---
title: "Prior Distributions for rstanarm Models"
author: "Jonah Gabry and Ben Goodrich"
date: "`r Sys.Date()`"
output: 
  html_vignette: 
    toc: yes
---

```{r, child="children/SETTINGS-knitr.txt"}
```
```{r, child="children/SETTINGS-gg.txt"}
```

# July 2020 Update

As of July 2020 there are a few changes to prior distributions: 

* Except for in default priors, `autoscale` now defaults to `FALSE`. This means
that when specifying custom priors you no longer need to manually set
`autoscale=FALSE` every time you use a distribution.

* There are minor changes to the default priors on the intercept and
(non-hierarchical) regression coefficients. See **Default priors and scale
adjustments** below.

We recommend the new book 
[Regression and Other Stories](https://avehtari.github.io/ROS-Examples/), 
which discusses the background behind the default priors in **rstanarm** and 
also provides examples of specifying non-default priors.

# Introduction

This vignette provides an overview of how the specification of prior 
distributions works in the __rstanarm__ package. It is still a work in progress 
and more content will be added in future versions of __rstanarm__. Before
reading this vignette it is important to first read the
[How to Use the __rstanarm__ Package](rstanarm.html) 
vignette, which provides a general overview of the package.

Every modeling function in __rstanarm__ offers a subset of the arguments in the 
table below which are used for specifying prior distributions for the model 
parameters.

| Argument       | Used in  | Applies to |
| ------------- | ------------- | ------------- |
| `prior_intercept` | All modeling functions except `stan_polr` and `stan_nlmer`| Model intercept, after centering predictors.|
| `prior` |  All modeling functions| Regression coefficients. Does _not_ include coefficients that vary by group in a multilevel model (see `prior_covariance`).|
| `prior_aux` | `stan_glm`\*, `stan_glmer`\*, `stan_gamm4`, `stan_nlmer`| Auxiliary parameter, e.g. error SD (interpretation depends on the GLM).|
| `prior_covariance` | `stan_glmer`\*, `stan_gamm4`, `stan_nlmer`| Covariance matrices in multilevel models with varying slopes and intercepts. See the [`stan_glmer` vignette](http://mc-stan.org/rstanarm/articles/glmer.html) for details on this prior.|
\* `stan_glm` also implies `stan_glm.nb`. `stan_glmer` implies `stan_lmer` and
`stan_glmer.nb`.

The `stan_polr`, `stan_betareg`, and `stan_gamm4` functions also provide
additional arguments specific only to those models:

| Argument       | Used only in  | Applies to |
| ------------- | ------------- | ------------- |
| `prior_smooth` | `stan_gamm4` | Prior for hyperparameters in GAMs (lower values yield less flexible smooth functions). |
| `prior_counts` | `stan_polr` | Prior counts of an _ordinal_ outcome (when predictors at sample means). |
| `prior_z`  |  `stan_betareg`| Coefficients in the model for `phi`.|
| `prior_intercept_z` | `stan_betareg`| Intercept in the model for `phi`. |
| `prior_phi` | `stan_betareg`| `phi`, if not modeled as function of predictors. |

To specify these arguments the user provides a call to one of the various
available functions for specifying priors (e.g., `prior = normal(0, 1)`, `prior = cauchy(c(0, 1), c(1, 2.5))`). 
The documentation for these functions can be found at `help("priors")`. The
__rstanarm__ documentation and the other [vignettes](index.html) provide many
examples of using these arguments to specify priors and the documentation for
these arguments on the help pages for the various __rstanarm__ modeling 
functions (e.g., `help("stan_glm")`) also explains which distributions can be
used when specifying each of the prior-related arguments.

# Default (Weakly Informative) Prior Distributions

With very few exceptions, the default priors in __rstanarm__ ---the priors used 
if the arguments in the tables above are untouched--- are _not_ flat priors. 
Rather, the defaults are intended to be _weakly informative_. That is, they are 
designed to provide moderate regularization and help stabilize computation. For
many (if not most) applications the defaults will perform well, but this is not
guaranteed (there are no default priors that make sense for every possible model
specification).

