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

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Presentation on theme: "The Uniform Prior and the Laplace Correction Supplemental Material not on exam."— Presentation transcript:

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


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