- Bayes’ Theorem: Pr(W, L, p) = Pr(W, L|p) * Pr(p)
- Expected value: Pr(W, L) = E(Pr(W, L|p))

Posterior Probability and Bayes' Theorem

Bayes' Theorem describes the probability of an event given prior knowledge of a related and/or previous event.


General formula: P(A|B) = [P(B|A) * P(A)]/[P(B)]

1. P(A|B)	=	probability of A given B is true
2. P(B|A)	=	probability of B given A is true
3. P(A), P(B)	=	the independent probabilities of A and B


Formula applied to globe example:
P(p|W, L) = [P(W, L|p)*P(p)]/[P(W, L)]

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The posterior distribution is a combination of data, likelihood, parameters, and prior. It takes on the form of a conditional probability. 

Posterior Distribution

Dive into Deep Learning

E indicates the expected value of a random variable.
Expectations are averages that are commonly called marginals.

Average Probability and Expectation

Bayes' Theorem 

After learning how to use Priors in multiple regression, we may use different methods to predict the posterior probability.

Posterior Predictions

Maximum a posterior estimation incorporates the prior distribution. It considers knowledge that is not in training data.
$$\theta_{MAP} = \argmax_{\theta} p(\theta|x)  = \argmax_{\theta} logp(x|\theta) + logP(\theta) $$

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