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Bayes Factors from Marginal Likelihoods

Source: R/bridgesampling.R
bayes_factor.brmsfit.Rd

Compute Bayes factors from marginal likelihoods.

Usage

# S3 method for class 'brmsfit'
bayes_factor(x1, x2, log = FALSE, ...)

Arguments

x1

A brmsfit object

x2

Another brmsfit object based on the same responses.

log

Report Bayes factors on the log-scale?

...

Additional arguments passed to bridge_sampler.

Details

Computing the marginal likelihood requires samples of all variables defined in Stan's parameters block to be saved. Otherwise bayes_factor cannot be computed. Thus, please set save_all_pars = TRUE in the call to brm, if you are planning to apply bayes_factor to your models.

The computation of Bayes factors based on bridge sampling requires a lot more posterior samples than usual. A good conservative rule of thumb is perhaps 10-fold more samples (read: the default of 4000 samples may not be enough in many cases). If not enough posterior samples are provided, the bridge sampling algorithm tends to be unstable, leading to considerably different results each time it is run. We thus recommend running bayes_factor multiple times to check the stability of the results.

More details are provided under bridgesampling::bayes_factor.

See also

bridge_sampler, post_prob

Examples

# \dontrun{
# model with the treatment effect
fit1 <- brm(
  count ~ zAge + zBase + Trt,
  data = epilepsy, family = negbinomial(),
  prior = prior(normal(0, 1), class = b),
  save_all_pars = TRUE
)
#> Warning: Argument 'save_all_pars' is deprecated. Please use argument 'all' in function 'save_pars()' instead.
#> Compiling Stan program...
#> Start sampling
#> 
#> SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 1).
#> Chain 1: 
#> Chain 1: Gradient evaluation took 2.8e-05 seconds
#> Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.28 seconds.
#> Chain 1: Adjust your expectations accordingly!
#> Chain 1: 
#> Chain 1: 
#> Chain 1: Iteration:    1 / 2000 [  0%]  (Warmup)
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#> Chain 1: Iteration: 2000 / 2000 [100%]  (Sampling)
#> Chain 1: 
#> Chain 1:  Elapsed Time: 0.112 seconds (Warm-up)
#> Chain 1:                0.12 seconds (Sampling)
#> Chain 1:                0.232 seconds (Total)
#> Chain 1: 
#> 
#> SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 2).
#> Chain 2: 
#> Chain 2: Gradient evaluation took 2.3e-05 seconds
#> Chain 2: 1000 transitions using 10 leapfrog steps per transition would take 0.23 seconds.
#> Chain 2: Adjust your expectations accordingly!
#> Chain 2: 
#> Chain 2: 
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#> Chain 2: Iteration: 2000 / 2000 [100%]  (Sampling)
#> Chain 2: 
#> Chain 2:  Elapsed Time: 0.112 seconds (Warm-up)
#> Chain 2:                0.127 seconds (Sampling)
#> Chain 2:                0.239 seconds (Total)
#> Chain 2: 
#> 
#> SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 3).
#> Chain 3: 
#> Chain 3: Gradient evaluation took 2.2e-05 seconds
#> Chain 3: 1000 transitions using 10 leapfrog steps per transition would take 0.22 seconds.
#> Chain 3: Adjust your expectations accordingly!
#> Chain 3: 
#> Chain 3: 
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#> Chain 3: 
#> Chain 3:  Elapsed Time: 0.114 seconds (Warm-up)
#> Chain 3:                0.115 seconds (Sampling)
#> Chain 3:                0.229 seconds (Total)
#> Chain 3: 
#> 
#> SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 4).
#> Chain 4: 
#> Chain 4: Gradient evaluation took 2.7e-05 seconds
#> Chain 4: 1000 transitions using 10 leapfrog steps per transition would take 0.27 seconds.
#> Chain 4: Adjust your expectations accordingly!
#> Chain 4: 
#> Chain 4: 
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#> Chain 4: 
#> Chain 4:  Elapsed Time: 0.112 seconds (Warm-up)
#> Chain 4:                0.102 seconds (Sampling)
#> Chain 4:                0.214 seconds (Total)
#> Chain 4: 
summary(fit1)
#>  Family: negbinomial 
#>   Links: mu = log 
#> Formula: count ~ zAge + zBase + Trt 
#>    Data: epilepsy (Number of observations: 236) 
#>   Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
#>          total post-warmup draws = 4000
#> 
#> Regression Coefficients:
#>           Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
#> Intercept     1.89      0.07     1.75     2.04 1.00     4970     2997
#> zAge          0.11      0.05     0.01     0.22 1.00     4770     3041
#> zBase         0.72      0.05     0.61     0.83 1.00     4868     2958
#> Trt1         -0.19      0.10    -0.39     0.01 1.00     4595     3228
#> 
#> Further Distributional Parameters:
#>       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
#> shape     2.35      0.32     1.80     3.04 1.00     4620     3111
#> 
#> Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
#> and Tail_ESS are effective sample size measures, and Rhat is the potential
#> scale reduction factor on split chains (at convergence, Rhat = 1).

