notation for this is P(A|B) and it is  read “the probability of A given B”.

So what is it? Well, it should be

eq03And if we divide both the numerator and the denominator by |U|

eq051we can rewrite it using the previously derived equations as

bayes-eq1What we’ve effectively done is change the Universe from U (all people), to B (people for whom the test is positive), but we are still dealing with probabilities defined in U.

venn-justb

Now let’s ask the converse question “given that a randomly selected individual has cancer (event A), what is the probability that the test is positive for that individual (event AB)?”. It’s easy to see that it is

bayes-eq2Now we have everything we need to derive Bayes’  theorem, putting those two equations together we get

bayes-eq3which is to say P(AB) is the same whether you’re looking at it from the point of view of A or B, and finally

bayes-eq4

Which is Bayes’ theorem. I have found that this Venn diagram method lets me re-derive Bayes’ theorem at any time without needing to memorize it. It also makes it easier to apply it.

Example

Take the following example from Yudowsky:

1% of women at age forty who participate in routine screening have breast cancer. 80% of women with breast cancer will get positive mammograms. 9.6% of women without breast cancer will also get positive mammograms. A woman in this age group had a positive mammography in a routine screening. What is the probability that she actually has breast cancer?

First of all, let’s consider the women with cancer

example14Now add the women with positive mammograms, note that we need to cover 80% of the area of event A and 9.6% of the area outside of event A.

example21It is clear from the diagram that if we restrict our universe to B (women with positive mammograms), only a small percentage actually have cancer.  According to the article, most doctors guessed that the answer to the question was around 80%, which is clearly impossible looking at the diagram!

Note that the efficacy of the test is given from the context of A, “80% of women with breast cancer will get positive mamograms”. This can be interpreted as “restricting the universe to just A, what is the probability of B?” or in other words P(B|A).

Even without an exact Venn diagram, visualizing the diagram can help us apply Bayes’ theorem:

  • 1% of women in the group have breast cancer → P(A) = 0.01
  • 80% of those women get a positive mammogram, and 9.6% of the women without breast cancer get a positive mammogram too → P(B) = 0.8 P(A) + 0.096 (1 – P(A)) = 0.008 + 0.09504 = 0.10304
  • we can get P(B|A) straight from the problem statement, remember 80% of women with breast cancer get a positive mammogram → P(B|A) = 0.8

Now let’s plug those values into Bayes’ theorem

eq06which is 0.0776 or about a 7.8% chance of actually having breast cancer given a positive mammogram.

Written by ob

May 1st, 2009 at 7:11 am

Posted in Math

Tagged with

128 Responses to 'Visualizing Bayes’ theorem'

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  1. Very well explained! I’ve been reviewing the fundamentals of Bayes theory recently to try to get my head around Bayesian networks from a non stats background. Don’t suppose you could point me in the direction of a similarly visual explanation for that, or maybe consider it in a future post?

    Rob

    1 May 09 at 1:37 pm

  2. Bravo! I actually did the same thing with Venn diagrams to explain it to myself awhile back – I’m glad you took the time to make it clear to others.

    Tom

    1 May 09 at 1:59 pm

  3. How can it be hard to derive Bayes formula if left and right part is essentially the same (A becomes B, B becomes A).

    James Bond

    1 May 09 at 2:06 pm

  4. Dmitriy Kropivnitskiy

    1 May 09 at 2:11 pm

  5. This was fantastic. Thank you.

    bumbledraven

    1 May 09 at 2:37 pm

  6. Best explanation of Bayesian, ever, period!!

    Doug

    1 May 09 at 3:01 pm

  7. Great job man! Visualizing this fascinating theorem was a great idea. I dunno though, I always found the actual examples for Bayes’s Theorem to be misleading… I guess that’s just the nature of statistics in odd contexts though!

    CJ Cenizal

    1 May 09 at 3:14 pm

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    story has entered the popular today section on popurls.com…

  9. [...] Visualizing Bayes’ theorem | Ramblings (tags: probability bayes visualization) [...]

