normal(location = 0, scale = NULL)
student_t(df = 1, location = 0, scale = NULL)
cauchy(location = 0, scale = NULL)
hs(df = 3)
hs_plus(df1 = 3, df2 = 3)
decov(regularization = 1, concentration = 1, shape = 1, scale = 1)
dirichlet(concentration = 1)
R2(location = NULL, what = c("mode", "mean", "median", "log"))
prior_options(prior_scale_for_dispersion = 5, min_prior_scale = 1e-12, scaled = TRUE)normal and student_t
(provided that df > 1) this is the prior mean. For cauchy
(which is equivalent to student_t with df=1), the mean does
not exist and location is the prior median. The default value is
$0$, except for R2 which has no default value for
location. For R2, location pertains to the prior
location of the $R^2$ under a Beta distribution, but the interpretation
of the location parameter depends on the specified value of the
what argument (see the "R2 family" section in Details).student_t, in which case it is equivalent to cauchy. For the
hierarchical shrinkage priors (hs and hs_plus) the degrees of
freedom parameter(s) default to $3$.decov prior. The default is $1$, implying a joint uniform
prior.decov prior. If shape and scale are both $1$ (the
default) then the gamma prior simplifies to the unit-exponential
distribution.'mode' (the default),
'mean', 'median', or 'log' indicating how the
location parameter is interpreted in the LKJ case. If
'log', then location is interpreted as the expected
logarithm of the $R^2$ under a Beta distribution. Otherwise,
location is interpreted as the what of the $R^2$
under a Beta distribution. If the number of predictors is less than
or equal to two, the mode of this Beta distribution does not exist
and an error will prompt the user to specify another choice for
what.TRUE. If TRUE the
prior_scale is further scaled by the range of the predictor if the
predictor has exactly two unique values and scaled by twice the standard
deviation of the predictor if it has more than two unique values.normal(location, scale)
student_t(df, location, scale)
cauchy(location, scale)
For the prior distribution for the intercept, location,
scale, and df should be scalars. For the prior for the other
coefficients they can either be vectors of length equal to the number of
coefficients (not including the intercept), or they can be scalars, in
which case they will be recycled to the appropriate length. As the
degrees of freedom approaches infinity, the Student t distribution
approaches the normal distribution and if the degrees of freedom are one,
then the Student t distribution is the Cauchy distribution.
If scale is not specified it will default to 10 for the intercept
and 2.5 for the other coefficients, unless the probit link function is
used, in which case these defaults are scaled by a factor of
dnorm(0)/dlogis(0), which is roughly 1.6.
Hierarchical shrinkage family Family members:
hs(df)
hs_plus(df1, df2)
The hierarchical shrinkage priors are normal with a mean of zero and a
standard deviation that is also a random variable. The traditional
hierarchical shrinkage prior utilizes a standard deviation that is
distributed half Cauchy with a median of zero and a scale parameter that is
also half Cauchy. This is called the "horseshoe prior". The hierarchical
shrinkage (hs) prior in the rstanarm package instead utilizes
a half Student t distribution for the standard deviation (with 3 degrees of
freedom by default), scaled by a half Cauchy parameter, as described by
Piironen and Vehtari (2015). It is possible to change the df
argument, the prior degrees of freedom, to obtain less or more shrinkage.
The hierarhical shrinkpage plus (hs_plus) prior is a normal with a
mean of zero and a standard deviation that is distributed as the product of
two independent half Student t parameters (both with 3 degrees of freedom
(df1, df2) by default) that are each scaled by the same
square root of a half Cauchy parameter.
These hierarchical shrinkage priors have very tall modes and very fat
tails. Consequently, they tend to produce posterior distributions that are
very concentrated near zero, unless the predictor has a strong influence on
the outcome, in which case the prior has little influence. Hierarchical
shrinkage priors often require you to increase the
adapt_delta tuning parameter in order to diminish the number
of divergent transitions. For more details on tuning parameters and
divergent transitions see the Troubleshooting section of the
How to Use the rstanarm Package vignette.
Dirichlet family Family members:
dirichlet(concentration)
The Dirichlet distribution is a multivariate generalization of the beta
distribution. It is perhaps the easiest prior distribution to specify
because the concentration parameters can be interpreted as prior counts
(although they need not be integers) of a multinomial random variable.
The Dirichlet distribution is used in stan_polr for an
implicit prior on the cutpoints in an ordinal regression model. More
specifically, the Dirichlet prior pertains to the prior probability of
observing each category of the ordinal outcome when the predictors are at
their sample means. Given these prior probabilities, it is straightforward
to add them to form cumulative probabilities and then use an inverse CDF
transformation of the cumulative probabilities to define the cutpoints.
If a scalar is passed to the concentration argument of the
dirichlet function, then it is replicated to the appropriate length
and the Dirichlet distribution is symmetric. If concentration is a
vector and all elements are $1$, then the Dirichlet distribution is
jointly uniform. If all concentration parameters are equal but greater than
$1$ then the prior mode is that the categories are equiprobable, and
the larger the value of the identical concentration parameters, the more
sharply peaked the distribution is at the mode. The elements in
concentration can also be given different values to represent that
not all outcome categories are a priori equiprobable.
Covariance matrices Family members:
decov(regularization, concentration, shape, scale)
Covariance matrices are decomposed into correlation matrices and
variances. The variances are in turn decomposed into the product of a
simplex vector and the trace of the matrix. Finally, the trace is the
product of the order of the matrix and the square of a scale parameter.
This prior on a covariance matrix is represented by the decov
function.
