Compute numerical estimate of Bayes factor.

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Description

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.

Usage

1
  bayesfactor (logw0, logw1)

Arguments

logw0

log-probabilities or log-importance weights under H0.

logw1

log-probabilities or log-importance weights under H1.

Value

The estimated Bayes factor.

Author(s)

Peter Carbonetto peter.carbonetto@gmail.com

References

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.

See Also

varbvs, normalizelogweights