Bayesian Approach Jake Blanchard Fall 2010

Introduction This is a methodology for combining observed data with expert judgment Treats all parameters are random variables

Discrete Case Suppose parameter  i has k discrete values Also, let p i represent the prior relative likelihoods (in a pmf) (based on old information) If we get new data, we want to modify the pmf to take it into account (systematically)

Terminology p i =P(  =  i )=prior relative likelihoods (data available prior to experiment providing  )  =observed outcome P(  =  i |  )=posterior probability of  =  I (after incorporating  ) P´(  =  i )=prior probability P´´ (  =  i )=posterior probability Estimator of parameter  is given by

Useful formulas

Example Variable is proportion of defective concrete piles Engineer estimates that probabilities are: Defective FractionProbability

Prior PMF

Find Posterior Probabilities Engineer orders one additional pile and it is defective Probabilities must be updated

Posterior PMF

What if next sample had been good? Switch to p representing good (rather than defective) “Good” FractionProbability

Find Posterior Probabilities Engineer orders one additional pile and it is good Probabilities must be updated

Continuous Case Prior pdf=f´(  )

Example Defective piles Assume uniform distribution Then, single inspection identifies defective pile

Solution

Sampling Suppose we have a population with a prior standard deviation (  ´) and mean (  ´) Assume we then sample to get sample mean (x)and standard deviation (  )

With Prior Information Weighted average of prior mean and sample mean