# model without the treatment effect
fit2 <- brm(
  count ~ zAge + zBase,
  data = epilepsy, family = negbinomial(),
  prior = prior(normal(0, 1), class = b),
  save_all_pars = TRUE
)
#> Warning: Argument 'save_all_pars' is deprecated. Please use argument 'all' in function 'save_pars()' instead.
#> Compiling Stan program...
#> Start sampling
#> 
#> SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 1).
#> Chain 1: 
#> Chain 1: Gradient evaluation took 2.8e-05 seconds
#> Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.28 seconds.
#> Chain 1: Adjust your expectations accordingly!
#> Chain 1: 
#> Chain 1: 
#> Chain 1: Iteration:    1 / 2000 [  0%]  (Warmup)
#> Chain 1: Iteration:  200 / 2000 [ 10%]  (Warmup)
#> Chain 1: Iteration:  400 / 2000 [ 20%]  (Warmup)
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#> Chain 1: Iteration: 1001 / 2000 [ 50%]  (Sampling)
#> Chain 1: Iteration: 1200 / 2000 [ 60%]  (Sampling)
#> Chain 1: Iteration: 1400 / 2000 [ 70%]  (Sampling)
#> Chain 1: Iteration: 1600 / 2000 [ 80%]  (Sampling)
#> Chain 1: Iteration: 1800 / 2000 [ 90%]  (Sampling)
#> Chain 1: Iteration: 2000 / 2000 [100%]  (Sampling)
#> Chain 1: 
#> Chain 1:  Elapsed Time: 0.105 seconds (Warm-up)
#> Chain 1:                0.093 seconds (Sampling)
#> Chain 1:                0.198 seconds (Total)
#> Chain 1: 
#> 
#> SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 2).
#> Chain 2: 
#> Chain 2: Gradient evaluation took 2e-05 seconds
#> Chain 2: 1000 transitions using 10 leapfrog steps per transition would take 0.2 seconds.
#> Chain 2: Adjust your expectations accordingly!
#> Chain 2: 
#> Chain 2: 
#> Chain 2: Iteration:    1 / 2000 [  0%]  (Warmup)
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#> Chain 2: Iteration: 2000 / 2000 [100%]  (Sampling)
#> Chain 2: 
#> Chain 2:  Elapsed Time: 0.103 seconds (Warm-up)
#> Chain 2:                0.108 seconds (Sampling)
#> Chain 2:                0.211 seconds (Total)
#> Chain 2: 
#> 
#> SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 3).
#> Chain 3: 
#> Chain 3: Gradient evaluation took 2.2e-05 seconds
#> Chain 3: 1000 transitions using 10 leapfrog steps per transition would take 0.22 seconds.
#> Chain 3: Adjust your expectations accordingly!
#> Chain 3: 
#> Chain 3: 
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#> Chain 3: 
#> Chain 3:  Elapsed Time: 0.106 seconds (Warm-up)
#> Chain 3:                0.102 seconds (Sampling)
#> Chain 3:                0.208 seconds (Total)
#> Chain 3: 
#> 
#> SAMPLING FOR MODEL 'anon_model' NOW (CHAIN 4).
#> Chain 4: 
#> Chain 4: Gradient evaluation took 2.7e-05 seconds
#> Chain 4: 1000 transitions using 10 leapfrog steps per transition would take 0.27 seconds.
#> Chain 4: Adjust your expectations accordingly!
#> Chain 4: 
#> Chain 4: 
#> Chain 4: Iteration:    1 / 2000 [  0%]  (Warmup)
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#> Chain 4: Iteration: 2000 / 2000 [100%]  (Sampling)
#> Chain 4: 
#> Chain 4:  Elapsed Time: 0.104 seconds (Warm-up)
#> Chain 4:                0.107 seconds (Sampling)
#> Chain 4:                0.211 seconds (Total)
#> Chain 4: 
summary(fit2)
#>  Family: negbinomial 
#>   Links: mu = log 
#> Formula: count ~ zAge + zBase 
#>    Data: epilepsy (Number of observations: 236) 
#>   Draws: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
#>          total post-warmup draws = 4000
#> 
#> Regression Coefficients:
#>           Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
#> Intercept     1.80      0.05     1.70     1.90 1.00     4117     2942
#> zAge          0.13      0.05     0.02     0.23 1.00     4044     2651
#> zBase         0.72      0.06     0.61     0.83 1.00     3983     2918
#> 
#> Further Distributional Parameters:
#>       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
#> shape     2.33      0.31     1.78     2.98 1.00     4171     3269
#> 
#> Draws were sampled using sampling(NUTS). For each parameter, Bulk_ESS
#> and Tail_ESS are effective sample size measures, and Rhat is the potential
#> scale reduction factor on split chains (at convergence, Rhat = 1).

# compute the bayes factor
bayes_factor(fit1, fit2)
#> Iteration: 1
#> Iteration: 2
#> Iteration: 3
#> Iteration: 4
#> Iteration: 1
#> Iteration: 2
#> Iteration: 3
#> Iteration: 4
#> Iteration: 5
#> Estimated Bayes factor in favor of fit1 over fit2: 0.54078
# }