The way __rstanarm__ attempts to make priors weakly informative by default is to
internally adjust the scales of the priors. How this works (and, importantly, 
how to turn it off) is explained below, but first we can look at the default
priors in action by fitting a basic linear regression model with the `stan_glm` 
function. For specifying priors, the `stan_glm` function accepts the arguments 
`prior_intercept`, `prior`, and `prior_aux`. To use the default priors we just 
leave those arguments at their defaults (i.e., we don't specify them):

```{r, default-prior-1, results="hide", eval = TRUE}
library("rstanarm")
default_prior_test <- stan_glm(mpg ~ wt + am, data = mtcars, chains = 1)
```

The `prior_summary` function provides a concise summary of the priors used: 

```{r, default-prior-summary, eval = TRUE}
prior_summary(default_prior_test)
```
```{r, echo=FALSE, eval = TRUE}
priors <- prior_summary(default_prior_test)
fr2 <- function(x) format(round(x, 2), nsmall = 2)
```

Starting from the bottom up, we can see that:

* __Auxiliary__: `sigma`, the error standard deviation, has a default prior that
is $\mathsf{exponential}(1)$. However, as a result of the automatic rescaling,
the actual scale used was `r fr2(priors$prior_aux$adjusted_scale)`.

* __Coefficients__: By default the regression coefficients (in this case the 
coefficients on the `wt` and `am` variables) are treated as a priori independent
with normal priors centered at 0 and with scale (standard deviation) $2.5$. Like
for `sigma`, in order for the default to be weakly informative __rstanarm__ will
adjust the scales of the priors on the coefficients. As a result, the prior
scales actually used were
`r fr2(priors$prior$adjusted_scale[1])` and 
`r fr2(priors$prior$adjusted_scale[2])`. 

* __Intercept__: For the intercept, the default prior is normal with mean $0$ 
and standard deviation $2.5$, but in this case the standard deviation was 
adjusted to `r fr2(priors$prior_intercept$adjusted_scale[1])`. There is also a
note in parentheses informing you that the prior applies to the intercept after
all predictors have been centered (a similar note can be found in the
documentation of the `prior_intercept` argument). In many cases the value of $y$
when $x=0$ is not meaningful and it is easier to think about the value when $x =
\bar{x}$. Therefore placing a prior on the intercept after centering the
predictors typically makes it easier to specify a reasonable prior for the
intercept. (Note: the user does _not_ need to manually center the predictors.)

To disable the centering of the predictors, you need to omit the intercept from
the model `formula` and include a column of ones as a predictor (which cannot
be named `"(Intercept)"` in the `data.frame`). Then you can specify a prior
"coefficient" for the column of ones.

The next two subsections describe how the rescaling works and how to easily
disable it if desired.


### Default priors and scale adjustments

Automatic scale adjustments happen in two cases: 

1. When the default priors are used.
2. When the user sets `autoscale=TRUE` when specifying their own prior (e.g., `normal(0, 3, autoscale=TRUE)`). See `help("priors")` for a list of distributions to see which have an `autoscale` argument.

Here we describe how the default priors work for the intercept, regression
coefficients, and (if applicable) auxiliary parameters. Autoscaling when not
using default priors works analogously (if `autoscale=TRUE`).

Assume we have outcome $y$ and predictors $x_1,\ldots,x_k$ and our model has linear 
predictor 

$$
\alpha + \beta_1 x_1 + \dots + \beta_K x_K.
$$

#### Regression coefficients 

The default prior on regression coefficients $\beta_k$ is

$$
\beta_k \sim \mathsf{Normal}(0, \, 2.5 \cdot s_y/s_x)
$$
where $s_x = \text{sd}(x)$ and 
$$ 
s_y =
\begin{cases}
\text{sd}(y) & \text{if } \:\: {\tt family=gaussian(link)}, \\
1 & \text{otherwise}.
\end{cases}
$$


This corresponds to `prior = normal(0, 2.5, autoscale = TRUE)` in **rstanarm** code. 


