Using Markov Chains to Estimate Marginal Distributions and Bayes factor

貝氏因子(Bayes factor) 是貝氏統計方法中對於推論假設檢定(hypothesis testing)所用的一個 統計量。它的定義可以表成兩個資料的邊際機率(marginal probability of data)的比值。然而， 貝氏因子的計算在實際應用上卻常常遇到困難。在過去要解決這個問題常使用Laplace method ；現在，拜電腦(軟硬體)快速發展所賜，則常使用Markov chain Monte Carlo (MCMC) 的方法。不過，直接運用MCMC 的方法只能提供我們有興趣之隨機變數(parameter of interest) 的事後樣本(posterior sample)，並不能直接得到貝氏因子的值。解決方法一是善加運用事後 機率樣本，直接估計在每一個模式下資料的邊際機率。簡單的說，我希望能夠找到事後樣 本中的某個值(如mode 或mean )，以及在該點的機率密度函數值，然後再估計資料的邊際 機率與貝氏因子。這個方法只是眾多貝氏因子估計值中的一個；但是，它提供另外一個簡 單好算的選擇。Bayes factor is a commonly used quantity in Bayesian testing hypotheses. Its definition is the ratio of posterior odds to prior odds in favor of the null hypothesis. This quantity can be rewritten, using Bayes’ theorem, as a ratio of marginal probabilities. Its computation is often complicated and difficult in practice, especially for complex hierarchical models such as random effect model. When the analytical integration is not feasible, one can use either the asymptotic Laplace approximation, or the numerical method. The Markov chain Monte Carlo (MCMC) method is one efficient way which has been used widely among statisticians. The first goal of the current research plan is to develop a numerical method using Markov chains. I intend to use the posterior samples, assumed coming easily from established software like BUGS, to estimate the mean (or the mode) and the value of the probability density function at that point. This approach was first mentioned in Raftery (1996) but has not yet been investigated. It would require the technique from nonparametric density estimate. The second goal of this research plan is to increase the feasibility of the pseudo-prior for more general cases