  10. Dmitriy: Linked in the intro.

    Oscar: “Of course there’s the wikipedia page, that long article by Yudkowsky, and a bunch of other explanations and tutorials. But none of them have any pictures.” <– the Java applets/visualizations don’t count as pictures?

    gwern

    1 May 09 at 6:31 pm

  11. [...] Visualizing Bayes’ theoremFormer Barney Frank staffer now top Goldman Sachs lobbyistThe Venture Boutique: ScalingWhy VC [...]

  12. Most beautiful and simple explanation. Thanks a lot

    Vinay

    1 May 09 at 9:40 pm

  13. Thanks a lot! This was the first time that Bayes actually made sense to me. It’s much easier to understand when visualized.

    BD

    1 May 09 at 9:47 pm

  14. You are right, memorizing this method gives a hard-to-forget method for rederiving and re-learning Bayes’ rule.

    Tom

    1 May 09 at 11:41 pm

  15. Thank you for this article!

    I really liked the last bit, where you showed the cancer example. That was a good example of “applying the theory shows why you should learn the theory”.

    Jens

    Jens

    2 May 09 at 2:33 am

  16. Nice. I created something similar a while back due to constantly having to explain this in #math on IRC:

    http://imagebin.org/47539

    The application to mammograms is a nice one though. Bookmarked!

    David House

    2 May 09 at 3:38 am

  17. This is the absolute best derivation of Bayes theorem I have ever seen, including a course in statistics I did last year. Your post definitely needs to go up on the wikipedia page, under the intuitive reasoning part.

    J

    2 May 09 at 5:17 am

  18. Hm. Bayesian networks… I don’t know of any “visual explanation” of those. I’ll take a look and see if I can come up with something. Thanks for the comment.

    ob

    2 May 09 at 10:19 am

  19. If I understand your question correctly, you’re saying “what happens when A and B are the same?”. In that case, P(AB) = P(A) = P(B), and P(A|B) = 1. Which is saying “given that event A occurred, what is the probability of event A occurring?”. It should be pretty obvious that it is 1.

    ob

    2 May 09 at 10:21 am

  20. True, the Yudkowsky article has some pictures, and if you keep Java off like I do, all you see are the pictures. My main problem with Yudkowsky’s article is that it’s really, really, really, long! If you are looking to just get the gist of what Bayes’ theorem is saying you are not going to read that much.

    ob

    2 May 09 at 10:24 am

  21. This is great. Very useful.

    I don’t want to be a pedant, and I may be wrong, but aren’t these technically Euler diagrams and not Venn diagrams?

    Patrick

    2 May 09 at 10:58 am

  22. [...] Good explanation of Bayes theorem [...]

  23. Well, if you consider the fact that the sets are only defined within the Universe, then they are indeed showing all the possible intersections. Thus they are Venn diagrams.

    ob

    2 May 09 at 6:05 pm

  24. [...] theorem, relating the conditional and marginal probabilities of two random events, this blog post does a great job through the use of [...]

  25. [...] Visualizing Bayes’ theorem | Ramblings [...]

  26. Wow.

    I’ve read quite a few explanations of Bayes’ theorem and this is the first which (in the space of really a few minutes!) has got the point across so succinctly and clearly I now actually understand it.

    My thanks.

    Clay

    4 May 09 at 2:59 am

  27. Brilliant! Thank you!

    Abhijith

    4 May 09 at 4:44 am

  28. Thanks a lot!!!

    Andries Inzé

    4 May 09 at 6:58 am

  29. Thank you for re-sparking my love for Discreet Mathematics.

    I fondly remember studying this, years ago, but had forgotten the enjoyment of it.

    I eagerly await any further of your works.

    Perhaps animated…?, perhaps interactive…?

    Again Thanks.

    Stephen Bell

    4 May 09 at 10:52 am

  30. [...] Ramblings Stuff that’s on my mind « Visualizing Bayes’ theorem [...]

  31. This should be made into a visualization app for Google’s new public data search and frankly any data set. Anyone want to work on it with me?

