Applied Bayesian Modelling - Free

INTRODUCTION 3distributed over all real values might be a Normal with mean zero and large variance.To adequately reflect prior ignorance while avoiding impropriety, Spiegelhalter et al.(1996) suggesting a prior standard deviation at least an order of magnitude greater thanthe posterior standard deviation.1.1.2 Posterior density vs. likelihoodIn classical approaches such as maximum likelihood, inference is based on thelikelihood of the data alone. In Bayesian models, the likelihood of the observed datay given parameters u, denoted f ( yju) or equivalently L(ujy), is used to modify theprior beliefs p(u), with the updated knowledge summarised in a posterior density,p(ujy). The relationship between these densities follows from standard probabilityequations. Thusf ( y, u) ˆ f ( yju)p(u) ˆ p(ujy)m( y)and therefore the posterior density can be writtenp(ujy) ˆ f ( yju)p(u)=m( y)The denominator m( y) is known as the marginal likelihood of the data and found byintegrating (or `marginalising') the likelihood over the prior densities…m( y) ˆ f ( yju)p(u)duThis quantity plays a central role in some approaches to Bayesian model choice, but forthe present purpose can be seen as a proportionality factor, so thatp(ujy) / f ( yju)p(u) (1:1)Thus, updated beliefs are a function of prior knowledge and the sample data evidence.From the Bayesian perspective the likelihood is viewed as a function of u given fixeddata y, and so elements in the likelihood that are not functions of u become part of theproportionality in Equation (1.1).1.1.3 PredictionsThe principle of updating extends to future values or predictions of `new data'.Before the study a prediction would be based on random draws from the priordensity of parameters and is likely to have little precision. Part of the goal of the anew study is to use the data as a basis for making improved predictions `out ofsample'. Thus, in a meta-analysis of mortality odds ratios (for a new as againstconventional therapy) it may be useful to assess the likely odds ratio z in ahypothetical future study on the basis of the observed study findings. Such aprediction is based is based on the likelihood of z averaged over the posterior densitybased on y:…f (zjy) ˆ f (zju)p(ujy)duwhere the likelihood of z, namely f (zju) usually takes the same form as adopted for theobservations themselves.