---
title: "Families, link functions and parameters"
author: "Robin Evans"
date: "`r Sys.Date()`"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Families}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r setup, include=FALSE}
knitr::opts_chunk$set(echo = TRUE, eval = TRUE, output = TRUE)
```

```{r package, message=FALSE}
library(causl)
```

## Families

The list of built-in families can be obtained by calling `family_vals`.  This
shows the name of each family, and the value that can be used to obtain it when
calling functions.  
```{family_vals, include=TRUE}
family_vals$val
```
The parameterizations for these families are based on 
generalized linear models.  Essentially, we have that 
\begin{align*}
\mathbb{E} [Y \mid X=x] = \mu(x) = g^{-1}(\eta(x))
\end{align*}
where $\eta(x) = x^T\beta$ for a vector of real parameters $\beta$.

Note that the beta and lognormal families are not actually exponential dispersion
families, so they do not share all the nice properties of full GLM distributions.
Here we include a table of the different models used.

```{r}
fams <- family_vals[-1,]
fams$val[5] <- "0/5"
fams$canon <- c("identity", "identity", "inverse", NA, "logistic", NA, "log-linear", "log-linear")
fams$default <- c("-", "-", "log", "logistic", "-", "exponential", "-", "-")
```