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Bayesian Inference – An introduction with an example

November 26, 2007 Posted by Emre S. Tasci

Suppose that Jo (of Example 2.3 in MacKay’s previously mentioned book), decided to take a test to see whether she’s pregnant or not. She applies a test that is 95% reliable, that is if she’s indeed pregnant, than there is a 5% chance that the test will result otherwise and if she’s indeed NOT pregnant, the test will tell her that she’s pregnant again by a 5% chance (The other two options are test concluding as positive on pregnancy when she’s indeed pregnant by 95% of all the time, and test reports negative on pregnancy when she’s actually not pregnant by again 95% of all the time). Suppose that she is applying contraceptive drug with a 1% fail ratio.

Now comes the question: Jo takes the test and the test says that she’s pregnant. Now what is the probability that she’s indeed pregnant?

I would definitely not write down this example if the answer was 95% percent as you may or may not have guessed but, it really is tempting to guess the probability as 95% the first time.

The solution (as given in the aforementioned book) is:

Formula: % MathType!MTEF!2!1!+-
% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn
% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr
% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9
% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x
% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaabbeaacaWGqb
% GaaiikaiaadshacaWGLbGaam4CaiaadshacaGG6aGaaGymaiaacYha
% caWGWbGaamOCaiaadwgacaWGNbGaaiOoaiaaigdacaGGPaGaeyypa0
% ZaaSaaaeaacaWGqbGaaiikaiaadchacaWGYbGaamyzaiaadEgacaGG
% 6aGaaGymaiaacYhacaWG0bGaamyzaiaadohacaWG0bGaaiOoaiaaig
% dacaGGPaGaamiuaiaacIcacaWGWbGaamOCaiaadwgacaWGNbGaaiOo
% aiaaigdacaGGPaaabaGaamiuaiaacIcacaWG0bGaamyzaiaadohaca
% WG0bGaaiOoaiaaigdacaGGPaaaaaqaaiabg2da9maalaaabaGaamiu
% aiaacIcacaWGWbGaamOCaiaadwgacaWGNbGaaiOoaiaaigdacaGG8b
% GaamiDaiaadwgacaWGZbGaamiDaiaacQdacaaIXaGaaiykaiaadcfa
% caGGOaGaamiCaiaadkhacaWGLbGaam4zaiaacQdacaaIXaGaaiykaa
% qaamaaqafabaGaamiuaiaacIcacaWG0bGaamyzaiaadohacaWG0bGa
% aiOoaiaaigdacaGG8bGaamiCaiaadkhacaWGLbGaam4zaiaacQdaca
% WGPbGaaiykaiaadcfacaGGOaGaamiCaiaadkhacaWGLbGaam4zaiaa
% cQdacaWGPbGaaiykaaWcbaGaamyAaaqab0GaeyyeIuoaaaaakeaacq
% GH9aqpdaWcaaqaaiaaicdacaGGUaGaaGyoaiaaiwdacqGHxdaTcaaI
% WaGaaiOlaiaaicdacaaIXaaabaGaaGimaiaac6cacaaI5aGaaGynai
% abgEna0kaaicdacaGGUaGaaGimaiaaigdacqGHRaWkcaaIWaGaaiOl
% aiaaicdacaaI1aGaey41aqRaaGimaiaac6cacaaI5aGaaGyoaaaaae
% aacqGH9aqpcaaIWaGaaiOlaiaaigdacaaI2aaaaaa!ADD3!
 P(test:1|preg:1) = \frac{{P(preg:1|test:1)P(preg:1)}}
{{P(test:1)}} \\ 
 = \frac{{P(preg:1|test:1)P(preg:1)}}
{{\sum\limits_i {P(test:1|preg:i)P(preg:i)} }} \\ 
 = \frac{{0.95 \times 0.01}}
{{0.95 \times 0.01 + 0.05 \times 0.99}} \\ 
 = 0.16 \\ 

 where P(b:bj|a:ai) represents the probability of b having the value bj given that a=ai. So Jo has P(test:1|preg=1) = 16% meaning that given that Jo is actually pregnant, the test would give the positive result by a probability of 16%. So we took into account both the test’s and the contra-ceptive’s reliabilities. If this doesn’t make sense and you still want to stick with the somehow more logical looking 95%, think the same example but this time starring John, Jo’s humorous boyfriend who as a joke applied the test and came with a positive result on pregnancy. Now, do you still say that John is 95% pregnant? I guess not  Just plug in 0 for P(preg:1) to the equation above and enjoy the outcoming likelihood of John being non-pregnant equaling to 0…

The thing to keep in mind is the probability of a being some value ai when b is bj is not equal to the probability of b being b when a is equal to ai.

One Response to “Bayesian Inference – An introduction with an example”

  1. Hex, Bugs and More Physics | Emre S. Tasci » Blog Archive » Bayesian Probabilities & Inference - Another example Says:

    […] John, Jo’s funny boyfriend in the previous bayesian example, used in context to show the unlikeliness of the "presumed" logical derivation, MacKay […]

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