Define priors for specific parameters or classes of parameters
set_prior(prior, class = "b", coef = "", group = "", resp = "",
dpar = "", nlpar = "", lb = NA, ub = NA, check = TRUE)prior(prior, ...)
prior_(prior, ...)
prior_string(prior, ...)
A character string defining a distribution in Stan language
The parameter class. Defaults to "b"
(i.e. population-level effects).
See 'Details' for other valid parameter classes.
Name of the (population- or group-level) parameter.
Grouping factor of group-level parameters.
Name of the response variable / category. Only used in multivariate models.
Name of a distributional parameter. Only used in distributional models.
Name of a non-linear parameter. Only used in non-linear models.
Lower bound for parameter restriction. Currently only allowed
for classes "b", "ar", "ma", and "arr".
Defaults to NULL, that is no restriction.
Upper bound for parameter restriction. Currently only allowed
for classes "b", "ar", "ma", and "arr".
Defaults to NULL, that is no restriction.
Logical; Indicates whether priors
should be checked for validity (as far as possible).
Defaults to TRUE. If FALSE, prior is passed
to the Stan code as is, and all other arguments are ignored.
Arguments passed to set_prior.
An object of class brmsprior to be used in the prior
argument of brm.
prior: Alias of set_prior allowing to
specify arguments as expressions without quotation marks.
prior_: Alias of set_prior allowing to specify
arguments as as one-sided formulas or wrapped in quote.
prior_string: Alias of set_prior allowing to
specify arguments as strings.
set_prior is used to define prior distributions for parameters
in brms models. The functions prior, prior_, and
prior_string are aliases of set_prior each allowing
for a different kind of argument specification.
prior allows specifying arguments as expression without
quotation marks using non-standard evaluation.
prior_ allows specifying arguments as one-sided formulas
or wrapped in quote.
prior_string allows specifying arguments as strings just
as set_prior itself.
Below, we explain its usage and list some common prior distributions for parameters. A complete overview on possible prior distributions is given in the Stan Reference Manual available at http://mc-stan.org/.
To combine multiple priors, use c(...) or the + operator
(see 'Examples'). brms does not check if the priors are written
in correct Stan language. Instead, Stan will check their
syntactical correctness when the model is parsed to C++ and
returns an error if they are not.
This, however, does not imply that priors are always meaningful if they are
accepted by Stan. Although brms trys to find common problems
(e.g., setting bounded priors on unbounded parameters), there is no guarantee
that the defined priors are reasonable for the model.
Below, we list the types of parameters in brms models,
for which the user can specify prior distributions.
1. Population-level ('fixed') effects
Every Population-level effect has its own regression parameter
represents the name of the corresponding population-level effect.
Suppose, for instance, that y is predicted by x1 and x2
(i.e., y ~ x1 + x2 in formula syntax).
Then, x1 and x2 have regression parameters
b_x1 and b_x2 respectively.
The default prior for population-level effects (including monotonic and
category specific effects) is an improper flat prior over the reals.
Other common options are normal priors or student-t priors.
If we want to have a normal prior with mean 0 and
standard deviation 5 for x1, and a unit student-t prior with 10
degrees of freedom for x2, we can specify this via
set_prior("normal(0,5)", class = "b", coef = "x1") and
set_prior("student_t(10,0,1)", class = "b", coef = "x2").
To put the same prior on all population-level effects at once,
we may write as a shortcut set_prior("<prior>", class = "b").
This also leads to faster sampling, because priors can be vectorized in this case.
Both ways of defining priors can be combined using for instance
set_prior("normal(0,2)", class = "b") and
set_prior("normal(0,10)", class = "b", coef = "x1")
at the same time. This will set a normal(0,10) prior on
the effect of x1 and a normal(0,2) prior
on all other population-level effects.
However, this will break vectorization and
may slow down the sampling procedure a bit.
In case of the default intercept parameterization
(discussed in the 'Details' section of brmsformula),
general priors on class "b" will not affect
the intercept. Instead, the intercept has its own parameter class
named "Intercept" and priors can thus be
specified via set_prior("<prior>", class = "Intercept").
Setting a prior on the intercept will not break vectorization
of the other population-level effects.
Note that technically, this prior is set on an intercept that
results when internally centering all population-level predictors
around zero to improve sampling efficiency. On this centered
intercept, specifying a prior is actually much easier and
intuitive than on the original intercept, since the former
represents the expected response value when all predictors
are at their means. To treat the intercept as an ordinary
population-level effect and avoid the centering parameterization,
use 0 + intercept on the right-hand side of the model formula.
