set_prior | R Documentation |
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, ...)
empty_prior()
prior |
A character string defining a distribution in Stan language |
class |
The parameter class. Defaults to |
coef |
Name of the coefficient within the parameter class. |
group |
Grouping factor for group-level parameters. |
resp |
Name of the response variable. Only used in multivariate models. |
dpar |
Name of a distributional parameter. Only used in distributional models. |
nlpar |
Name of a non-linear parameter. Only used in non-linear models. |
lb |
Lower bound for parameter restriction. Currently only allowed
for classes |
ub |
Upper bound for parameter restriction. Currently only allowed
for classes |
check |
Logical; Indicates whether priors
should be checked for validity (as far as possible).
Defaults to |
... |
Arguments passed to |
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 https://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.
Below, we provide details for the individual parameter classes that you can
set priors on. 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
.
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.
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.
The same goes for specifying priors for specific distributional
parameters in the context of distributional regression, for example,
set_prior("<prior>", coef = "x", dpar = "sigma")
.
For most other parameter classes (see below), you need to indicate
non-linear and distributional parameters in the same way as shown here.
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. 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 2.5.
This prior is used (a) to be only 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>")
.
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. Smoothing Splines
Smoothing 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), sderr
(standard
deviation of latent residuals in latent ARMA models), cosy
(compound
symmetry correlation), 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
;
cosy
, car
, lagsar
, and errorsar
are bounded
between 0
and 1
. The default priors are flat over the
respective definition areas.
7. Parameters of measurement error terms
Latent variables induced via measurement error me
terms
require both mean and standard deviation parameters, whose prior classes
are named "meanme"
and "sdme"
, respectively. If multiple
latent variables are induced this way, their correlation matrix will
be modeled as well and corresponding priors can be specified via the
"corme"
class. All of the above parameters have flat priors over
their respective definition spaces by default.
8. 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.
9. 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.
10. Shrinkage priors
To reduce the danger of overfitting in models with many predictor terms fit
on comparably sparse data, brms supports special shrinkage priors, namely
the (regularized) horseshoe
and the R2D2
prior.
These priors can be applied on many parameter classes, either directly on
the coefficient classes (e.g., class b
), if directly setting priors
on them is supported, or on the corresponding standard deviation
hyperparameters (e.g., class sd
) otherwise. Currently, the following
classes support shrinkage priors: b
(overall regression
coefficients), sds
(SDs of smoothing splines), sdgp
(SDs of
Gaussian processes), ar
(autoregressive coefficients), ma
(moving average coefficients), sderr
(SD of latent residuals),
sdcar
(SD of spatial CAR structures), sd
(SD of varying
coefficients).
11. Fixing parameters to constants
Fixing parameters to constants is possible by using the constant
function, for example, constant(1)
to fix a parameter to 1.
Broadcasting to vectors and matrices is done automatically.
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.
empty_prior()
: Create an empty brmsprior
object.
get_prior
## 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
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))
# 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 = bprior)
# use the horseshoe prior to model sparsity in regression coefficients
make_stancode(count ~ zAge + zBase * Trt,
data = epilepsy, family = poisson(),
prior = set_prior("horseshoe(3)"))
# fix certain priors to constants
bprior <- prior(constant(1), class = "b") +
prior(constant(2), class = "b", coef = "zBase") +
prior(constant(0.5), class = "sd")
make_stancode(count ~ zAge + zBase + (1 | patient),
data = epilepsy, prior = bprior)
# 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)
# define priors in a vectorized manner
# useful in particular for categorical or multivariate models
set_prior("normal(0, 2)", dpar = c("muX", "muY", "muZ"))
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