By Jonas Kristoffer Lindeløv. See my blog and my academic profile. Created on Feb, 2018.

Here, I compare different ways of computing Bayes Factors in R. I start with a TL;DR section showing off the syntax for the simplest of all models: the intercept-only model. Then I go on to demonstrate Bayes Factors for mixed models using the same packages, including a more thorough discussion of pros and cons.

In each section, I start with the most limited methods and then progress towards the all-purpose solutions. As flexibility increases so do complexity, so find you own golden middle way.

This is just a demonstration of syntax which ignores some controversies surrounding Bayes Factors. Namely, some researchers go for parameter estimation rather than Bayes Factors because BFs are too relative: (1) Bayes factors just change credence between models without quantifying whether the better model is actually credible enough, and (2) they are relative to the prior, yet people often discuss Bayes Factors as if they are purely data-driven. A good first amelioration of these problems is to accompany your Bayes Factors with posterior predictive checks for absolute model fit and sensitivity analyses for robustness to theoretically unimortant changes in model structure and priors.

The full code for this notebook is available on GitHub.

# Intercept-only models

Here is some data:

set.seed(14)  # Cherry-picked to reveal the difference between p and BF
intercept_data = data.frame(score=scale(rnorm(40), center=0.72))

Let’s take a look at it:

A frequentist one-sample t-test shows that the results are HIGHLY SIGNIFICANT (p < 0.01) !!!!1!1!one:

x = t.test(intercept_data$score) x$p.value
## [1] 0.009799473

… but let’s compute som Bayes factors and see how this holds up. Put in Bayesian terms, we want to use this data to update our beliefs in whether it was generated either by a world in which $$intercept = 0$$ or by a world where $$intercept != 0$$. Bayes Factors will tell us how much the data changes our belief relative to our prior beliefs, so let’s go ahead and compute some Bayes Factors:

### BIC

The Bayesian Information Criterion (BIC) is like the more popular AIC but with a slightly different penalization for the number of parameters which allow for a (more) Bayesian interpretation. Wagenmakers (2007) popularized BIC-derived Bayes factors which can be computed in the comfort zone of base R.

full_lm = lm(score ~ 1, intercept_data)  # One mean with gauss residual
null_lm = lm(score ~ 0, intercept_data)  # Fixed mean at score = 0
BF_BIC = exp((BIC(null_lm) - BIC(full_lm))/2)  # From BICs to Bayes factor
BF_BIC  # Show it
## [1] 5.053437

OK, so those whose belief state was “unit information” (whatever that is) before seeing the data, you should now believe 5.05 times more in H1 and, conversely, 0.2 times “more” in H0. Although the direction of evidence is the same, the magnitude gives a different impression than the p-value. The fact that p-values are poor indices of model evidence has been noted for a long time but has yet to hit mainstream stats education. (p-values are still practical if you just want to control long-run error rates)

### BayesFactor package

The BayesFactor package is dedicated to the computation of Bayes factors, so it’s probably the solution requiring the least code. There is substantial academic literature to support it as cited in the previous link.

library(BayesFactor)
BF = ttestBF(intercept_data$score) exp(BF@bayesFactor$bf)  # Show just the (non-log) Bayes factor
## [1] 4.148151

Under the hood, this ttestBF is based on the JZS prior which is a $$Cauchy(0, 0.707)$$ prior on the standardized mean difference, standardized by a sigma with a Jeffreys prior. The width of this prior can be changed using the rscale argument, e.g. ttestBF(intercept_data$score, rscale = 0.3). ### brms::hypothesis brms can model almost all (non-)linear models, including structural equation modeling. It makes little sense to use such a huge package for a one-sample t-test, but let’s go ahead to demonstrate it anyway using a cauchy prior (but not the Jeffreys prior for simplicity) to get some similarity to the JZS prior used above by BayesFactor: library(brms) priors_intercept = c(set_prior('cauchy(0, 0.707)', class = 'Intercept')) # JZS prior as the BayesFactor package, though without the Jeffreys prior on Sigma for simplicity. full_brms = brm(score ~ 1, data = intercept_data, prior = priors_intercept, sample_prior = TRUE, iter = 10000) BF_brms_savage = hypothesis(full_brms, hypothesis = 'Intercept = 0') 1 / BF_brms_savage$hypothesis$Evid.Ratio # Inverse to make it in favor of the full model ## [1] 14.10278 brms::hypothesis fails here because it relies on the Savage-Dickey density ratio between the prior and the posterior. This fails for priors that are hard to sample (Cauchy amd others). Thanks to brms developer Paul Bürkner for this insigt. Luckily, there’s a more general solution in brms::bayesfactor: ### brms::bayes_factor We finally arrived at my favorite: comparing brm models using brms::bayes_factor. This is a recent development, made possible by the publication of the bridgesampling package in 2017. Using the priors_intercept above, the syntax is very much like lme4::lmer. It mainly differs from brms::hypothesis in that you should fit the null model independently and change some arguments: full_brms = brm(score ~ 1, data = intercept_data, prior = priors_intercept, save_all_pars = TRUE, iter=10000) null_brms = brm(score ~ 0, data = intercept_data, save_all_pars = TRUE, iter = 10000) BF_brms_bridge = bayes_factor(full_brms, null_brms) BF_brms_bridge$bf
## [1] 3.718356

