---
title: "Priors for dynamic multivariate panel models"
link-citations: yes
output:
  bookdown::html_document2:
    base_format: rmarkdown::html_vignette
pkgdown:
  as_is: true
vignette: >
  %\VignetteIndexEntry{Priors for dynamic multivariate panel models} 
  %\VignetteEngine{knitr::rmarkdown} 
  %\VignetteEncoding{UTF-8}
---

```{r, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)
```

```{r srr, eval = FALSE, echo = FALSE}
#' @srrstats {BS1.2b} The package contains a vignette.
```

```{r setup, echo = FALSE, warning=FALSE}
library("dynamite")
library("ggplot2")
suppressPackageStartupMessages(library("dplyr"))
theme_set(theme_bw())
options(dplyr.summarise.inform = FALSE)
options(crayon.enabled = FALSE)
options(dplyr.summarise.inform = FALSE)
set.seed(0)
data.table::setDTthreads(1) # For CRAN
```

# Default prior distributions in dynamite

The default priors in `dynamite` are chosen to be relatively uninformative (i.e., weakly informative) such that they provide computational stability by slightly regularizing the posterior. The motivation behind the default priors is thus similar to other popular Stan-based R packages such as `brms`^[https://CRAN.R-project.org/package=brms] and `rstanarm`^[https://CRAN.R-project.org/package=rstanarm].

Define $\sigma_x=\max(1, \text{SD}(x))$, where $\mbox{SD}(x)$ is the standard deviation of the predictor variable $x$ over groups and non-fixed time points (see section "Lagged responses and predictors" in the main vignette for more information: `vignette("dynamite", package = "dynamite")`). Define also
\[
\sigma_y = \begin{cases}
\max(1, \text{SD}(y)), &\text{if family is gaussian or student}\\ 
1, &\text{otherwise}
\end{cases},
\] 
where $\mbox{SD}(y)$ is the standard deviation of the response variable as over groups and non-fixed time points.

## Regression coefficients

We define the default prior for a time-invariant regression coefficients $\beta$ as well for the first time-varying coefficient $\delta_1$ as zero-mean normal distribution with standard deviation of 
$2\sigma_y/\sigma_x$. The two maximums used in definitions of $\sigma_y$ and $\sigma_x$ ensure that the prior standard deviation is at least 2.

## Intercept

As the posterior correlations between the intercept $\alpha$ and the regression coefficients $\beta$ and $\delta$ can be cause computational inefficiencies with gradient-based sampling algorithms of Stan, sampling of the intercept is performed indirectly via parameter $a$, so that the intercept $\alpha$ (or $\alpha_1$ in case of time-varying intercept) is constructed as 
\[
\alpha = a - \bar X^\beta_1\beta - \bar X_1^\delta\delta_1,
\]
where $\bar X^\beta$ and $\bar X^\delta$ are the means of the corresponding predictors at first (non-fixed) time point. The prior is then defined for $a$ as
\[
a \sim \mbox{N}(\bar y, 2\sigma_y),
\]
where $\bar y$ is mean of the response variable values at first time point after applying the link function (except in case of categorical and multinomial response where $\bar y$ is set to zero).

## Standard deviation parameters

The prior for the standard deviation parameter $\tau$ of the random walk prior of the spline coefficients is half-normal with the standard deviation of $2\sigma_y/\sigma_x$ (with $\sigma_x=1$ in case of a time-varying intercept). The same prior distribution is also used for the the standard deviations of the random effects. Note that this prior can be too uninformative in some cases especially for the $\tau$ parameters. A boundary (zero) avoiding $Gamma(2, z)$ prior with suitable value of $z$ can often work better for $\tau$ when the default prior leads to divergences (alternatively, in some occasions switching between centered and noncentered parameterization in `splines` can help).

## Correlation matrices

The correlation matrix of the random effects and latent factors uses a $\mbox{LKJ}(1)$ prior with the Cholesky parameterization, see Stan documentation of `lkj_corr_cholesky` for details^[https://mc-stan.org/docs/functions-reference/cholesky-lkj-correlation-distribution.html]. The default corresponds to uniform distribution over valid correlation matrices.

## Parameters related to latent factors

When defining latent factor term with `nonzero_lambda = TRUE`, priors are set for $\zeta$ and $\kappa$, with defaults $\kappa \sim N(0, 1)$ and $\kappa \sim Beta(2, 2)$. These are used to define standard deviation $\sigma_\lambda$ of the loadings as $\sigma_\lambda = \kappa \zeta$, and $\tau_\psi = (1 - \kappa)\zeta$, the standard deviation of the random walk prior of $\psi_t$.

In case where `nonzero_lambda = FALSE`, $\tau_\psi$ is fixed to one, and prior is set directly on $\sigma_\lambda $ as $N(0, 1)$. In both cases, prior on first $\psi_1 \sim N(0, 1)$.

## Family-specific parameters

For standard deviation parameter of gaussian and student's $t$ responses, we use exponential prior with a rate parameter $\frac{1}{2\max(1,\sigma_y)}$. The degrees-of-freedom parameter of the student's $t$-distribution has a $\mbox{Gamma}(2, 0.1)$ prior, whereas for other family specific parameters we set $\phi \sim \mbox{Exponential}(1)$.

# Defining priors of dynamite models

While `dynamite()` can be used with the default prior choices, we recommend to check whether the defaults make sense for your particular problem and to codify them accordingly based on the domain knowledge. 

Any univariate unbounded continuous distributions supported by Stan can be used as a prior for univariate model parameters (in case the parameter is constrained to be positive, the distribution is automatically truncated). Naturally, any univariate distribution bounded to the positive real line can be used as a prior for parameters constrained to be positive (e.g., a standard deviation parameter).
See the Stan function reference at \url{https://mc-stan.org/users/documentation/} for possible distributions. For custom priors, you should first get the default priors with the `get_priors()` function, and then modify the `priors` column of the obtained data frame before supplying it to the `dynamite()` function:

```{r}
f <- obs(y ~ -1 + z + varying(~ x + lag(y)) + random(~1 + z), "gaussian") +
  random_spec(correlated = TRUE) + splines(df = 20)
p <- get_priors(f, data = gaussian_example, time = "time", group = "id")
p
```
```{r}
p$prior[p$type == "sigma_nu"] <- "normal(0, 1)" # change prior for sigma_nu
p$prior[p$parameter == "sigma_y"] <- "student_t(3, 0, 2)" # prior for sigma_y
p
```
```{r, eval = FALSE}
fit <- dynamite(f, data = gaussian_example, time = "time",  group = "id",
                priors = p)
```
