1. The Uniform Prior and the Laplace Correction Supplemental Material not on exam

2. Bayesian Inference We start with P(  ) - prior distribution about the values of  P(x 1, …, x n |  ) - likelihood of examples given a known value  Given examples x 1, …, x n, we can compute posterior distribution on  Where the marginal likelihood is

3. Binomial Distribution: Laplace Est. In this case the unknown parameter is  = P(H) Simplest prior P(  ) = 1 for 0<  <1 Likelihood where h is number of heads in the sequence Marginal Likelihood:

4. Marginal Likelihood Using integration by parts we have: Multiply both side by n choose h, we have

5. Marginal Likelihood - Cont The recursion terminates when h = n Thus We conclude that the posterior is

6. Bayesian Prediction How do we predict using the posterior? We can think of this as computing the probability of the next element in the sequence – Assumption: if we know , the probability of X n+1 is independent of X 1, …, X n

7. Bayesian Prediction Thus, we conclude that