  32. Great job, short, clear and effective – thanks for taking the time to create it!

    Avi

    5 May 09 at 2:24 pm

  33. The exact problem at the end is a good example of how stats can confuse the lay man. Test accuracy should be reported in the rate of false negatives, rather than true positives. The cost of a large number of false negatives can be great relative to a large number of false positives. However, depending on your situation, you could have a lot of false negatives AND false positives, indicating that your test is not well-correlated for the condition being tested for. In fact, just calculating the correlation coefficient is an important step.

    Veggie

    5 May 09 at 3:34 pm

  34. Great Job, and excellent example :P | Muy Buen trabajo y excelentes ejemplos!

    Edwin Pardo

    7 May 09 at 7:24 pm

  35. Great explanation! I teach high school mathematics and am stealing your visualization, example and all, to use in one of my next lessons! :-)

    Andrea

    10 May 09 at 2:18 pm

  36. hmm… i think the easiest way to understand bayes theorem is to multiply both sides to get..

    P(B) P(A|B) = P(A) P(B|A)

    both of these are just P(A intersect B) – probability of A and B is the same as the probability of A times the probability of B given A, or the probability of B times the probability of A given B

    jeff wu

    28 May 09 at 3:10 pm

  37. David Newman’s book, Hippocrates’ Shadow, Simon&Schuster 2008 has an good treatment of of this in relation to medical interventions and clinical trials. He uses a bar graph variant to demonstrate things Baysean. Good, but not as good as your demonstration! His discussion of this topic sent me to the web where I finally and gratefully got to your visualization! Thanks.

    AlexPirie

    30 May 09 at 6:51 pm

  38. i n really convay to this method of learning of process

    ael

    5 Jun 09 at 12:26 am

  39. this is freagin awesome explanation. thank u :)

    Milz

    8 Jun 09 at 6:45 pm

  40. I too teach mathematics and am impressed with your clear explaination.
    Well done, and thanks for taking the time to lay it out.

    Ken walker

    10 Jun 09 at 7:09 am

  41. An addendum that “completes” the Bayes theorem so that the denominator is the same as in the Yudkowsky article would be nice. Graphically, that would be saying that the area of B is the “union” of two sections: 1. the non-overlap, which is areas in B “starting with” the areas that are NOT in A (the ~A part of the universe) i.e. P(B|~A) * P(~A), 2. the overlap, which is areas in B “starting with” the areas that ARE in A i.e. P(B|A) * P(A). It’s more helpful for applications in which the data makes it easier to figure out probabilities of A and relate those to B, than to figure out probabilities of B.

    Reedo

    11 Jun 09 at 4:24 am

  42. Professor Oscar,

    You are a great prof! You set a benchmark for us trying-to-be’s.

    A question. When can I write

    P(AB/R) = P(A/R) + P(B/R) ?

    What conditions need to be imposed on A, B and R?

    Thanks Professor!

    Tapan Bagchi at the Indian Institute of Technology Kharagpur

    Tapan Bagchi

    16 Jul 09 at 6:48 pm

  43. thank you very much…very nice article…

    desiNerd,
    IITKGP

    desiNerd

    19 Jul 09 at 3:46 am

  44. that’s a really appealing way of expressing Bayes’ theorem
    how would the venn-diagram be if the conditional probability of A
    didn’t depend on B i.e P(A/B)=P(A) ?

    pinki

    14 Aug 09 at 2:23 am

  45. Very nice artice article in every meaning of the word. Have you publised elsewhere?

    jozsef

    14 Aug 09 at 12:31 pm

  46. Great work dude….I was banging my head to develop intuition on Bayes theorem. Now I got the idea clearly…

    Thank u very much..

    CHARLY

    22 Aug 09 at 3:19 am

  47. May want to check the book by Wonnocott and Wonnocott. They have a visual way of explaining Bayes Theorem

    Bill

    23 Aug 09 at 3:57 pm

  48. Beautiful explanation. I was searching for something that would help me explain it to undergrads…thanks a bunch.

    Usman

    16 Sep 09 at 10:35 am

  49. This is one of the best explanations I’ve found. Perhaps we can see if I really understand it by trying a real world problem I’m wrestling with.