A special shrinkage prior to be applied on population-level effects
is the horseshoe prior. See horseshoe
for details. Another shrinkage prior is the so-called lasso prior.
See lasso for details.
In non-linear models, population-level effects are defined separately
for each non-linear parameter. Accordingly, it is necessary to specify
the non-linear parameter in set_prior so that priors
we can be assigned correctly.
If, for instance, alpha is the parameter and x the predictor
for which we want to define the prior, we can write
set_prior("<prior>", coef = "x", nlpar = "alpha").
As a shortcut we can use set_prior("<prior>", nlpar = "alpha")
to set the same prior on all population-level effects of alpha at once.
If desired, population-level effects can be restricted to fall only
within a certain interval using the lb and ub arguments
of set_prior. This is often required when defining priors
that are not defined everywhere on the real line, such as uniform
or gamma priors. When defining a uniform(2,4) prior,
you should write set_prior("uniform(2,4)", lb = 2, ub = 4).
When using a prior that is defined on the positive reals only
(such as a gamma prior) set lb = 0.
In most situations, it is not useful to restrict population-level
parameters through bounded priors
(non-linear models are an important exception),
but if you really want to this is the way to go.
2. Standard deviations of group-level ('random') effects
Each group-level effect of each grouping factor has a standard deviation named
sd_<group>_<coef>. Consider, for instance, the formula
y ~ x1 + x2 + (1 + x1 | g).
We see that the intercept as well as x1 are group-level effects
nested in the grouping factor g.
The corresponding standard deviation parameters are named as
sd_g_Intercept and sd_g_x1 respectively.
These parameters are restricted to be non-negative and, by default,
have a half student-t prior with 3 degrees of freedom and a
scale parameter that depends on the standard deviation of the response
after applying the link function. Minimally, the scale parameter is 10.
This prior is used (a) to be only very weakly informative in order to influence
results as few as possible, while (b) providing at least some regularization
to considerably improve convergence and sampling efficiency.
To define a prior distribution only for standard deviations
of a specific grouping factor,
use set_prior("<prior>", class = "sd", group = "<group>").
To define a prior distribution only for a specific standard deviation
of a specific grouping factor, you may write
set_prior("<prior>", class = "sd", group = "<group>", coef = "<coef>").
Recommendations on useful prior distributions for
standard deviations are given in Gelman (2006), but note that he
is no longer recommending uniform priors, anymore.
When defining priors on group-level parameters in non-linear models,
please make sure to specify the corresponding non-linear parameter
through the nlpar argument in the same way as
for population-level effects.
3. Correlations of group-level ('random') effects
If there is more than one group-level effect per grouping factor,
the correlations between those effects have to be estimated.
The prior "lkj_corr_cholesky(eta)" or in short
"lkj(eta)" with eta > 0
is essentially the only prior for (Cholesky factors) of correlation matrices.
If eta = 1 (the default) all correlations matrices
are equally likely a priori. If eta > 1, extreme correlations
become less likely, whereas 0 < eta < 1 results in
higher probabilities for extreme correlations.
Correlation matrix parameters in brms models are named as
cor_<group>, (e.g., cor_g if g is the grouping factor).
To set the same prior on every correlation matrix,
use for instance set_prior("lkj(2)", class = "cor").
Internally, the priors are transformed to be put on the Cholesky factors
of the correlation matrices to improve efficiency and numerical stability.
The corresponding parameter class of the Cholesky factors is L,
but it is not recommended to specify priors for this parameter class directly.
4. Splines
Splines are implemented in brms using the 'random effects'
formulation as explained in gamm).
Thus, each spline has its corresponding standard deviations
modeling the variability within this term. In brms, this
parameter class is called sds and priors can
be specified via set_prior("<prior>", class = "sds",
coef = "<term label>"). The default prior is the same as
for standard deviations of group-level effects.
5. Gaussian processes
Gaussian processes as currently implemented in brms have
two parameters, the standard deviation parameter sdgp,
and characteristic length-scale parameter lscale
(see gp for more details). The default prior
of sdgp is the same as for standard deviations of
group-level effects. The default prior of lscale
is an informative inverse-gamma prior specifically tuned
to the covariates of the Gaussian process (for more details see
https://betanalpha.github.io/assets/case_studies/gp_part3/part3.html).
