---
title: "Specifying priors for Bayesian regression"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Specifying priors for Bayesian regression}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

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

## General principle

`morie_bayes_lm()` fits a Bayesian linear regression by MCMC. Each
regression coefficient is given an independent zero-mean Normal prior
whose standard deviation is the `prior_sd` hyperparameter. The prior
encodes how large the coefficients are expected to be *before* seeing
the data: a large `prior_sd` is weakly informative, a small one shrinks
estimates toward zero.

## Applied example

```{r}
library(rmorie)
d <- data.frame(x = rnorm(100))
d$y <- 1 + 2 * d$x + rnorm(100)

# weakly informative
fit <- morie_bayes_lm(y ~ x, d, prior_sd = 10, chains = 2, iter = 1000)
print(fit)

# a per-coefficient prior vector is also accepted
fit2 <- morie_bayes_lm(y ~ x, d, prior_sd = c(5, 1))
```

Tighter priors (small `prior_sd`) regularise the fit; inspect
convergence with `morie_bayes_diagnostics(fit)` and the posterior with
`morie_bayes_plot(fit)`.