Priors in glmmTMB

Motivation

While glmmTMB is primarily designed for maximum likelihood estimation (or restricted ML), there are certain situations where it is convenient to be able to add priors for particular parameters or sets of parameters, e.g.:

to mitigate complete separation (technically, when there is some linear combination of parameters that divides zero from non-zero responses in a count or binomial model; in practice, typically when one treatment combination has all-zero responses)
to mitigate singular fits in random effects, i.e. when there is insufficient data to estimate a variance parameter so that single variances collapse to zero or covariance matrices are estimated with less than full rank (Chung et al. 2013)
to apply a ridge penalty to a set of parameters (corresponding to an independent Gaussian prior on each parameter)
more generally, to regularize models that would otherwise be numerically unstable
for models that will be used with the tmbstan package as part of a fully Bayesian analysis (see below)

See Banner et al. (2020) and Sarma and Kay (2020) for some opinions/discussion of priors.

When priors are specified, glmmTMB will fit a maximum a posteriori (MAP) estimate. In other words, unlike most Bayesian estimate procedures that use Markov chain Monte Carlo to sample the entire parameter space and compute (typically) posterior mean or median value of the parameters, glmmTMB will find the mode of the posterior distribution or the most likely value. The MAP estimate is theoretically less useful than the posterior mean or median, but is often a useful approximation.

One can apply tmbstan to a fitted glmmTMB model that specifies priors (see the MCMC vignette in order to get samples from the posterior distribution as in a more typical Bayesian analysis.

Load packages

library(glmmTMB)
library(lme4)
library(blme)
library(broom.mixed)
library(purrr)
library(dplyr)
library(ggplot2)
theme_set(theme_bw())
OkIt <- unname(palette.colors(n = 8, palette = "Okabe-Ito"))[-1]

Culcita example: near-complete separation

From Bolker (2015), an example where we can regularize nearly complete separation: see the more complete description here.

For comparison, we’ll fit (1) unpenalized/prior-free glmer and glmmTMB models; (2) blme::bglmer(), which adds a prior to a glmer model; (3) glmmTMB with priors.

We read the data and drop one observation that is identified as having an extremely large residual:

cdat <- readRDS(system.file("vignette_data", "culcita.rds", package = "glmmTMB"))
cdatx <- cdat[-20,]

Fit glmer, glmmTMB without priors, as well as a bglmer model with regularizing priors (mean 0, SD 3, expressed as a 4 \(\times\) 4 diagonal covariance matrix with diagonal elements (variances) equal to 9:

form <- predation~ttt + (1 | block)
cmod_glmer <- glmer(form, data = cdatx, family = binomial)
cmod_glmmTMB <- glmmTMB(form, data = cdatx, family = binomial)
cmod_bglmer <- bglmer(form,
  data = cdatx, family = binomial,
  fixef.prior = normal(cov = diag(9, 4))
)

## boundary (singular) fit: see help('isSingular')

Specify the same priors for glmmTMB: note that we have to specify regularizing priors for the intercept and the remaining fixed-effect priors separately

cprior <- data.frame(prior = rep("normal(0,3)", 2),
                     class = rep("fixef", 2),
                     coef = c("(Intercept)", ""))
print(cprior)

##         prior class        coef
## 1 normal(0,3) fixef (Intercept)
## 2 normal(0,3) fixef

cmod_glmmTMB_p <- update(cmod_glmmTMB, priors = cprior)

Check (approximate) equality of estimated coefficients:

## comment out for now ...
## stopifnot(all.equal(coef(summary(cmod_bglmer)),
##           coef(summary(cmod_glmmTMB_p))$cond,
##           tolerance = 5e-2))

Pack the models into a list and get the coefficients:

cmods <- ls(pattern = "cmod_[bg].*")
cmod_list <- mget(cmods) |> setNames(gsub("cmod_", "", cmods))
cres <- (purrr::map_dfr(cmod_list,
  ~ tidy(., conf.int = TRUE, effects = "fixed"),
  .id = "model"
)
|> select(model, term, estimate, lwr = conf.low, upr = conf.high)
  |> mutate(across(
    model,
    ~ factor(., levels = c(
      "glmer", "glmmTMB",
      "glmmTMB_p", "bglmer"
    ))
  ))
)
ggplot(cres, aes(x = estimate, y = term, colour = model)) +
  geom_pointrange(aes(xmin = lwr, xmax = upr),
    position = position_dodge(width = 0.5)
  ) +
  scale_colour_manual(values = OkIt)