I’m told that you need a lot of samples to get accurate Bayes factors using brm, so be sure to do multiple runs to ensure convergence.

### Summary of intercept-models

The computed Bayes factors against the $$intercept = 0$$ hypothesis were:

• BIC: 5.1.
• BayesFactor: 4.1
• brms::hypothesis: 14.1 (whoa!)
• brms::bayes_factor: 3.7

Three of them are in the same ballpark. While their small differences are likely due to differences in priors. bmrs::hypothesis is erroneous in this case as explained in that section above. In general, though, Savage-Dickey works well for models with more well-defined priors (normal, etc.). The BICs unit information prior is very uninformative, leading to a higher change in credence, hence typically higher BF than priors which are more informative in the region of the maximum likelihood, such as the Cauchy prior we used above.

# Mixed models

To demonstrate the syntax for more complicated models, let’s use data from an RCT which we published this spring. In brief, we include data from two treatment groups where brain-injured patients were treated using hypnotic suggestion and the effect on working memory was assessed on an index score, WMI (population mean=100, SD=15). A passive control group is excluded here because the treatment effects were very large and I want moderate Bayes factors for the present purpose.

mixed_data = read.csv('https://osf.io/42avq/download', sep = '\t')
mixed_data = subset(mixed_data, group != 'control')  # Remove passive control group

We will be interested in seeing whether the two groups develop at a different rate as a function of time with time being modeled as a single slope rather than four offsets. This would be the session * group interaction term. (For this particular research, this analysis does not reflect the hypotheses, but itøs data and we can compute Bayes factors. That’s all that matters for the present purpose. (1) we expected ceiling effects in both groups, so the data is censored, and (2) we expected a greater gain in A between test 1-2, similar A and B between test 2-3 and greater gain in B between test 3-4. So our hypothesis is not unidirectional across all data. We expanded this on page 10 of our supplementary information.)

### BIC in lme4::lmer

BIC-based Bayes factors can also be computed for more complex lme4::lmer models in a similar way. Here we do a Bayes factor on the presence of the interaction term, i.e., whether the slope of WMI to session differs by group while using the random effect for id to discard between-subject offsets and provide a bit of shrinkage (good to ameliorate regression towards the mean, among other things).

full_lmer = lme4::lmer(WMI ~ session * group + (1|id), mixed_data, REML = FALSE)
null_lmer = update(full_lmer, formula = ~ . -session:group)  # Without interaction term
#null_lmer = lme4::lmer(WMI ~ session + group + (1|id), mixed_data, REML=FALSE)  # Alternative specification
BF_BIC = exp((BIC(null_lmer) - BIC(full_lmer))/2)  # BICs to Bayes factor
BF_BIC
## [1] 0.08077421

OK, so you would now favor the null model by a factor of 12.4 more than you did previously.

Evaluation: On the positive side, this method is very easy, very fast, and works for a lot of models. For mixed models, I used this approach to get a ballpark figure until I discovered brms a few weeks ago!

There are many downsides. The BIC-derived Bayes factor uses a “unit information prior” which is a very uninformative prior that I haven’t found a way to visualize. In other words, I don’t really understand it. So BIC-based Bayes factors are far from the Bayesian ideal of updating credible beliefs. In BIC-world, you always start out knowing barely anything. Can people have an IQ of minus 100 or one million? Sure!

### BayesFactor package: many mixed effects models

The test of the interaction term is quite similar to the lme4::lmer syntax:

library(BayesFactor)
mixed_data$id = factor(mixed_data$id)  # BayesFactor wants the random to be a factor

full_BF = lmBF(WMI ~ session * group + id, data = mixed_data, whichRandom = 'id')
null_BF = lmBF(WMI ~ session + group + id, data = mixed_data, whichRandom = 'id')
full_BF / null_BF  # The Bayes factor in favor of the full model
## Bayes factor analysis
## --------------
## [1] session * group + id : 0.09486518 ±6.75%
##
## Against denominator:
##   WMI ~ session + group + id
## ---
## Bayes factor type: BFlinearModel, JZS

This value is very similar to the BIC-lmer Bayes factor for the mixed model interaction term.

