The prior_summary method provides a summary of the prior distributions
used for the parameters in a given model. In some cases the user-specified
prior does not correspond exactly to the prior used internally by
rstanarm (see the sections below). Especially in these cases, but also
in general, it can be much more useful to visualize the priors. Visualizing
the priors can be done using the posterior_vs_prior function,
or alternatively by fitting the model with the prior_PD argument set
to TRUE (to draw from the prior predictive distribution instead of
conditioning on the outcome) and then plotting the parameters.
# S3 method for stanreg prior_summary(object, digits = 2, ...)
| object | A fitted model object returned by one of the
rstanarm modeling functions. See |
|---|---|
| digits | Number of digits to use for rounding. |
| ... | Currently ignored by the method for stanreg objects. |
A list of class "prior_summary.stanreg", which has its own print method.
For rstanarm modeling functions that accept a prior_intercept
argument, the specified prior for the intercept term applies to the
intercept after rstanarm internally centers the predictors so they
each have mean zero. The estimate of the intercept returned to the user
correspond to the intercept with the predictors as specified by the user
(unmodified by rstanarm), but when specifying the prior the
intercept can be thought of as the expected outcome when the predictors are
set to their means. The only exception to this is for models fit with the
sparse argument set to TRUE (which is only possible with a
subset of the modeling functions and never the default).
For some models you may see "adjusted scale"
in the printed output and adjusted scales included in the object returned
by prior_summary. These adjusted scale values are the prior scales
actually used by rstanarm and are computed by adjusting the prior
scales specified by the user to account for the scales of the predictors
(as described in the documentation for the autoscale
argument). To disable internal prior scale adjustments set the
autoscale argument to FALSE when setting a prior using one of
the distributions that accepts an autoscale argument. For example,
normal(0, 5, autoscale=FALSE) instead of just normal(0, 5).
For the models fit with an rstanarm modeling function that supports
the QR argument (see e.g, stan_glm), if QR is
set to TRUE then the prior distributions for the regression
coefficients specified using the prior argument are not relative to
the original predictor variables \(X\) but rather to the variables in the
matrix \(Q\) obtained from the \(QR\) decomposition of \(X\).
In particular, if prior = normal(location,scale), then this prior on
the coefficients in \(Q\)-space can be easily translated into a joint
multivariate normal (MVN) prior on the coefficients on the original
predictors in \(X\). Letting \(\theta\) denote the coefficients on
\(Q\) and \(\beta\) the coefficients on \(X\) then if \(\theta
\sim N(\mu, \sigma)\) the corresponding prior on
\(\beta\) is \(\beta \sim MVN(R\mu, R'R\sigma^2)\), where \(\mu\) and \(\sigma\) are vectors of the
appropriate length. Technically, rstanarm uses a scaled \(QR\)
decomposition to ensure that the columns of the predictor matrix used to
fit the model all have unit scale, when the autoscale argument
to the function passed to the prior argument is TRUE (the
default), in which case the matrices actually used are
\(Q^\ast = Q \sqrt{n-1}\) and \(R^\ast =
\frac{1}{\sqrt{n-1}} R\). If autoscale = FALSE
we instead scale such that the lower-right element of \(R^\ast\) is
\(1\), which is useful if you want to specify a prior on the coefficient
of the last predictor in its original units (see the documentation for the
QR argument).
If you are interested in the prior on \(\beta\) implied by the prior on \(\theta\), we strongly recommend visualizing it as described above in the Description section, which is simpler than working it out analytically.
The priors help page and the Prior Distributions vignette.