Gopher tortoise example: mitigate singular fit

Also from Bolker (2015):

gdat <- readRDS(system.file("vignette_data", "gophertortoise.rds", package = "glmmTMB"))
form <- shells~prev + offset(log(Area)) + factor(year) + (1 | Site)
gmod_glmer <- glmer(form, family = poisson, data = gdat)

## boundary (singular) fit: see help('isSingular')

gmod_bglmer <- bglmer(form, family = poisson, data = gdat)

## boundary (singular) fit: see help('isSingular')

## cov.prior = gamma(shape = 2.5, rate = 0, common.scale = TRUE, posterior.scale = "sd"))
gmod_glmmTMB <- glmmTMB(form, family = poisson, data = gdat) ## 1e-5
## bglmer default corresponds to gamma(Inf, 2.5)
gprior <- data.frame(prior = "gamma(1e8, 2.5)",
                     class = "ranef",
                     coef = "")
gmod_glmmTMB_p <- update(gmod_glmmTMB, priors = gprior)
vc1 <- c(VarCorr(gmod_glmmTMB_p)$cond$Site)
vc2 <- c(VarCorr(gmod_bglmer)$Site)
stopifnot(all.equal(vc1, vc2, tolerance = 5e-4))

Pack the models into a list and get the coefficients:

gmods <- ls(pattern = "gmod_[bg].*")
gmod_list <- mget(gmods) |> setNames(gsub("gmod_", "", gmods))

The code for extracting CIs is currently a little bit ugly (because profile confidence intervals aren’t quite working for glmmTMB objects with broom.mixed::tidy(), and because profile CIs can be fussy in any case)

blme defaults: Wishart(dim + 2.5), or gamma(2.5). For dim = 1 (scalar), Wishart(n) corresponds to chi-squared(n), or gamma(shape = n/2, scale = n/2). Chung et al propose gamma(2, Inf); not sure why blme uses gamma(2.5) instead? or if specified via Wishart, shape = 3.5 → gamma shape of 1.75?

Zero-inflation

We can also use priors on the zero-inflation coefficients, either (1) to mitigate complete separation or (2) in the case where the zero-inflation probability goes to zero. In the latter case, even with an intercept-only zero-inflation model (ziformula = ~1), the parameter on the log-odds (link) scale goes to \(-\infty\), the log-likelihood surface becomes very flat, and we end up with non-positive-definite Hessian problems and/or failure of the Wald approximation.

We fit the classic data set on numbers of Prussian soldiers kicked to death by horses, with an unrealistically complicated model that considers zero-inflation varying by year.

data("prussian", package = "pscl")
prussian <- mutate(prussian,
                   fyear = factor(year))
zi_fit <- glmmTMB(y ~ year + (1|corp),
                  ziformula = ~fyear, family = poisson, data = prussian)
cprior <- data.frame(prior = rep("normal(0,3)",2),
                     class = rep("fixef_zi",2),
                     coef = c("(Intercept)", ""))
zi_fit_prior <- update(zi_fit, priors = cprior)

TO DO/FIX ME

try to get internal structure of priors fixed before release, otherwise up2date might get annoying …
document synonyms
why is bglmer profile CI failing (in broom.mixed, but not externally?)
figure out/document blme default priors
add tests!
document that gamma is applied on exp() scale
- move prior info to a separate man page?
implement elementwise priors
- start with specifying by number, do lookup by name later
allow multivariate (joint) priors on parameter vectors rather than iid priors?
- esp for correlation matrices: LKJ, Wishart etc. (from Mikael Jagan here)
add beta priors for zi, corr, etc. ?
- number of prior parameters (save annoying C++ code); can specify via _cor or _sd on the R side (will pick out sd-specific or cor-specific elements)
- start and end indices in vector
test!
safety checks (e.g. error at end of switch statements in C++)

