The Bayes factor is the ratio of the marginal likelihoods
under two different models (see Kass & Raftery, 1995). Function
varbvsbf provides a convenient interface for computing the
Bayes factor comparing the fit of two different varbvs models.
varbvsbf (fit0, fit1) bayesfactor (logw0, logw1)
| fit0 | An output returned from |
|---|---|
| fit1 | Another output returned from |
| logw0 | log-probabilities or log-importance weights under H0. |
| logw1 | log-probabilities or log-importance weights under H1. |
The estimated Bayes factor.
Computes numerical estimate of $$ BF = Pr(data | H1) / Pr(data | H0), $$ the probability of the data given the "alternative" hypothesis (H1) over the probability of the data given the "null" hypothesis (H0). This is also known as a Bayes factor (see Kass & Raftery, 1995). Here we assume that although these probabilities cannot be computed analytically because they involve intractable integrals, we can obtain reasonable estimates of these probabilities with a simple numerical approximation over some latent variable assuming the prior over this latent variable is uniform. The inputs are the log-probabilities $$ Pr(data, Z0 | H0) = Pr(data | Z0, H0) x Pr(Z0 | H0), Pr(data, Z1 | H1) = Pr(data | Z1, H1) x Pr(Z1 | H1), $$ where Pr(Z0 | H0) and Pr(Z1 | H1) are uniform over all Z0 and Z1.
Alternatively, this function can be viewed as computing an importance sampling estimate of the Bayes factor; see, for example, R. M. Neal, "Annealed importance sampling", Statistics and Computing, 2001. This formulation described above is a special case of importance sampling when the settings of the latent variable Z0 and A1 are drawn from the same (uniform) distribution as the prior, Pr(Z0 | H0) and Pr(Z1 | H1), respectively.
P. Carbonetto and M. Stephens (2012). Scalable variational inference for Bayesian variable selection in regression, and its accuracy in genetic association studies. Bayesian Analysis 7, 73--108.
R. E. Kass and A. E. Raftery (1995). Bayes Factors. Journal of the American Statistical Association 90, 773--795.
R. M. Neal (2001). Annealed importance sampling. Statistics and Computing 11, 125--139.