#> Priors for model 'example_model' #> ------ #> Intercept (after predictors centered) #> ~ normal(location = 0, scale = 2.5) #> #> Coefficients (in Q-space) #> ~ normal(location = [0,0,0,...], scale = [2.5,2.5,2.5,...]) #> #> Covariance #> ~ decov(reg. = 1, conc. = 1, shape = 1, scale = 1) #> ------ #> See help('prior_summary.stanreg') for more details#> [1] "prior" "prior_intercept" "prior_covariance"priors$prior$scale#> [1] 2.5 2.5 2.5 2.5priors$prior$adjusted_scale#> NULL# for a glm with adjusted scales (see Details, above), compare # the default (rstanarm adjusting the scales) to setting # autoscale=FALSE for prior on coefficients fit <- stan_glm(mpg ~ wt + am, data = mtcars, prior = normal(0, c(2.5, 4)), prior_intercept = normal(0, 5), iter = 10, chains = 1) # only for demonstration#> #> SAMPLING FOR MODEL 'continuous' NOW (CHAIN 1). #> Chain 1: #> Chain 1: Gradient evaluation took 2e-05 seconds #> Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.2 seconds. #> Chain 1: Adjust your expectations accordingly! #> Chain 1: #> Chain 1: #> Chain 1: WARNING: No variance estimation is #> Chain 1: performed for num_warmup < 20 #> Chain 1: #> Chain 1: Iteration: 1 / 10 [ 10%] (Warmup) #> Chain 1: Iteration: 2 / 10 [ 20%] (Warmup) #> Chain 1: Iteration: 3 / 10 [ 30%] (Warmup) #> Chain 1: Iteration: 4 / 10 [ 40%] (Warmup) #> Chain 1: Iteration: 5 / 10 [ 50%] (Warmup) #> Chain 1: Iteration: 6 / 10 [ 60%] (Sampling) #> Chain 1: Iteration: 7 / 10 [ 70%] (Sampling) #> Chain 1: Iteration: 8 / 10 [ 80%] (Sampling) #> Chain 1: Iteration: 9 / 10 [ 90%] (Sampling) #> Chain 1: Iteration: 10 / 10 [100%] (Sampling) #> Chain 1: #> Chain 1: Elapsed Time: 0.000122 seconds (Warm-up) #> Chain 1: 0.000435 seconds (Sampling) #> Chain 1: 0.000557 seconds (Total) #> Chain 1:#> Warning: The largest R-hat is 1.9, indicating chains have not mixed. #> Running the chains for more iterations may help. See #> http://mc-stan.org/misc/warnings.html#r-hat#> Warning: Markov chains did not converge! Do not analyze results!prior_summary(fit)#> Priors for model 'fit' #> ------ #> Intercept (after predictors centered) #> ~ normal(location = 0, scale = 5) #> #> Coefficients #> ~ normal(location = [0,0], scale = [2.5,4.0]) #> #> Auxiliary (sigma) #> Specified prior: #> ~ exponential(rate = 1) #> Adjusted prior: #> ~ exponential(rate = 0.17) #> ------ #> See help('prior_summary.stanreg') for more detailsfit2 <- update(fit, prior = normal(0, c(2.5, 4), autoscale=FALSE), prior_intercept = normal(0, 5, autoscale=FALSE))#> #> SAMPLING FOR MODEL 'continuous' NOW (CHAIN 1). #> Chain 1: #> Chain 1: Gradient evaluation took 2e-05 seconds #> Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.2 seconds. #> Chain 1: Adjust your expectations accordingly! #> Chain 1: #> Chain 1: #> Chain 1: WARNING: No variance estimation is #> Chain 1: performed for num_warmup < 20 #> Chain 1: #> Chain 1: Iteration: 1 / 10 [ 10%] (Warmup) #> Chain 1: Iteration: 2 / 10 [ 20%] (Warmup) #> Chain 1: Iteration: 3 / 10 [ 30%] (Warmup) #> Chain 1: Iteration: 4 / 10 [ 40%] (Warmup) #> Chain 1: Iteration: 5 / 10 [ 50%] (Warmup) #> Chain 1: Iteration: 6 / 10 [ 60%] (Sampling) #> Chain 1: Iteration: 7 / 10 [ 70%] (Sampling) #> Chain 1: Iteration: 8 / 10 [ 80%] (Sampling) #> Chain 1: Iteration: 9 / 10 [ 90%] (Sampling) #> Chain 1: Iteration: 10 / 10 [100%] (Sampling) #> Chain 1: #> Chain 1: Elapsed Time: 0.000267 seconds (Warm-up) #> Chain 1: 0.000415 seconds (Sampling) #> Chain 1: 0.000682 seconds (Total) #> Chain 1:#> Warning: There were 1 chains where the estimated Bayesian Fraction of Missing Information was low. See #> http://mc-stan.org/misc/warnings.html#bfmi-low#> Warning: Examine the pairs() plot to diagnose sampling problems#> Warning: The largest R-hat is 1.9, indicating chains have not mixed. #> Running the chains for more iterations may help. See #> http://mc-stan.org/misc/warnings.html#r-hat#> Warning: Markov chains did not converge! Do not analyze results!prior_summary(fit2)#> Priors for model 'fit2' #> ------ #> Intercept (after predictors centered) #> ~ normal(location = 0, scale = 5) #> #> Coefficients #> ~ normal(location = [0,0], scale = [2.5,4.0]) #> #> Auxiliary (sigma) #> Specified prior: #> ~ exponential(rate = 1) #> Adjusted prior: #> ~ exponential(rate = 0.17) #> ------ #> See help('prior_summary.stanreg') for more details