    Here’s the data:
    – The odds of a chest pain (CP) being caused by a heart attack is 40%.
    – The odds of a CP being caused by other factors (anxiety, depression, etc.) is 60%.
    – The odds of a heart attack occurring to a female above age 50 is 80%.
    – The odds of a heart attack occurring to a female under age 50 is 20%.

    I am presented with a 24 year old female who says she is having chest pain. What is the probability that her chest pain is caused by a heart attack? Is it 0.4 x 0.2 = 0.08?

    Also, 78% of patients having heart attacks present with diaphoresis (sweating), so 22% of patients having heart attacks don’t sweat. This female is not sweating, so are the odds of her having a heart attack 0.22 x 0.08 = 0.0176?

    Thank you!

    Dan Weisberg

    2 Oct 09 at 10:47 am

  50. You beat me to it! I got the idea of thinking about Bayes’ theorem by Venn diagrams by reading Reza’s An Introduction to Information Theory.

    Truecrimson

    11 Oct 09 at 4:26 am

  51. [...] want to explain the calculation by Venn diagrams but someone else beats me to it http://oscarbonilla.com/2009/05/visualizing-bayes-theorem/ There’re links to others’ presentations so pick your favorite. What I think in terms [...]

  52. Can someone help me with this question?
    One fifth of customers entering a certain Future
    Shop store are under 20 years old. 5% of these under 20 year olds make a purchase over $500,
    and 10% of the customers 20 years or older make a purchase over $500. What is the probability
    that is a major purchase is made, it was by a person 20 years or older?

    Riley

    26 Nov 09 at 3:09 pm

  53. Just wanted to convey my thanks.
    I am teaching this subject to undergrads at a Chinese university and found your explanations and visuals a huge help.

    Mark

    8 Dec 09 at 6:58 pm

  54. Good visual explanation. I used it to work out a probability but don’t know if the professor is going to reject it.
    Thanks.

    Franco

    13 Dec 09 at 6:02 pm

  55. this proof was awesome.

    viky

    9 Feb 10 at 10:57 pm

  56. Really “Great” Job. Keep doing such good job, humanity will be enriched by people like you. Hats off to You!!

    Amlan

    15 Jun 10 at 6:01 am

  57. Great! It’s subtle and effective. Thanks a lot.

    Rishav

    24 Aug 10 at 12:28 am

  58. Very nice illustration of how to think of the probabilities.

    Well done.

    Christian

    2 Nov 10 at 8:40 am

  59. i want bayes theorem application

    shalu

    15 Nov 10 at 8:24 am

  60. Thanks a lot. It was useful for me as a Bioinformatics student.
    Good work! Wish you inspiration for making more good stuff!

    Dada

    22 Nov 10 at 9:12 am

  61. great awesome .no words

    munish dadhwal

    29 Nov 10 at 9:45 am

  62. U = me
    A = Satisfaction with the explanation

    p(A) = |A|/|U| = 1

    The3rd_Chimp

    13 Dec 10 at 3:34 pm

  63. Very helpful demonstration.

    For a nearly identical explanation of Bayes theorem using Venn diagrams, see Bolstad’s book, Introduction to Bayesian Statistics.

    freethoughtful

    19 Dec 10 at 7:17 pm

  64. [...] science I cannot recommend it enough[2] . Yeah, that Yudkowsky,  perhaps you remember him from here. [↩]With just one caveat, the work isn't finished. Don't expect the story to end. [↩] [...]

  65. Excellent explanation!

    plarser48

    20 Jan 11 at 7:17 pm

  66. Cool! I just tried to do something like this myself using a “sieve” analogy. Check it out HERE. Someone linked to your example and I agree it’s more intuitive, though I commented on one aspect of the sieve I like a little better HERE. Thanks for putting this together!

    Hendy

    9 Apr 11 at 9:07 pm

  67. very useful. Thank you

    antonio

    9 May 11 at 5:32 pm

  68. A great blogpost, I just passed this onto a colleague who was doing a little analysis on this. And he in fact bought me breakfast because I found it for him. smile. So let me reword that: Thank you for the treat! But yeah Thanks for taking the time to discuss this, I feel strongly about it and enjoy learning more on this topic. If possible, as you gain expertise, would you mind updating your blog with more details? It is extremely helpful for me. Two thumb up for this blog!

