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)

## Arguments

fit0 An output returned from varbvs. Another output returned from varbvs. log-probabilities or log-importance weights under H0. log-probabilities or log-importance weights under H1.

## Value

The estimated Bayes factor.

## Details

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.

varbvs, normalizelogweights