---
title: "Statistical Rethinking Chapter 2 Problems"
author: "John Fox"
output: html_document
---

#### `r as.character(Sys.Date())`

```{r echo=FALSE}
# include this code chunk as-is to set options
knitr::opts_chunk$set(comment=NA, prompt=TRUE, fig.height=8, fig.width=8) 
#    use fig.height=6.5, fig.width=6.5 for word output
```

2H1
----

The parameter is the unknown species, either A or B. Prior probabilities are equal (to 1/2). The datum is the observation of a twin birth, and the likelihood function is the probability of a twin birth for each value of p. The posterior distribution is the normalized product of the two:

```{r}
prior1 <- rep(0.5, 2) # Pr(p = A) = Pr(p = B) = 1/2
names(prior1) <- c("A", "B")
likelihood1 <- c(0.1, 0.2)  # Pr(twin | p) for p = A, B
names(likelihood1) <- c("A", "B")
posterior1 <- prior1*likelihood1
library(MASS)  # for fractions()
fractions(posterior1 <- posterior1/sum(posterior1)) # Pr(p | twin), normalized
```
Then by the law of total probability:

```{r}
fractions(sum(posterior1*likelihood1))  # Pr(p = A) Pr(twin | p = A) + Pr(p = B) Pr(twin | p = B)
```


2H2
----

The posterior probability of species A was already computed:

```{r}
fractions(posterior1["A"])
```
2H3
----

The updated prior is the previous posterior, and the new datum is the observation of a singleton birth:

```{r}
prior2 <- posterior1  
likelihood2 <- 1 - likelihood1  # Pr(singleton | p) = 1 - Pr(twin | p)
posterior2 <- prior2*likelihood2
(posterior2 <- posterior2/sum(posterior2))["A"]
```

2H4
----

Ignoring the information on births returns us to the equal-probability prior for species A or B. The datum is the test result for A.

```{r}
likelihood3 <- c(0.8, 1 - 0.65)  # Pr (test is A | p) for p = A, B
names(likelihood3) <-  c("A", "B")
posterior3 <- prior1*likelihood3
(posterior3 <- posterior3/sum(posterior3))["A"]
```

Using the information on births, we take the posterior after the second birth (i.e., twin followed by singleton) as the prior:

```{r}
prior3 <- posterior2
posterior4 <- prior3*likelihood3
(posterior4 <- posterior4/sum(posterior4))["A"]
```