  69. It is so great explanation.Now I have bookmarked your site.Now I regularly visit your site to enhance my understanding.

    raghaw

    27 May 11 at 4:16 am

  70. This is the most concise and comprehensive explanation of Bayes theorem that I’ve ever read. Thanks a lot!

    Joemar

    25 Jun 11 at 2:52 pm

  71. Thank you for your post, I enjoyed reading it.

    May I ask… what software did you draw your diagrams with?

    Carlos

    4 Jul 11 at 9:49 pm

  72. I drew the diagrams with OmniGraffle: http://www.omnigroup.com/products/omnigraffle/

    ob

    5 Jul 11 at 1:45 pm

  73. This explanation is exactly what I was looking for. Hats off to you.

    saleem

    6 Jul 11 at 6:41 pm

  74. How about a water/hydraulics method? People sometimes speak of ‘probability-mass’ or ‘probability-fluid’, so you could depict the network as a series of tubes changing in size and then the volume of output is the answer.

    gwern

    19 Jul 11 at 3:30 pm

  75. Great common sense here. Wish I’d thought of that.

    Early

    22 Jul 11 at 2:50 pm

  76. I’d like to use your explanation in an on-line high school class. Can I get you permission to copy it into our course material, with credit given to you as the source?

    David Nelson

    14 Aug 11 at 7:07 pm

  77. I hate giving a “me too” answer, but I really must. This is the best explanation of Bayes’ theorem that I’ve ever seen. Great job!

    Graham Percival

    4 Sep 11 at 7:37 am

  78. outstanding, very useful, will start using with my students

    Ricardo

    4 Sep 11 at 4:00 pm

  79. [...] Visualizing Bayes’ Theorem by Oscar Bonilla. [...]

  80. [...] BrainPickings James Burke theFWA (BenTheBodyGuard) ArtDirectorsClub Tatt.ly Zach Holmann Oscar Bonilla Next » « Previous Add a [...]

  81. I drew up a similar explanation about a year ago to explain why, even though there are more poor whites than poor blacks in the world, the perception that most black people are poor is not incorrect. I used current census data at the time, and realized the person registering the complaint on twitter was making the very natural error of inverting probabilities. Here was my illustration: http://cl.ly/V2Y

    Jeffrey Horn

    22 Sep 11 at 10:31 pm

  82. I should have said this was the natural error of ignoring base rates, not inverting probabilities.

    Jeffrey Horn

    22 Sep 11 at 10:38 pm

  83. [...] an incredible article! I wish I had this explanation in the high school. As a fully qualified mathematician I want to say [...]

  84. I think your explanation is flawed. You said P(A) = | A | / |U|, but that is only true if all elements of A have the same probability. In general that is a false statement.

    luis

    20 Oct 11 at 11:43 am

  85. >80% of women with breast cancer will get positive mammograms. 9.6% of women
    >without breast cancer will also get positive mammograms.

    80% have cancer and 9.6% don’t have cancer (when you have positive mammogram). Why doesn’t 80 + 9.6 addup to 100%? What probabilities are remaining?

    Damn! I wish I had slept in my math classes

    Ganesh Krishnan

    21 Oct 11 at 8:46 pm

  86. I agree, I can’t understand how to figure out Pr (B) :(

    I’ve got a stats module in my masters and having come from a non mathematical background… I’m struggling a bit.

    Mary

    22 Oct 11 at 11:57 am

  87. P(B) is the total probability of B, it’s the chance of B happening regardless of whether A happens. Thus, P(B) = P(B|A)P(A) + P(B|¬A)P(¬A).

    ob

    27 Oct 11 at 11:42 am

  88. 80% of women with breast cancer will get a positive mammogram means that P(positive mammogram | woman has cancer) = 0.8.
    The remaining 20% is the probability of getting a negative mammogram given that the woman has cancer.

