This vignette explains how to estimate linear models using the
stan_lm function in the rstanarm
package.
Steps 3 and 4 are covered in more depth by the vignette entitled “How to Use the rstanarm Package”. This vignette focuses on Step 1 when the likelihood is the product of independent normal distributions.
The goal of the rstanarm package is to make Bayesian estimation of common regression models routine. That goal can be partially accomplished by providing interfaces that are similar to the popular formula-based interfaces to frequentist estimators of those regression models. But fully accomplishing that goal sometimes entails utilizing priors that applied researchers are unaware that they prefer. These priors are intended to work well for any data that a user might pass to the interface that was generated according to the assumptions of the likelihood function.
It is important to distinguish between priors that are easy for applied researchers to specify and priors that are easy for applied researchers to conceptualize. The prior described below emphasizes the former but we outline its derivation so that applied researchers may feel more comfortable utilizing it.
The likelihood for one observation under a linear model can be written as a conditionally normal PDF $$\frac{1}{\sigma_{\epsilon} \sqrt{2 \pi}} e^{-\frac{1}{2} \left(\frac{y - \mu}{\sigma_{\epsilon}}\right)^2},$$ where μ = α + x⊤β is a linear predictor and σϵ is the standard deviation of the error in predicting the outcome, y. The likelihood of the entire sample is the product of N individual likelihood contributions.
It is well-known that the likelihood of the sample is maximized when the sum-of-squared residuals is minimized, which occurs when $$ \widehat{\boldsymbol{\beta}} = \left(\mathbf{X}^\top \mathbf{X}\right)^{-1} \mathbf{X}^\top \mathbf{y}, $$ $$ \widehat{\alpha} = \overline{y} - \overline{\mathbf{x}}^\top \widehat{\boldsymbol{\beta}}, $$ $$ \widehat{\sigma}_{\epsilon}^2 = \frac{\left(\mathbf{y} - \widehat{\alpha} - \mathbf{X} \widehat{ \boldsymbol{\beta}}\right)^\top \left(\mathbf{y} - \widehat{\alpha} - \mathbf{X} \widehat{ \boldsymbol{\beta}}\right)}{N},$$ where $\overline{\mathbf{x}}$ is a vector that contains the sample means of the K predictors, X is a N × K matrix of centered predictors, y is a N-vector of outcomes and $\overline{y}$ is the sample mean of the outcome.
The lm function in R actually performs a QR
decomposition of the design matrix, X = QR,
where Q⊤Q = I
and R is upper
triangular. Thus, the OLS solution for the coefficients can be written
as (X⊤X)−1X⊤y = R−1Q⊤y.
The lm function utilizes the QR decomposition for numeric
stability reasons, but the QR decomposition is also useful for thinking
about priors in a Bayesian version of the linear model. In addition,
writing the likelihood in terms of Q allows it to be evaluated
in a very efficient manner in Stan.
The key innovation in the stan_lm function in the
rstanarm package is the prior for the parameters in the
QR-reparameterized model. To understand this prior, think about the
equations that characterize the maximum likelihood solutions before
observing the data on X and especially y.
What would the prior distribution of θ = Q⊤y be? We can write its k-th element as $\theta_k = \rho_k \sigma_Y \sqrt{N - 1}$ where ρk is the correlation between the kth column of Q and the outcome, σY is the standard deviation of the outcome, and $\frac{1}{\sqrt{N-1}}$ is the standard deviation of the k column of Q. Then let $\boldsymbol{\rho} = \sqrt{R^2}\mathbf{u}$ where u is a unit vector that is uniformly distributed on the surface of a hypersphere. Consequently, R2 = ρ⊤ρ is the familiar coefficient of determination for the linear model.
An uninformative prior on R2 would be standard uniform, which is a special case of a Beta distribution with both shape parameters equal to 1. A non-uniform prior on R2 is somewhat analogous to ridge regression, which is popular in data mining and produces better out-of-sample predictions than least squares because it penalizes β⊤β, usually after standardizing the predictors. An informative prior on R2 effectively penalizes ρ⊤ρ, which encourages β = R−1θ to be closer to the origin.
Lewandowski, Kurowicka, and Joe (2009) derives a distribution for a
correlation matrix that depends on a single shape parameter η > 0, which implies the variance
of one variable given the remaining K variables is $\mathrm{Beta}\left(\eta,\frac{K}{2}\right)$.
