---
title: "STAT 491 - Lecture 1"
date: January 11, 2018
output: pdf_document
---

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

## Bayesian Thought Experiment

There are two foundational elements in a Bayesian Analysis:

1. Bayesian inference is a re-allocation of credibility across possibilities
2. These possibilities are parameter values in meaningful mathematical models

### Guess Who Exercise
Consider the game Guess Who, where the goal is to ask questions to identify an opposing player's character.

![Guess Who Faces](images/guesswho.jpg)

\pagebreak

Given the line up of suspects above, construct a set of probabilities for each character. Note these should sum to one and constitute the first foundation element in Bayesian Analysis. The first set of probabilities are known as *prior* probabilities.

```{r echo=FALSE}
players <- c('Megan','Donna','Clark','Ally','Grace','Wyatt')
label <- 1:6
prob <- rep(1/6,6)
plot(label,prob, type='n', axes=F, ylab='Probability', xlab='Character')
box()
axis(1, labels = players, at = 1:6)
```

Formally these probabilities are parameter values in a multinomial model with one selection or object. The mathematical notation associated with this is known as a probability mass function (more details in a few weeks). 

> Follow the conversation and update your probabilities accordingly

> - **You**: Does your character have a hat
> - **Your adversary**: Yes

```{r echo=FALSE}
players <- c('Megan','Donna','Clark','Ally','Grace','Wyatt')
label <- 1:6
prob <- rep(1/6,6)
plot(label,prob, type='n', axes=F, ylab='Probability', xlab='Character')
box()
axis(1, labels = players, at = 1:6)
```

\newpage

> Follow the conversation and update your probabilities accordingly, again

> - **You**: Is your character wearing glasses
> - **Your adversary**: Yes

```{r echo=FALSE}
players <- c('Megan','Donna','Clark','Ally','Grace','Wyatt')
label <- 1:6
prob <- rep(1/6,6)
plot(label,prob, type='n', axes=F, ylab='Probability', xlab='Character')
box()
axis(1, labels = players, at = 1:6)
```

> Follow the conversation and update your probabilities accordingly, again

> - **You**: Is your character wearing purple glasses
> - **Your adversary**: No

```{r echo=FALSE}
players <- c('Megan','Donna','Clark','Ally','Grace','Wyatt')
label <- 1:6
prob <- rep(1/6,6)
plot(label,prob, type='n', axes=F, ylab='Probability', xlab='Character')
box()
axis(1, labels = players, at = 1:6)
```

\newpage

### Key Points From This Exercise

- *Thinking with distributions:*
 \vfill

- *Specifying a prior distribution:*
\vfill

- *Update distribution with additional data or evidence:*
\vfill