    The 9.6 comes from the other part of the population, i.e. women without cancer that also get a positive mammogram.

    In general, P(B | A) + P(¬B | A) will sum to 1, but P(B|A) + P(B|¬A) will not as they are unrelated probabilities.

    ob

    27 Oct 11 at 11:46 am

  89. I didn’t want to overcomplicate the example.

    ob

    27 Oct 11 at 11:46 am

  90. very useful. thank you

    Ann

    1 Nov 11 at 12:24 am

  91. Dear Oscar,

    A clear presentation – a pleasure to read…

    E. Fitzgerald

    14 Nov 11 at 4:33 pm

  92. Nicely done. A similar (and equally intuitive and effective) visual explanation of conditional probability and Bayes theorem based on Venn diagrams can be found in Bolstad’s textbook on Bayesian Statistics.

    freethoughtful

    25 Nov 11 at 10:41 pm

  93. Very usefull article

    Wiki page must have link of Such a good and easy to under stand article.

    Thank you.

    Hemant Patel

    8 Dec 11 at 2:27 pm

  94. Outstanding! I’ve been trying to get my head around this for some time and your explanation made everything fall into place. Nicely done. Thanks for posting this.

    Mark

    22 Jan 12 at 8:14 am

  95. If this is the basis of Bayes Theory than we can simplify all of this to say that, in any given circumstance, if no explicit variables are defined than the outcome of the equation, whether inversely or conversely calculated always equals an equal percentage of the number of circumstances… Or any equal division of the given number of arguments.

    In your cancer example, you give explicitly defined variables which makes is possible to calculate an explicit answer.

    I take this to mean that Bayes Theory is simply a definition of how to form the equation.

    Am I right in my assumption?

    BCPower

    9 Feb 12 at 5:20 am

  96. Furthermore, the simplified formula that I mentioned above could, consequently be compounded by the number of given circumstances. Assuming no variables are defines as the graphic at the top suggests, it could be said that the Bayes formula would compound in similar, possible equations much like possible combinations of a lock. That is to say, if given a combination lock with 4 dials, each with 2 numbers, your number of possible combinations are mathematically derived because we are in fact given two variables. The actual answer is not what I’m after but merely to say that it is, in fact, calculable.

    To bring my previous derision full circle, we were given two variables. The first was A and the second was U.

    My assertion is that Bayes Theory is simple a rule-set by which to write a formula. A formula for a formula if you will. :o)

    Kind of redundant in a matter of thinking but a useful teaching tool none the less.

    BCPower

    9 Feb 12 at 5:49 am

  97. Thanx Oscar , that was really intuitive ..

    Sree

    18 Mar 12 at 2:01 am

  98. Thanks for the explanation

    Masud

    16 Apr 12 at 5:10 am

  99. Thanks! You proved how truly understanding something means being able to teach it in a way that grade school kids can understand :)

    Chris

    8 May 12 at 4:40 am

  100. Thank you so much!!! Very intuitive diagrams!!

    Inder

    19 May 12 at 6:37 am

  101. [...] Thinking about Bayes theorem graphically: http://oscarbonilla.com/2009/05/visualizing-bayes-theorem/ [...]

  102. Hi, I got the concept cleared. Very much Simplified. I think it is the best way to explain this concept.

    Ravikanth

    30 May 12 at 2:50 am

  103. [...] http://oscarbonilla.com/2009/05/visualizing-bayes-theorem/ Share this:TwitterFacebookLike this:LikeBe the first to like this post. Posted in: programming | Tagged: machine learning, R [...]

  104. Misaki Yoshitake

    24 Jun 12 at 1:03 pm

  105. Great Post.

    Thanks for writing it up

    Fernando Nava

    9 Aug 12 at 9:04 pm

  106. Wow, that was really well explained! i finally get it :) yay! thanks a lot.

    Jess

    11 Aug 12 at 12:36 am

  107. Outstanding! Thanks!

    Jack

    29 Aug 12 at 3:16 am

  108. [...] ST101 and Probability @ Khan Academy followed or accompanied by Visualizing Bayes’ theorem and Eliezer Yudkowsky’s An Intuitive (and Short) Explanation of Bayes’ Theorem should get [...]