Thus, the R2 is
distributed $\mathrm{Beta}\left(\frac{K}{2},\eta\right)$
and any prior information about the location of R2 can be used to choose
a value of the hyperparameter η. The
R2(location, what) function in the
rstanarm package supports four ways of choosing η:
what = "mode" and location is some prior
mode on the (0, 1) interval. This is
the default but since the mode of a $\mathrm{Beta}\left(\frac{K}{2},\eta\right)$
distribution is $\frac{\frac{K}{2} -
1}{\frac{K}{2} + \eta - 2}$ the mode only exists if K > 2. If K ≤ 2, then the user must specify
something else for what.what = "mean" and location is some prior
mean on the (0, 1) interval, where the
mean of a $\mathrm{Beta}\left(\frac{K}{2},\eta\right)$
distribution is $\frac{\frac{K}{2}}{\frac{K}{2} +
\eta}$.what = "median" and location is some prior
median on the (0, 1) interval. The
median of a $\mathrm{Beta}\left(\frac{K}{2},\eta\right)$
distribution is not available in closed form but if K > 2 it is approximately equal
to $\frac{\frac{K}{2} -
\frac{1}{3}}{\frac{K}{2} + \eta - \frac{2}{3}}$. Regardless of
whether K > 2, the
R2 function can numerically solve for the value of η that is consistent with a given
prior median utilizing the quantile function.what = "log" and location is some
(negative) prior value for $\mathbb{E} \ln R^2
= \psi\left(\frac{K}{2}\right)-
\psi\left(\frac{K}{2}+\eta\right)$, where ψ(⋅) is the digamma
function. Again, given a prior value for the left-hand side it is easy
to numerically solve for the corresponding value of η.There is no default value for the location argument of
the R2 function. This is an informative prior on
R2, which must be
chosen by the user in light of the research project. However, specifying
location = 0.5 is often safe, in which case $\eta = \frac{K}{2}$ regardless of whether
what is "mode", "mean", or
"median". In addition, it is possible to specify
NULL, in which case a standard uniform on R2 is utilized.
We set σy = ωsy where sy is the sample standard deviation of the outcome and ω > 0 is an unknown scale parameter to be estimated. The only prior for ω that does not contravene Bayes’ theorem in this situation is Jeffreys prior, $f\left(\omega\right) \propto \frac{1}{\omega}$, which is proportional to a Jeffreys prior on the unknown σy, $f\left(\sigma_y\right) \propto \frac{1}{\sigma_y} = \frac{1}{\omega \widehat{\sigma}_y} \propto \frac{1}{\omega}$. This parameterization and prior makes it easy for Stan to work with any continuous outcome variable, no matter what its units of measurement are.
It would seem that we need a prior for σϵ, but our prior beliefs about $\sigma_{\epsilon} = \omega s_y \sqrt{1 - R^2}$ are already implied by our prior beliefs about ω and R2. That only leaves a prior for $\alpha = \overline{y} - \overline{\mathbf{x}}^\top \mathbf{R}^{-1} \boldsymbol{\theta}$. The default choice is an improper uniform prior, but a normal prior can also be specified such as one with mean zero and standard deviation $\frac{\sigma_y}{\sqrt{N}}$.
The previous sections imply a posterior distribution for ω, α, u, and R2. The parameters of interest can then be recovered as generated quantities:
The implementation actually utilizes an improper uniform prior on ln ω. Consequently, if ln ω = 0, then the marginal standard deviation of the outcome implied by the model is the same as the sample standard deviation of the outcome. If ln ω > 0, then the marginal standard deviation of the outcome implied by the model exceeds the sample standard deviation, so the model overfits the data. If ln ω < 0, then the marginal standard deviation of the outcome implied by the model is less than the sample standard deviation, so the model underfits the data or that the data-generating process is nonlinear. Given the regularizing nature of the prior on R2, a minor underfit would be considered ideal if the goal is to obtain good out-of-sample predictions. If the model badly underfits or overfits the data, then you may want to reconsider the model.
We will utilize an example from the HSAUR3 package by Brian S. Everitt and Torsten Hothorn, which is used in their 2014 book A Handbook of Statistical Analyses Using R (3rd Edition) (Chapman & Hall / CRC). This book is frequentist in nature and we will show how to obtain the corresponding Bayesian results.
The model in section 5.3.1 analyzes an experiment where clouds were
seeded with different amounts of silver iodide to see if there was
increased rainfall. This effect could vary according to covariates,
which (except for time) are interacted with the treatment
variable. Most people would probably be skeptical that cloud hacking
could explain very much of the variation in rainfall and thus the prior
mode of the R2
would probably be fairly small.
The frequentist estimator of this model can be replicated by executing
data("clouds", package = "HSAUR3")
ols <- lm(rainfall ~ seeding * (sne + cloudcover + prewetness + echomotion) +
time, data = clouds)
round(coef(ols), 3)Note that we have not looked at the estimated R2 or σ for the ordinary least squares
model. We can estimate a Bayesian version of this model by prepending
stan_ to the lm call, specifying a prior mode
for R2, and
optionally specifying how many cores the computer may utilize:
library(rstanarm)
post <-
stan_lm(
rainfall ~ seeding * (sne + cloudcover + prewetness + echomotion) + time,
data = clouds,
prior = R2(location = 0.2),
seed = 12345
)
postIn this case, the “Bayesian point estimates”, which are represented
by the posterior medians, appear quite different from the ordinary least
squares estimates. However, the log-fit_ratio (i.e. ln ω) is quite small, indicating
that the model only slightly overfits the data when the prior derived
above is utilized. Thus, it would be safe to conclude that the ordinary
least squares estimator considerably overfits the data since there are
only 24 observations to estimate 12 parameters with and no prior information
on the parameters.
