demo6_3.Rmd

---
title: "Bayesian data analysis demo 6.3"
author: "Aki Vehtari, Markus Paasiniemi"
date: "`r format(Sys.Date())`"
output:
  html_document:
    theme: readable
    code_download: true
---
## Posterior predictive checking demo

Light speed example with poorly chosen test statistic

ggplot2 is used for plotting, tidyr for manipulating data frames

```{r setup, message=FALSE, error=FALSE, warning=FALSE}
library(ggplot2)
theme_set(theme_minimal())
library(tidyr)
library(latex2exp)
library(rprojroot)
root<-has_file(".BDA_R_demos_root")$make_fix_file()
```

Data

```{r }
y <- read.table(root("demos_ch6","light.txt"))$V1
```

Sufficient statistics

```{r }
n <- length(y)
s <- sd(y)
my <- mean(y)
```

Replicated data

```{r }
sampt <- replicate(1000, rt(n, n-1)*sqrt(1+1/n)*s+my) %>%
  as.data.frame()
```

Test statistic here is variance, which is not a good choice as the
model has variance parameter and the posterior of that parameter
has been updated by the same data used now in checking.

```{r }
sampt_vars <- data.frame(x = sapply(sampt, var))
```

Plot test statistics for the data and replicates.
Vertical line corresponds to the original data, and
the histogram to the replicate data.

```{r }
title1 <- 'Light speed example with poorly chosen test statistic
Pr(T(yrep,theta) <= T(y,theta)|y)=0.42'
ggplot(data = sampt_vars) +
  geom_histogram(aes(x = x), fill = 'steelblue',
                 color = 'black', binwidth = 6) +
  geom_vline(aes(xintercept = x), data = data.frame(x = s^2),
             color = 'red') +
  labs(x = TeX('Variance of \\mathit{y} and \\mathit{y}^{\\mathrm{rep}}'),
       y = '', title = title1) +
  scale_y_continuous(breaks=NULL)
```