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.

1 | ```
bayesfactor (logw0, logw1)
``` |

`logw0` |
log-probabilities or log-importance weights under H0. |

`logw1` |
log-probabilities or log-importance weights under H1. |

The estimated Bayes factor.

Peter Carbonetto peter.carbonetto@gmail.com

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.

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