Also, it is not obvious what the estimated average treatment effect
is since the treatment variable, seeding, is interacted
with four other correlated predictors. However, it is easy to estimate
or visualize the average treatment effect (ATE) using
rstanarm’s posterior_predict function.
clouds_cf <- clouds
clouds_cf$seeding[] <- "yes"
y1_rep <- posterior_predict(post, newdata = clouds_cf)
clouds_cf$seeding[] <- "no"
y0_rep <- posterior_predict(post, newdata = clouds_cf)
qplot(x = c(y1_rep - y0_rep), geom = "histogram", xlab = "Estimated ATE")As can be seen, the treatment effect is not estimated precisely and is as almost as likely to be negative as it is to be positive.
The prior derived above works well in many situations and is quite
simple to use since it only requires the user to specify the
prior location of the R2. Nevertheless, the
implications of the prior are somewhat difficult to
conceptualize. Thus, it is perhaps worthwhile to compare to
another estimator of a linear model that simply puts independent Cauchy
priors on the regression coefficients. This simpler approach can be
executed by calling the stan_glm function with
family = gaussian() and specifying the priors:
simple <-
stan_glm(
rainfall ~ seeding * (sne + cloudcover + prewetness + echomotion) + time,
data = clouds,
family = gaussian(),
prior = cauchy(),
prior_intercept = cauchy(),
seed = 12345
)We can compare the two approaches using an approximation to
Leave-One-Out (LOO) cross-validation, which is implemented by the
loo function in the loo package.
The results indicate that the first approach is expected to produce better out-of-sample predictions but the Warning messages are at least as important. Many of the estimated shape parameters for the Generalized Pareto distribution are above 0.5 in the model with Cauchy priors, which indicates that these estimates are only going to converge slowly to the true out-of-sample deviance measures. Thus, with only 24 observations, they should not be considered reliable. The more complicated prior derived above is stronger — as evidenced by the fact that the effective number of parameters is about half of that in the simpler approach and 12 for the maximum likelihood estimator — and only has a few of the 24 Pareto shape estimates in the “danger zone”. We might want to reexamine these observations
because the posterior is sensitive to them but, overall, the results seem tolerable.
In general, we would expect the joint prior derived here to work better when there are many predictors relative to the number of observations. Placing independent, heavy-tailed priors on the coefficients neither reflects the beliefs of the researcher nor conveys enough information to stabilize all the computations.
This vignette has discussed the prior distribution utilized in the
stan_lm function, which has the same likelihood and a
similar syntax as the lm function in R but adds the ability
to expression prior beliefs about the location of the R2, which is the familiar
proportion of variance in the outcome variable that is attributable to
the predictors under a linear model. Since the R2 is a well-understood
bounded scalar, it is easy to specify prior information about it,
whereas other Bayesian approaches require the researcher to specify a
joint prior distribution for the regression coefficients (and the
intercept and error variance).
However, most researchers have little inclination to specify all
these prior distributions thoughtfully and take a short-cut by
specifying one prior distribution that is taken to apply to all the
regression coefficients as if they were independent of each other (and
the intercept and error variance). This short-cut is available in the
stan_glm function and is described in more detail in other
rstanarm vignettes for Generalized Linear Models
(GLMs), which can be found by navigating up one level.
We are optimistic that this prior on the R2 will greatly help in
accomplishing our goal for rstanarm of making Bayesian
estimation of regression models routine. The same approach is used to
specify a prior in ANOVA models (see stan_aov) and
proportional-odds models for ordinal outcomes (see
stan_polr).
Finally, the stan_biglm function can be used when the
design matrix is too large for the qr function to process.
The stan_biglm function inputs the output of the
biglm function in the biglm package, which
utilizes an incremental QR decomposition that does not require the
entire dataset to be loaded into memory simultaneously. However, the
biglm function needs to be called in a particular way in
order to work with stan_biglm. In particular, The means of
the columns of the design matrix, the sample mean of the outcome, and
the sample standard deviation of the outcome all need to be passed to
the stan_biglm function, as well as a flag indicating
whether the model really does include an intercept. Also, the number of
columns of the design matrix currently cannot exceed the number of rows.
Although stan_biglm should run fairly quickly and without
much memory, the resulting object is a fairly plain stanfit
object rather than an enhanced stanreg object like that
produced by stan_lm. Many of the enhanced capabilities of a
stanreg object depend on being able to access the full
design matrix, so doing posterior predictions, posterior checks, etc.
with the output of stan_biglm would require some custom R
code.
Lewandowski, D., Kurowicka D., and Joe, H. (2009). Generating random correlation matrices based on vines and extended onion method. Journal of Multivariate Analysis. 100(9), 1989–2001.