7. Likelihood function We don’t know the parameters (for example mean μ or variance σ2) We have known data From known data, we can calculate missing parameter
8. Bayesian estimation What is Bayesian estimator? Terminology Squared error loss Absolute value loss Example
9. What is Bayesian estimator Bayesian estimator is an estimator that minimizes the expected loss (Bayes risk) of a given posterior distribution π(θ|D) over parameter θ.
10. Terminology Prior distributionπ(θ): initial beliefs about some unknown quantity Likelihood function p(x|θ): information in the data Given data D, the posterior densitywhere
11. Terminology - example Prior distribution: uniform distribution on (0,1) Likelihood function Data
12. Terminology The mean of discrete random variable: The mean of the prior distribution: The mean of the posterior distribution:
13. Terminology Bayesian estimator: True value: θ Loss function - to find a lower value that aindicate estimate is better estimate of θ Expected loss (Bayes risk):
15. Squared error loss (MSE) Other name is Minimum Squared Error (MSE) Loss function:= (true value – Bayesian estimator)2 Bayes risk: Minimize the risk by taking the 1st derivation = 0
16. The Bayes estimator of a parameter θ ̂ with respect to squared loss is the mean of the posterior density
25. What is HDR Highest Density Regions (HDR’s) are intervals containing a specified posterior probability. The figure below plots the 95% highest posterior density region. HDR
27. Pros Incorporating prior knowledge into an analysis Loss functions allow a range of outcomes rather only 2 (the null & alternative hypothesis) Present data Past data
Only 1 example to demonstrate our group presentation
Continuous distribution
ore technically, although all posterior quantities are automatically defined as integrals with respect to the posterior distribution, it may be quite difficult to provide a numerical value in practice, and, in particular, an explicit form of the posterior distribution cannot always be derived.