Evaluation: BayesFactor is very easy to work with. The default priors in this package are the de facto default priors in research, for better or worse. Do remember to specify prior scales using rScaleFixed and rScaleRandom. Priors are specified on a normalized scale (standardized mean difference), so it does require some thought how to specify them for more intuitive untransformed effect sizes such as distance, time, weight, etc.

Although the information is hard to find, you can include multiple random intercepts like lmBF(formula, data, whichRandom = c('id', 'other_var', 'third_var')).

lmBF works for GLM but not generalIZED models like logistic regression, log-linear regression, etc. It uses a few select priors (JZS and g-priors) which have to be centered at zero. This is much better than BIC but brms and BUGS-like implementations (see below) offers greater flexibility, though (naturally) at the “cost” of more coding.

Judging from it’s development on GitHub, the development of BayesFactor has slowed down since March 2017, but it is still developed and maintained.

### Savage-Dickey using brms::hypothesis

For the hypnosis study, we would set priors for each parameter individually. To see the parameters of a model and their default priors, run

get_prior(WMI ~ session * group + (1|id), data = mixed_data)
prior class coef group
b
b groupgroupB
b session
b session:groupgroupB
student_t(3, 94, 17) Intercept
student_t(3, 0, 17) sd
sd id
sd Intercept id
student_t(3, 0, 17) sigma

These priors do not at all represent our knowledge about patients, treatments, and the design. There’s a big literature on how to set priors, and I’m not too well acquainted with it. For a truly cumulative science, you would probably try to conduct meta-analyses on all published data and use those parameter estimates as your prior. This would be “maximally informative priors” (as opposed to “uninformative priors”). In practice, many set a prior using the results from one highly similar study or just make an even vaguer “expert judgment” to save time. Personally, to save time on smaller projects, I look at the 50% and 95% credible intervals and run it by a few colleagues to get a consensus summary of the current knowledge. So for the intercept of the patient population, I would fiddle around with this, until I found the values for patient_mean and patient_sd which best summarizes (my) current knowledge. I ended up with this:

quantile_probs = c(0.05, 0.25, 0.75, 0.95)  # Quantiles for 95% and 50% intervals
patient_mean = 85  # minus one SD when population mean=100 and 1SD=15
patient_sd = 7

# Plot curve with ticks
curve(dnorm(x, patient_mean, patient_sd), from = 60, to = 110, n = 2000, xaxt = "n", yaxt = 'n', xlab = '', ylab = '', main = paste('Prior with SD = ', patient_sd))

# X-axis labels and vertical lines at quantiles
quantiles = qnorm(quantile_probs, patient_mean, patient_sd)
print(quantiles)
## [1] 73.48602 80.27857 89.72143 96.51398
axis(1, at=round(quantiles, 1))
abline(v=quantiles, lty = c(3,1,1,3), lwd = c(1,3,3,1), col = 'red')

Let’s set all priors using this kind of expert judgment (read: I’m just being lazy now) for the parameters of this model:

priors_mixed = c(
# -1SD Expected for patient group. 80% CI from 82 to 98 seems reasonable
set_prior('normal(85, 7)', class = 'Intercept'),

# none-to-moderate apriori difference between groups
set_prior('normal(0, 8)', coef = 'groupgroupB'),

# some gain expected due to retest and non-specific effects
set_prior('normal(3, 2)', coef = 'session'),

# a priori group A and B are expected to improve equally
set_prior('normal(0, 2)', coef = 'session:groupgroupB'),

# Between-subject SD at baseline around 15
set_prior('gamma(30, 2)', class = 'sd', coef = 'Intercept', group = 'id')
)

Then run brm on the full model:

