文档介绍：Outline
Some Issues of Likelihoods and Posteriors
Approximations to Likelihoods and Posteriors
Chapter 3: Likelihoods, Posteriors and Their
Approximations
Jiangsheng Yu
School of Electronics Engineering puter Science
Peking University, Beijing 100871, China
Statistical Machine Learning, 2008
Jiangsheng Yu Likelihoods, Posteriors and Their Approximations
Outline
Some Issues of Likelihoods and Posteriors
Approximations to Likelihoods and Posteriors
Outline of topics
1 Some Issues of Likelihoods and Posteriors
Likelihood function and posterior density
Some useful distributions
Examples of likelihoods and posteriors
Specification of the prior
Fisher information and Rao-Cramer´ inequality
Noninformative priors — data-translated format and Jeffreys’ rules
Conjugate priors
Newton-Raphson algorithm for MLE
2 Approximations to Likelihoods and Posteriors
Normal-based inference
Posterior moments and marginalization: Numerical methods
Random simulation: Monte Carlo methods
Importance sampling
Rejection-acceptance algorithm
Jiangsheng Yu Likelihoods, Posteriors and Their Approximations
Outline Likelihood function and posterior density
Some Issues of Likelihoods and Posteriors Specification of the prior
Approximations to Likelihoods and Posteriors Newton-Raphson algorithm for MLE
Characteristic function
Let X be a random variable, the function
ϕ(t) = E eitX (1)
is called the characteristic function of X, where t ∈ R.
Let (X, Y)T be a random vector, the function
h i
ϕ(s, t) = E ei(sX+tY) (2)
is called the characteristic function of (X, Y)T, where
s, t ∈ R.
Characteristic function of rv X ↔ distribution of X
Jiangsheng Yu Likelihoods, Posteriors and Their Approximations
Outline Likelihood function and posterior density
Some Issues of Likelihoods and Posteriors Specification of the prior
Approximations to Likelihoods and Posteriors Newton-Raphson algorithm for MLE
Gamma function
For α> 0, we