The prior for a correlation matrix is called LKJ whose density is
proportional to the determinant of the correlation matrix raised to the
power of a positive regularization parameter minus one. If
regularization = 1 (the default), then this prior is jointly
uniform over all correlation matrices of that size. If
regularization > 1, then the identity matrix is the mode and in the
unlikely case that regularization < 1, the identity matrix is the
trough.
The trace of a covariance matrix is equal to the sum of the variances. We
set the trace equal to the product of the order of the covariance matrix
and the square of a positive scale parameter. The particular
variances are set equal to the product of a simplex vector --- which is
non-negative and sums to $1$ --- and the scalar trace. In other words,
each element of the simplex vector represents the proportion of the trace
attributable to the corresponding variable.
A symmetric Dirichlet prior is used for the simplex vector, which has a
single (positive) concentration parameter, which defaults to
$1$ and implies that the prior is jointly uniform over the space of
simplex vectors of that size. If concentration > 1, then the prior
mode corresponds to all variables having the same (proportion of total)
variance, which can be used to ensure the the posterior variances are not
zero. As the concentration parameter approaches infinity, this
mode becomes more pronounced. In the unlikely case that
concentration < 1, the variances are more polarized.
If all the variables were multiplied by a number, the trace of their
covariance matrix would increase by that number squared. Thus, it is
reasonable to use a scale-invariant prior distribution for the positive
scale parameter, and in this case we utilize a Gamma distribution, whose
shape and scale are both $1$ by default, implying a
unit-exponential distribution. Set the shape hyperparameter to some
value greater than $1$ to ensure that the posterior trace is not zero.
If regularization, concentration, shape and / or
scale are positive scalars, then they are recycled to the
appropriate length. Otherwise, each can be a positive vector of the
appropriate length, but the appropriate length depends on the number of
covariance matrices in the model and their sizes. A one-by-one covariance
matrix is just a variance and thus does not have regularization or
concentration parameters, but does have shape and
scale parameters for the prior standard deviation of that
variable.
R2 family Family members:
R2(location, what)
The stan_lm, stan_aov, and
stan_polr functions allow the user to utilize a function
called R2 to convey prior information about all the parameters.
This prior hinges on prior beliefs about the location of $R^2$, the
proportion of variance in the outcome attributable to the predictors,
which has a Beta prior with first shape
hyperparameter equal to half the number of predictors and second shape
hyperparameter free. By specifying what to be the prior mode (the
default), mean, median, or expected log of $R^2$, the second shape
parameter for this Beta distribution is determined internally. If
what = 'log', location should be a negative scalar; otherwise it
should be a scalar on the $(0,1)$ interval.
For example, if $R^2 = 0.5$, then the mode, mean, and median of
the Beta distribution are all the same and thus the
second shape parameter is also equal to half the number of predictors.
The second shape parameter of the Beta distribution
is actually the same as the shape parameter in the LKJ prior for a
correlation matrix described in the previous subsection. Thus, the smaller
is $R^2$, the larger is the shape parameter, the smaller are the
prior correlations among the outcome and predictor variables, and the more
concentrated near zero is the prior density for the regression
coefficients. Hence, the prior on the coefficients is regularizing and
should yield a posterior distribution with good out-of-sample predictions
if the prior location of $R^2$ is specified in a reasonable
fashion.
Gelman, A., Jakulin, A., Pittau, M. G., and Su, Y. (2008). A weakly informative default prior distribution for logistic and other regression models. Annals of Applied Statistics. 2(4), 1360--1383.
Piironen, J., and Vehtari, A. (2015). Projection predictive variable selection using Stan+R. http://arxiv.org/abs/1508.02502/
Stan Development Team. (2015). Stan Modeling Language Users Guide and Reference Manual. http://mc-stan.org/documentation/
fmla <- mpg ~ wt + qsec + drat + am
# Draw from prior predictive distribution (by setting prior_PD = TRUE)
prior_pred_fit <- stan_glm(fmla, data = mtcars, chains = 1, prior_PD = TRUE,
prior = student_t(df = 4, 0, 2.5),
prior_intercept = cauchy(0,10),
prior_ops = prior_options(prior_scale_for_dispersion = 2))
## Not run:
# # Can assign priors to names
# N05 <- normal(0, 5)
# fit <- stan_glm(fmla, data = mtcars, prior = N05, prior_intercept = N05)
# ## End(Not run)
# Visually compare normal, student_t, and cauchy
library(ggplot2)
compare_priors <- function(scale = 1, df_t = 2, xlim = c(-10, 10)) {
dt_loc_scale <- function(x, df, location, scale) {
# t distribution with location & scale parameters
1 / scale * dt((x - location) / scale, df)
}
ggplot(data.frame(x = xlim), aes(x)) +
stat_function(fun = dnorm,
args = list(mean = 0, sd = scale),
color = "purple", size = .75) +
stat_function(fun = dt_loc_scale,
args = list(df = df_t, location = 0, scale = scale),
color = "orange", size = .75) +
stat_function(fun = dcauchy,
args = list(location = 0, scale = scale),
color = "skyblue", size = .75, linetype = 2) +
ggtitle("normal (purple) vs student_t (orange) vs cauchy (blue)")
}
# Cauchy has fattest tails, then student_t, then normal
compare_priors()
# The student_t with df = 1 is the same as the cauchy
compare_priors(df_t = 1)
# Even a scale of 5 is somewhat large. It gives plausibility to rather
# extreme values
compare_priors(scale = 5, xlim = c(-20,20))
# If you use a prior like normal(0, 1000) to be "non-informative" you are
# actually saying that a coefficient value of e.g. -500 is quite plausible
compare_priors(scale = 1000, xlim = c(-1000,1000))
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