Weighted distributions viewed in the context of model selection: A Bayesian perspective

Summary: Closed form expressions and Monte Carlo estimates for the Bayes factor are obtained for selection among weighted and unweighted models. Weighted distributions occur naturally in contexts where the probability that a particular observation enters the sample gets multiplied by some non-negative weight function. Suppose a realizationy of Y under the generalized densityf(y|ϑ) enters the investigator’s record with probability proportional tow(y,τ). Clearly, the recorded y is not an observation onY, but on the random variableY ^w , say, having pdf:<table><tbody><tr><td> $$f^w (y|\theta ,\tau ) = w(y|\tau )f(y|\theta )/E_{y|\theta } [w(y,\tau )],$$ </td></tr></tbody></table> which is called a weighted distribution. Closed form expressions for the Bayes factor are obtained for models arising from the exponential family for commonly used weight functions, and the behavior of these expressions is analyzed. Unknown weight functions are also considered. A convenient form for Monte Carlo estimation of the Bayes factor is presented, and a computational example is discussed, which uses this method to compare weighted mixture models of some aircraft data from Proschan (1963).