The Uniform Prior and the Laplace Correction

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(x1, …, xn|) - likelihood of examples given a known value  Given examples x1, …, xn, 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 his 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 Xn+1 is independent of X1, …, Xn

7. Bayesian Prediction • Thus, we conclude that