This tuned prior may be overly informative in some cases, so please
consider other priors as well to make sure inference is
robust to the prior specification. If tuning fails, a half-normal prior
is used instead.
6. Autocorrelation parameters
The autocorrelation parameters currently implemented are named
ar (autoregression), ma (moving average),
arr (autoregression of the response), car
(spatial conditional autoregression), as well as lagsar
and errorsar (Spatial simultaneous autoregression).
Priors can be defined by set_prior("<prior>", class = "ar")
for ar and similar for other autocorrelation parameters.
By default, ar and ma are bounded between -1
and 1, car, lagsar, and errorsar are
bounded between 0, and 1, and arr is unbounded
(you may change this by using the arguments lb and ub).
The default prior is flat over the definition area.
7. Distance parameters of monotonic effects
As explained in the details section of brm,
monotonic effects make use of a special parameter vector to
estimate the 'normalized distances' between consecutive predictor
categories. This is realized in Stan using the simplex
parameter type. This class is named "simo" (short for
simplex monotonic) in brms.
The only valid prior for simplex parameters is the
dirichlet prior, which accepts a vector of length K - 1
(K = number of predictor categories) as input defining the
'concentration' of the distribution. Explaining the dirichlet prior
is beyond the scope of this documentation, but we want to describe
how to define this prior syntactically correct.
If a predictor x with K categories is modeled as monotonic,
we can define a prior on its corresponding simplex via
prior(dirichlet(<vector>), class = simo, coef = mox1).
The 1 in the end of coef indicates that this is the first
simplex in this term. If interactions between multiple monotonic
variables are modeled, multiple simplexes per term are required.
For <vector>, we can put in any R expression
defining a vector of length K - 1. The default is a uniform
prior (i.e. <vector> = rep(1, K-1)) over all simplexes
of the respective dimension.
8. Parameters for specific families
Some families need additional parameters to be estimated.
Families gaussian, student, skew_normal,
lognormal, and gen_extreme_value need the parameter
sigma to account for the residual standard deviation.
By default, sigma has a half student-t prior that scales
in the same way as the group-level standard deviations.
Further, family student needs the parameter
nu representing the degrees of freedom of students-t distribution.
By default, nu has prior "gamma(2, 0.1)"
and a fixed lower bound of 1.
Families gamma, weibull, inverse.gaussian, and
negbinomial need a shape parameter that has a
"gamma(0.01, 0.01)" prior by default.
For families cumulative, cratio, sratio,
and acat, and only if threshold = "equidistant",
the parameter delta is used to model the distance between
two adjacent thresholds.
By default, delta has an improper flat prior over the reals.
The von_mises family needs the parameter kappa, representing
the concentration parameter. By default, kappa has prior
"gamma(2, 0.01)".
Every family specific parameter has its own prior class, so that
set_prior("<prior>", class = "<parameter>") is the right way to go.
All of these priors are chosen to be weakly informative,
having only minimal influence on the estimations,
while improving convergence and sampling efficiency.
Often, it may not be immediately clear,
which parameters are present in the model.
To get a full list of parameters and parameter classes for which
priors can be specified (depending on the model)
use function get_prior.
Gelman A. (2006). Prior distributions for variance parameters in hierarchical models. Bayesian analysis, 1(3), 515 -- 534.
# NOT RUN {
## use alias functions
(prior1 <- prior(cauchy(0, 1), class = sd))
(prior2 <- prior_(~cauchy(0, 1), class = ~sd))
(prior3 <- prior_string("cauchy(0, 1)", class = "sd"))
identical(prior1, prior2)
identical(prior1, prior3)
## check which parameters can have priors
get_prior(rating ~ treat + period + carry + (1|subject),
data = inhaler, family = cumulative())
## define some priors
prior <- 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))
## verify that the priors indeed found their way into Stan's model code
make_stancode(rating ~ treat + period + carry + (1|subject),
data = inhaler, family = cumulative(),
prior = prior)
## use the horseshoe prior to model sparsity in population-level effects
make_stancode(count ~ zAge + zBase * Trt,
data = epilepsy, family = poisson(),
prior = set_prior("horseshoe(3)"))
## alternatively use the lasso prior
make_stancode(count ~ zAge + zBase * Trt,
data = epilepsy, family = poisson(),
prior = set_prior("lasso(1)"))
## pass priors to Stan without checking
prior <- prior_string("target += normal_lpdf(b[1] | 0, 1)", check = FALSE)
make_stancode(count ~ Trt, data = epilepsy, prior = prior)
# }
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