#### Intercept 

The intercept is assigned a prior indirectly. The `prior_intercept` argument
refers to the intercept after all predictors have been centered (internally by
**rstanarm**). That is, instead of placing the prior on the expected value of
$y$ when $x=0$, we place a prior on the expected value of $y$ when 
$x = \bar{x}$. The default prior for this centered intercept, say $\alpha_c$, is

$$
\alpha_c \sim \mathsf{Normal}(m_y, \, 2.5 \cdot s_y)
$$
where 

$$
m_y =
\begin{cases}
\bar{y} & \text{if } \:\: {\tt family=gaussian(link="identity")}, \\
0 & \text{otherwise}
\end{cases}
$$
and $s_y$ is the same as above (either 1 or $\text{sd(y)}$).

#### Auxiliary parameters

The default prior on the auxiliary parameter (residual standard deviation for
Gaussian, shape for gamma, reciprocal dispersion for negative binomial, etc.) is
an exponential distribution with rate $1/s_y$


$$
\text{aux} \sim \mathsf{Exponential}(1/s_y)
$$
where $s_y$ is the same as above (either 1 or $\text{sd(y)}$). 

This corresponds to `prior_aux = exponential(1, autoscale=TRUE)` in **rstanarm**
code.

#### Note on data-based priors

Because the scaling is based on the scales of the predictors (and possibly the
outcome) these are technically data-dependent priors. However, since these
priors are quite wide (and in most cases rather conservative), the amount of 
information used is weak and mainly takes into account the order of magnitude of
the variables. This enables __rstanarm__ to offer defaults that are reasonable 
for many models.


### Disabling prior scale adjustments

To disable automatic rescaling simply specify a prior other than the default.
**rstanarm** versions up to and including version `2.19.3` used to require you
to explicitly set the `autoscale` argument to `FALSE`, but now autoscaling only
happens by default for the default priors. To use autoscaling with manually
specified priors you have to set `autoscale = TRUE`. For example, this prior
specification will not include any autoscaling:

```{r, no-autoscale, results="hide", eval = TRUE}
test_no_autoscale <-
  update(
    default_prior_test,
    prior = normal(0, 5),
    prior_intercept = student_t(4, 0, 10),
    prior_aux = cauchy(0, 3)
  )
```

We can verify that the prior scales weren't adjusted by checking
`prior_summary`:

```{r, no-autoscale-prior-summary, eval = TRUE}
prior_summary(test_no_autoscale)
```


# How to Specify Flat Priors (and why you typically shouldn't)

### Uninformative is usually unwarranted and unrealistic (flat is frequently frivolous and fictional)

When "non-informative" or "uninformative" is used in the context of prior 
distributions, it typically refers to a flat (uniform) distribution or a nearly 
flat distribution. Sometimes it may also be used to refer to the 
parameterization-invariant Jeffreys prior. Although __rstanarm__ does not
prevent you from using very diffuse or flat priors, unless the data is
very strong it is wise to avoid them. 

Rarely is it appropriate in any applied setting to use a prior that gives the
same (or nearly the same) probability mass to values near zero as it gives
values bigger than the age of the universe in nanoseconds. Even a much narrower
prior than that, e.g., a normal distribution with $\sigma = 500$, will tend to
put much more probability mass on unreasonable parameter values than reasonable
ones. In fact, using the prior 
$\theta \sim \mathsf{Normal(\mu = 0, \sigma = 500)}$ 
implies some strange prior beliefs. For example, you believe a priori that 
$P(|\theta| < 250) < P(|\theta| > 250)$, 
which can easily be verified by doing the calculation with the normal CDF

```{r, eval = TRUE}
p <- 1 - 2 * pnorm(-250, mean = 0, sd = 500)
print(paste("Pr(-250 < theta < 250) =", round(p, 2)))
```

or via approximation with Monte Carlo draws:

```{r, fig.cap="_There is much more probability mass outside the interval (-250, 250)._", eval = TRUE}
theta <- rnorm(1e5, mean = 0, sd = 500)
p_approx <- mean(abs(theta) < 250)
print(paste("Pr(-250 < theta < 250) =", round(p_approx, 2)))

d <- data.frame(theta, clr = abs(theta) > 250)
library(ggplot2)
ggplot(d, aes(x = theta, fill = clr)) + 
  geom_histogram(binwidth = 5, show.legend = FALSE) + 
  scale_y_continuous(name = "", labels = NULL, expand = c(0,0)) + 
  scale_x_continuous(name = expression(theta), breaks = c(-1000, -250, 250, 1000))
```