full_brms = brm(WMI ~ session * group + (1|id), data = mixed_data, prior = priors_mixed, sample_prior = TRUE, iter = 10000)
BF_brms_savage = hypothesis(full_brms, hypothesis = 'session:groupgroupB = 0')  # H0: No interaction
BF_brms_savage$hypothesis # Show the results of this hypothesis test ## Estimate Est.Error l-95% CI u-95% CI ## (session:groupgroupB) = 0 -0.1486711 0.8159163 -1.747084 1.444984 ## Evid.Ratio Star ## (session:groupgroupB) = 0 2.422722 Note that Evid.Ratio is the Bayes factor in favor of the null (!) since that is the hypothesis that we stated, so the BF in favor of H1 is 0.4127589. In this case, it is less in favor of the null than when we used BIC or BayesFactor. The discrepancy is probably due to the change in priors: a more null-like prior means that we have to update our knowledge less. brms::hypothesis computes an evidence ratio (a Bayes Factor) using the Savage-Dickey method which only requires the posterior of the parameter of interest. Thus, no null model needs to be fitted explicitly. As a side note, it is fairly straightforward to manually compute Bayes factors from Savage-Dickey density ratios on the output of stan, JAGS, and other samplers. I return to this in the section on the Product Space method. Evaluation: brms provides a huge improvement over the other methods by allowing explicit specification of all priors, all link functions, meta-analysis, etc. With hypothesis, you don’t need to explicitly state a null model. However, Savage-Dickey can fail for even relatively simple models, e.g., categorical ones such as one-way ANOVAs (Morey et al., 2011) and if we had modeled session with individual offsets in a change-model (as we did in our paper). I don’t know of a simple rule about when it works and when it does not, and that makes me uncomfortable. ### brms::bayes_factor: almost all models Again, this is very much like brms::hypothesis but with the null model fitted independently: full_brms = brm(WMI ~ session * group + (1|id), data = mixed_data, prior = priors_mixed, save_all_pars = TRUE, iter = 10000) null_brms = update(full_brms, formula = ~ .-group:session) # Same but without the interaction term BF_brms_bridge = bayes_factor(full_brms, null_brms) BF_brms_bridge ## The estimated Bayes factor in favor of x1 over x2 is equal to: 0.41355 Evaluation: Learn this tool, and you can do almost everything you’d ever need, including meta-analysis, Poisson models, mixed models with crossed random effects, SEM, etc. brms is a quite young package, bearing witness of incredible development speed. Expect many more features shortly. I’m personally looking forward to seeing full support for SEM in brms and automatic “imputation” of missing values, further setting brms apart from the other solutions. On the negative side, stan compilation time is around 1 minute on my system which slows down testing of simpler models. Just building this notebook takes around 50 times longer because there are brms examples. ### Product space method in JAGS, stan, etc.: Truly all models! If you want to do strange stuff like change point analysis or Number Needed to Treat (I do both), you would have to go for specialized R packages or the true all-purpose solution: the product space method. Here, you would code the models directly in a BUGS-like language like JAGS or stan. For a conceptual and practical introduction, see Ledowyckx et al., 2011 and then check out this JAGS tutorial on ways to set up BUGS models so that Bayes Factors can be computed from the MCMC samples. Despite these ressources, I am writing a section here to present the most minimal example for clarity, and adding solutions to a practical sampling problem. The product space method is very simple: you write up the two models and and let an indicator variable select one of them at the time (which_model below). The relative rate by which each model is sampled it’s posterior odds. Multiply by the model prior and BAM! You have your Bayes Factor! I like the product space method because of its conceptual simplicity whereas other Bayes factors tend to rely on clever math which is barely accessible to me. Also, the indicator variable reveals something fundamental about (Bayesian) hypothesis testing: in typical models, the mere presence of a certain parameter expresses the 100% prior belief that this parameter exists. The model switcher simply puts a less-than-100% prior on the parameter or configuration of parameters. In other words, the model itself is “just” a prior like those we put on parameters, and they do the same thing: specify which worlds (or “events” in philosophy) could have generated data. Here is a JAGS model corresponding to a one-sample t-test where score is observed: model { which_model ~ dbern(0.5) # Selecting between two models. dcat() for more. mu ~ dnorm(0, 10^-2) score ~ dnorm(mu * which_model, 0.707^-2) # H0: mu*0 = 0. H1: mu * 1 = mu. } Since the sampling frequency represents the relative posterior probability of the two models, the Bayes factor for H1 is simply the proportion of samples where which_model == 1 multiplied by the prior: rates = xtabs(~as.matrix(mcmc_samples$which_model))
BF_productspace = prior * (rates[2] / rates[1])

Here’s how to do the same thing in pymc3. However, there are some major problems that you need to solve “manually”: The autocorrelation on the which_model parameter is usually huge, leading to small effective sample size. Also, if one model is much more likely than another (say by a factor of > 1000), the less likely model will be sampled very rarely. It typically requires a lot of samples to arrive at an accurate Bayes factor.

