---
title: "Chapter 3 Beta-Binomial Bayesian Model Notes"
author: "Emanuel Rodriguez"
execute:
    message: false
    warning: false
format:
    html:
        monofont: "Cascadia Mono"
        highlight-style: gruvbox-dark
        css: styles.css
        callout-icon: false
        callout-apperance: simple
        toc: false
        html-math-method: katex
---

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

The chapter is set up with an example of polling results. We are put into 
the scenario where we are managig the campaing for a candidate. We know 
that on average her support based on recent polls is around 45%. In the 
next few sections we'll work through our Bayesian framework and incorporate
a new tool the **Beta-Binomial** model. This model will take develop a 
continuous prior, as opposed to the discrete one's we've been working with 
so far. 


## The Beta prior

:::{.callout-note}
## Probability Density Function

Let $\pi$ be a continuous random variable with probability density
function (pdf) $f(\pi)$. Then $f(\pi)$ has the following properties:

1. $f(\pi) \geq 0$
2. $\int_{\pi}f(\pi)d\pi = 1$ (this is analogous to $\sum$ in the case of pmfs)
3. $P(a < \pi < b) = \int_a^bf(\pi)d\pi$ when $a\leq b$
:::

:::{.callout-tip icon="true"}
a quick note on (1) above. Note that it does not place a restriction on
$f(\pi)$ being less than 1. This means that we can't interpret values of 
$f$ as probabilities, we can however use to interpret plausability of 
two different events, the greater the value of $f$ the more plausible.
To calculate probabilities using $f$ we must determine the area under the
curve it defines, as shown in (3).  
:::

