---
title: "Conjugate Families"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Conjugate Families}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

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

```{r setup}
library(bayesrules)
```

The **bayesrules** package has a set of functions that support exploring Bayesian models from three conjugate families: **Beta-Binomial**, **Gamma-Poisson**, and **Normal-Normal**. 
The functions either help with plotting (prior, likelihood, and/or posterior) or summarizing the descriptives (mean, mode, variance, and sd) of the prior and/or posterior. 

## The Beta-Binomial Model

We use the Beta-Binomial model to show the different set of functions and the arguments.

### Prior

```{r fig.align='center', fig.height=4, fig.width=5}
plot_beta(alpha = 3, beta = 13, mean = TRUE, mode = TRUE)
```

```{r}
summarize_beta(alpha = 3, beta = 13)
```


### Likelihood

In addition, `plot_binomial_likelihood()` helps users visualize the Binomial likelihood function and shows the maximum likelihood estimate.

```{r fig.align='center', fig.height=4, fig.width=5, message = FALSE}
plot_binomial_likelihood(y = 3, n = 15, mle = TRUE)
```



### Prior-Likelihood-Posterior

The two other functions `plot_beta_binomial()` and `summarize_beta_binomial()` require both the prior parameters and the data for the likelihood. 

```{r fig.align='center', warning = FALSE, fig.height=4, fig.width=5}
plot_beta_binomial(alpha = 3, beta = 13, y = 5, n = 10, 
                   prior = TRUE, #the default 
                   likelihood = TRUE,  #the default
                   posterior = TRUE #the default
                   )
```

```{r fig.align='center', warning = FALSE, fig.height=4, fig.width=5}
summarize_beta_binomial(alpha = 3, beta = 13, y = 5, n = 10)
```

## Other Conjugate-Families

For Gamma-Poisson and Normal-Normal models, the set of functions are similar but the arguments are different for each model. Arguments of the Gamma-Poisson functions include the `shape` and `rate` of the Gamma prior and `sum_y` and `n` arguments related to observed data which represent the sum of observed data values and number of observations respectively.

```{r fig.align='center', warning = FALSE, fig.height=4, fig.width=5}
plot_gamma_poisson(
  shape = 3,
  rate = 4,
  sum_y = 3,
  n = 9,
  prior = TRUE,
  likelihood = TRUE,
  posterior = TRUE
)
```

For the Normal-Normal model functions, the prior Normal model has the `mean` and `sd` argument. The observed data has `sigma`, `y_bar`, and `n` which indicate the standard deviation, mean, and sample size of the data respectively.

```{r}
summarize_normal_normal(mean = 3.8, sd = 1.12, sigma = 5.8, y_bar = 3.35, n = 8)
```