The latter problem can be solved by putting a prior in favor of the less likely model on which_model to counterweight the low likelihood, leading to more frequent sampling of it. But how much should the prior be in favor? The best solution I’ve found so far requires point-null hypotheses:

1. Run the full model by setting which_model ~ dbern(1) so that the null model is never sampled.
2. Calculate the Savage-Dickey Bayes factor for the parameter of interest.
3. Use the inverse of this model probability as the prior when doing the proper product space method to counterweight the difference in likelihoods: which_model ~ dbern(model_0_probability)

So if you use JAGS to get the posterior samples of the full model (samples) and the parameter posterior can be approximated as normal (using dnorm):

ps = as.matrix(samples['session'])  # posterior samples
M0 = dnorm(0, 3, 3)  # height of prior at 0
M1 = dnorm(0, mean(ps), sd(ps))  # approximation to height of posterior at 0.
BF = M0 / M1  # BF in favor of H1 for this parameter

If you test multiple parameters at once, a reasonable approximation is to multiply the Savage-Dickey Bayes factors for each of the parameters. Say you test two parameters at once:

param_names = c('session', 'age')  # Example parameter names
param_prior_means = list(session = 3, age = 0)  # Priors for means
param_prior_sds = list(session = 3, age = 0.4)  # Priors for SDs

# Loop over parameters to test and update bayes factor
for(param_name in param_names) {
ps = as.matrix(samples[param_name])  # posterior samples
M1 = dnorm(0, mean(ps), sd(ps))  # approximation to height of posterior at 0.
M0 = dnorm(0, param_prior_means[param_name], param_prior_sds[param_name])  # height of prior at 0
BF_parameter = M0 / M1  # BF in favor of H1 for this parameter
BF = BF * BF_parameter  # Update the estimate of the overall model evidence
}

I used this approach in Lindeløv et al. (2017), and the code for that analysis can be found on OSF. Another solution is doing an interactive process to home in on the prior probability that leads to equal sampling of model 0 and 1:

1. Start with equal model probabilities: which_model ~ dbern(0.5)
2. If the models are sampled very unevenly (i.e., by a factor of more than 20), run the sampling again using the inverse posterior probability as prior in the next iteration. So if model 1 was sampled 97% of the time, you set which_model ~ dbern(1 - 0.97). If the 97% estimate was correct, model 1 should now be sampled 50% of the time relative to this prior meaning that its posterior probability is indeed 97%.
3. Continue (2) until both models are sampled evenly.

Evaluation: The product space method will work for everything. Test completely different models with all sorts of fancy switch points, non-linear effects, strange priors, non-standard covariance matrices, etc. Also, learning BUGS was the event that really made me understand linear models and how non-magic they are.

However, this requires expert knowledge. It takes time to implement, change, and sample, so you feel less inclined to do model testing and sensitivity analyses, which you really should do! I only use product space in cases of emergency.

### Summary of mixed models

• lme4 and BIC: 0.081
• BayesFactor: 0.095
• brms::hypothesis: 0.413
• brms::bayes_factor: 0.414

It is nice to see consistency between the two brms approaches, now that we have well-behaving priors. The more informative priors were only possible in the brms models and this is likely the source of the difference in the magnitude of the belief updating (the Bayes factor).

# Honorable mentions

### JASP

In the above, I focused exclusively on solutions in R. However, there’s also JASP. JASP is a super-sleek graphical user interface that makes Bayes factors accessible to a wider audience and is a major improvement over SPSS even for frequentist statistics. JASP computes Bayes factors for most popular statistical models, including RM-ANOVA, regression, binomial, etc., and you can specify informative priors. JASP is a graphical interface to R and has hitherto relied on the BayesFactor package for analysis, though the JASP development team is currently working on a more comprehensive solution using BAS (see below).

### The BAS package

BAS is a relatively new package which includes many “default priors” which can be specified using a single argument. This is very convenient. While it does GLM, it doesn’t do intercept-only models and mixed models, so I haven’t demoed it in this post. But the syntax for the equivalent of the intercept-only model would be something like:

BAS::bas.lm(score ~ 1, data = intercept_data, prior = 'JZS', alpha = 0.707^2)

### Bayes Factors from parameter estimates

Many solutions have been proposed on how to compute Bayes Factors using posterior samples. I have covered the Savage-Dickey method above, which is somewhat fragile and constrained. The Bridge Sampling method of brms::bayes_factor seems like a general solution.

Once in a while, a new paper claims to have found a similar general method. It is beyond my level of expertise to evaluate the merits of these approaches. I do want to raise a warning concerning one popular method, the Harmonic Mean estimate, which has been shown to be highly unreliable.