This will almost never correspond to the prior beliefs of a researcher about a
parameter in a well-specified applied regression model and yet priors like 
$\theta \sim \mathsf{Normal(\mu = 0, \sigma = 500)}$ (and more extreme) remain quite popular. 

Even when you know very little, a flat or very wide prior will almost never be 
the best approximation to your beliefs about the parameters in your model that 
you can express using __rstanarm__ (or other software). _Some_ amount of prior 
information will be available. For example, even if there is nothing to suggest 
a priori that a particular coefficient will be positive or negative, there is 
almost always enough information to suggest that different orders of magnitude
are not equally likely. Making use of this information when setting a prior
scale parameter is simple ---one heuristic is to set the scale an order of
magnitude bigger than you suspect it to be--- and has the added benefit of
helping to stabilize computations.

A more in-depth discussion of non-informative vs weakly informative priors is 
available in the case study 
[_How the Shape of a Weakly Informative Prior Affects  Inferences_](http://mc-stan.org/users/documentation/case-studies/weakly_informative_shapes.html).

### Specifying flat priors

__rstanarm__ will use flat priors if `NULL` is specified rather than a
distribution. For example, to use a flat prior on regression coefficients you
would specify `prior=NULL`:

```{r, flat-prior-1, echo=FALSE, results="hide", eval = TRUE}
flat_prior_test <- stan_glm(mpg ~ wt, data = mtcars, prior = NULL, iter = 10, chains = 1)
```
```{r, flat-prior-2, eval=FALSE, eval = TRUE}
flat_prior_test <- stan_glm(mpg ~ wt, data = mtcars, prior = NULL)
```

In this case we let __rstanarm__ use the default priors for the intercept and
error standard deviation (we could change that if we wanted), but the 
coefficient on the `wt` variable will have a flat prior. To double check that
indeed a flat prior was used for the coefficient on `wt` we can call
`prior_summary`:

```{r, flat-prior-summary, eval = TRUE}
prior_summary(flat_prior_test)
```


# Informative Prior Distributions

Although the default priors tend to work well, prudent use of more informative
priors is encouraged. For example, suppose we have a linear regression model 
$$y_i \sim \mathsf{Normal}\left(\alpha + \beta_1 x_{1,i} + \beta_2 x_{2,i}, \, \sigma\right)$$ 
and we have evidence (perhaps from previous research on
the same topic) that approximately $\beta_1 \in (-15, -5)$ and 
$\beta_2 \in (-1, 1)$. An example of an informative prior for 
$\boldsymbol{\beta} = (\beta_1, \beta_2)'$ could be

$$
\boldsymbol{\beta} \sim \mathsf{Normal} \left( 
  \begin{pmatrix} -10 \\ 0 \end{pmatrix},
  \begin{pmatrix} 5^2 & 0 \\ 0 & 2^2 \end{pmatrix}
\right),
$$
which sets the prior means at the midpoints of the intervals and then allows for
some wiggle room on either side. If the data are highly informative about the
parameter values (enough to overwhelm the prior) then this prior will yield
similar results to a non-informative prior. But as the amount of data
and/or the signal-to-noise ratio decrease, using a more informative prior
becomes increasingly important.

If the variables `y`, `x1`, and `x2` are in the data frame `dat` then this
model can be specified as
```{r, eval=FALSE}
my_prior <- normal(location = c(-10, 0), scale = c(5, 2))
stan_glm(y ~ x1 + x2, data = dat, prior = my_prior)
```
We left the priors for the intercept and error standard deviation at their 
defaults, but informative priors can be specified for those parameters in 
an analogous manner.

