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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The posterior distribution is the conditional probability distribution of a parameter given observed data. It combines the prior distribution and the likelihood via Bayes' theorem: $$P(\theta \mid D) = \frac{P(D \mid \theta)\,P(\theta)}{P(D)}$$ where $$P(\theta)$$ is the prior, $$P(D \mid \theta)$$ is the likelihood, and $$P(D)$$ is the average (marginal) probability of the data across all parameter values.

Posterior Distribution

Goodfellow, I., Bengio, Y., & Courville, A. (2016). $\mathit{Deep \ Learning.}$ MIT Press. Retrieved from [www.deeplearningbook.org](https://www.deeplearningbook.org) 

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 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)]

Bayes' Theorem 

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

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