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User:Thepigdog/Bayes' theorem

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Bayesian Inference

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So far we have a theory for finding the most likely model, but not how likely the model is. Bayes' theorem tells us how to calculate the probability of a model.

Bayes' theorem is about conditional probabilities so I will explain these first. Probability is about sets of outcomes. We start by assuming that these outcomes are equally likely. Suppose we have a bag full of balls, each ball is either red or blue. Each ball is also either Small or Big. Taking a ball from the bag is an outcome.


Red Blue Total
Small
20
40
 60
Big
10
30
 40
Total
30
70
100

The conditional probability of a ball taken from the bag being Red if we already know it is Big is 10/40. This is written,

These are conditional probabilities. means,

 First I found that the ball was Big.  What then is the probability of it being red.

The probabilities for a a ball being red is,

Note that has no meaning by itself. Instead probability has two sets,

  • The set of events that register success.
  • The domain from which those events are taken.

Note that,

The probabilities for a a ball being Big is,

Now the probability of a ball being Red and Big is,

or,

so,

similarly,

so the result is,

This is Bayes' theorm usually written as,

it is also true that,

so,