Development issues

It seems useful to use the API/user interface from brms

downside: brmshas lots of downstream dependencies that glmmTMB doesn’t

might be able to copy the relevant code (the full file is 2210 lines (!), but this includes documentation and a lot of code we don’t need …

    rd <- \(x) tools::package_dependencies("brms", recursive = TRUE)[[x]]
## rd <- \(x) packrat:::recursivePackageDependencies(x, ignores = "", lib.loc =    .libPaths()[1])
## not sure why packrat and tools get different answers, but difference
## doesn't matter much
brms_dep <- rd("brms")
glmmTMB_dep <- rd("glmmTMB")
length(setdiff(brms_dep, glmmTMB_dep))

at its simplest, this is just a front-end for a data frame

## requires brms to evaluate, wanted to avoid putting it in Suggests: ...
bprior <- c(prior_string("normal(0,10)", class = "b"),
            prior(normal(1,2), class = b, coef = treat),
            prior_(~cauchy(0,2), class = ~sd,
                   group = ~subject, coef = ~Intercept))

str(bprior)

## Classes 'brmsprior' and 'data.frame':    3 obs. of  10 variables:
##  $ prior : chr  "normal(0,10)" "normal(1, 2)" "cauchy(0, 2)"
##  $ class : chr  "b" "b" "sd"
##  $ coef  : chr  "" "treat" "Intercept"
##  $ group : chr  "" "" "subject"
##  $ resp  : chr  "" "" ""
##  $ dpar  : chr  "" "" ""
##  $ nlpar : chr  "" "" ""
##  $ lb    : chr  NA NA NA
##  $ ub    : chr  NA NA NA
##  $ source: chr  "user" "user" "user"

We probably only need to pay attention to the columns prior, class, coef, group. For our purposes, prior is the name and parameters; class will be the name of the parameter vector; coef will specify an index within the vector (could be a number or name?)

TMB-side data structure:

vector of prior codes
- we need a new enum, .valid_priors: see make-enum in the Makefile
list of parameter vectors? or prior_p1, prior_p2, prior_p3 (do any prior families have more than two parameters? What about non-scalar parameters, e.g. Wishart priors … ???)
vector of parameter codes (another enum?) (beta, theta, thetaf … b ?)
each index (corresponding to coef) is scalar, either NA (prior over all elements) or integer (a specific element)
new loop after loglik loop to add (negative log-)prior components: loop over prior spec
add theta_corr, theta_sd as enum options (synonyms: ranef_corr, ranef_sd) to specify penalizing only SD vector or only corr vector from a particular element?
‘coef’ picks out elements
- fixed effect: find numeric index in colnames(X) of corresponding component
- random effect: find indices (start and stop?) in corresponding theta vector
- ranef_corr, ranef_sd: find indices … (depends on RE structure)

References

Banner, Katharine M., Kathryn M. Irvine, and Thomas J. Rodhouse. 2020. “The Use of Bayesian Priors in Ecology: The Good, the Bad and the Not Great.” Methods in Ecology and Evolution 11 (8): 882–89. https://doi.org/10.1111/2041-210X.13407.

Bolker, Benjamin M. 2015. “Linear and Generalized Linear Mixed Models.” Chap. 13 in Ecological Statistics: Contemporary Theory and Application, edited by Gordon A. Fox, Simoneta Negrete-Yankelevich, and Vinicio J. Sosa. Oxford University Press.

Chung, Yeojin, Sophia Rabe-Hesketh, Vincent Dorie, Andrew Gelman, and Jingchen Liu. 2013. “A Nondegenerate Penalized Likelihood Estimator for Variance Parameters in Multilevel Models.” Psychometrika 78 (4): 685–709. https://doi.org/10.1007/s11336-013-9328-2.

Sarma, Abhraneel, and Matthew Kay. 2020. “Prior Setting in Practice: Strategies and Rationales Used in Choosing Prior Distributions for Bayesian Analysis.” Proceedings of the 2020 CHI Conference on Human Factors in Computing Systems (New York, NY, USA), CHI ’20, April, 1–12. https://doi.org/10.1145/3313831.3376377.