demo2_1.Rmd

---
title: "Bayesian data analysis demo 2.1"
author: "Aki Vehtari, Markus Paasiniemi"
date: "`r format(Sys.Date())`"
output:
  html_document:
    theme: readable
    code_download: true
---
## Probability of a girl birth given placenta previa (BDA3 p. 37).

437 girls and 543 boys have been observed. Calculate and plot the
posterior distribution of the proportion of girls $\theta$, using
uniform prior on $\theta$.

ggplot2 is used for plotting
to install new packages, type e.g. install.packages('ggplot2')

```{r setup, message=FALSE, error=FALSE, warning=FALSE}
library(ggplot2)
theme_set(theme_minimal())
```

Posterior is Beta(438,544)

```{r }
# seq creates evenly spaced values
df1 <- data.frame(theta = seq(0.375, 0.525, 0.001)) 
a <- 438
b <- 544
# dbeta computes the posterior density
df1$p <- dbeta(df1$theta, a, b)
```

compute also 95% central interval

```{r }
# seq creates evenly spaced values from 2.5% quantile
# to 97.5% quantile (i.e., 95% central interval)
# qbeta computes the value for a given quantile given parameters a and b
df2 <- data.frame(theta = seq(qbeta(0.025, a, b), qbeta(0.975, a, b), length.out = 100))
# compute the posterior density
df2$p <- dbeta(df2$theta, a, b)
```

Plot posterior (Beta(438,544))
and 48.8% line for population average

```{r }
ggplot(mapping = aes(theta, p)) +
  geom_line(data = df1) +
  # Add a layer of colorized 95% posterior interval
  geom_area(data = df2, aes(fill='1')) +
  # Add the proportion of girl babies in general population
  geom_vline(xintercept = 0.488, linetype='dotted') +
  # Decorate the plot a little
  labs(title='Uniform prior -> Posterior is Beta(438,544)', y = '') +
  scale_y_continuous(expand = c(0, 0.1), breaks = NULL) +
  scale_fill_manual(values = 'lightblue', labels = '95% posterior interval') +
  theme(legend.position = 'bottom', legend.title = element_blank())
```