  109. This is definitely the best visualization of Bayes’ theorem as far!

    Fengyuan Hu

    3 Sep 12 at 6:43 am

  110. Watch this and you will understand Bayes’ theorem in less than 2.5 minutes! http://www.youtube.com/watch?v=D8VZqxcu0I0

    Luana Micallef

    8 Nov 12 at 10:24 pm

  111. Apreciado Oscar

    Tu trabajo primero lo adiviné.
    Luego lo busqué bajo el rubro
    ” Venn diagrams and Bayes’s theorem”
    y lo encontré !!!.
    Y era mejor de lo que imaginé. De hecho , en mi país, la campaña vs el cancer de mama usa extensamente el color rosa y la tuya es una hermosa explicación que aclara el procedimiento.

    Finalmente: ” prefiero hallar una buena explicación que ser el rey de los persas… ” Heráclito

    jose luis castro

    20 Nov 12 at 12:41 pm

  112. This is similar to how Sebastian Thrun talks about it in his Udacity stats class.

    Paul K

    28 Nov 12 at 3:10 pm

  113. Great job. Nw hw can we explain bayesian theorems as explaines in bayes event, and also give an individual illustrative example of real life event

    Ope

    2 Dec 12 at 9:50 am

  114. Great job! I am happy many people like it.

    Just in case you couldn’t follow, here is another approach offered by Gerd Giigerenzer in his book entitled “reckoning with risk:learning to live with uncertainty”. Penguin books. 2002.

    Without going through the details but merely using Oscar’s example:

    We take as an example a population of 100 000 women at age forty who participated in the screening. We were given the fact that 1000 among them (1% of 100 000) actually have cancer, out of whom 800 (80% of 1000) were diagnosed positive.

    We were also given the fact that out of those 99 000 women who went for the screening and who actually did not have cancer (99% of 100 000), 9 504 women were still diagnosed positive.

    Now, out of the 10 304 women who had been diagnosed positive in the screening, only 800 actually have cancer. What is the chance?

    Note: The use of 100 000 women as an example is only for convenience. You can use any number as we are working with ratios.

    Ng

    15 Dec 12 at 6:45 pm

  115. How about a Venn Pie Chart?
    A pie chart with overlapping sectors helps to understand the Bayes’ theorem.

    http://oracleaide.wordpress.com/2012/12/26/a-venn-pie/

    oracleaide

    27 Dec 12 at 4:24 pm

  116. An interactive Venn Pie chart showing relations between prior and posterior probabilities (HTML5, requires a modern browser):

    http://dl.dropbox.com/u/133074120/venn_pie.html

    oracleaide

    1 Jan 13 at 9:27 pm

  117. [...] posterior probability are completely sound). Bayes' Theorem Illustrated (My Way) – Less Wrong Visualizing Bayes’ theorem | oscarbonilla.com Reply With [...]

  118. Made a new video which visualizes things in a new way. Curious what you think:
    http://www.youtube.com/watch?v=Zxm4Xxvzohk

    Brit Cruise

    24 Jan 13 at 1:32 am

  119. Thanks so much.

    Geno

    25 Oct 13 at 8:00 am

  120. I was trying bayes theorem with venn and was tying up myself in knots. Finally, I googled and found this awesome explanation. Thank you

    prasanth

    26 Oct 13 at 9:51 pm

  121. well done, determine probability of intermittent low level noise in circuits?

    smith

    4 Nov 13 at 11:37 pm

  122. Wish this had a share button, statistics classes would love this explanation.

    prawn

    7 Dec 13 at 12:01 am

  123. Venn diagram analysis is always helpful when analysing probabilities and you have done a great job! Great explanation and thank you.

    Rudin

    26 Jan 14 at 2:18 am

  124. […] […]

  125. Very clear explanation, so helpful. Thank you.

    Mike

    26 Feb 14 at 11:49 pm

  126. DataMax

    23 Mar 14 at 4:32 pm

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