bayes.Rmd

---
title: "Regression and Other Stories: Elections Economy - Bayes"
author: "Andrew Gelman, Jennifer Hill, Aki Vehtari"
date: "`r format(Sys.Date())`"
output:
  html_document:
    theme: readable
    toc: true
    toc_depth: 2
    toc_float: true
    code_download: true
---
Demonstration of Bayesian information aggregation. See Chapter 9 in
Regression and Other Stories.

-------------


```{r setup, include=FALSE}
knitr::opts_chunk$set(message=FALSE, error=FALSE, warning=FALSE, comment=NA)
# switch this to TRUE to save figures in separate files
savefigs <- FALSE
```

#### Load packages

```{r }
library("rprojroot")
root<-has_file(".ROS-Examples-root")$make_fix_file()
```

## Calculations

Prior based on a previously-fitted model using economic and
political condition.

```{r }
theta_hat_prior <- 0.524
se_prior <- 0.041
```

Survey of 400 people, of whom 190 say they will vote for the
Democratic candidate

```{r }
n <- 400
y <- 190
```

#### Data estimate

```{r }
theta_hat_data <- y/n
se_data <- sqrt((y/n)*(1-y/n)/n)
```

#### Bayes estimate

```{r }
theta_hat_bayes <- (theta_hat_prior/se_prior^2 + theta_hat_data/se_data^2) / (1/se_prior^2 + 1/se_data^2)
se_bayes <- sqrt(1/(1/se_prior^2 + 1/se_data^2))
```

## Figures


```{r eval=FALSE, include=FALSE}
if (savefigs) pdf(root("ElectionsEconomy/figs","prior_data_posterior_a.pdf", height=3, width=5.5))
```
```{r }
par(mar=c(3,1,1,1), mgp=c(1.5, 0.5, 0), tck=-.02)
plot(0, 0, xlim=c(0.37, 0.67), ylim=c(0, 20), xlab=expression(theta), xaxt="n", ylab="", yaxs="i", yaxt="n", bty="n", cex.lab=1.2)
axis(1, seq(0.3, 0.7, 0.1))
curve(dnorm(x, theta_hat_prior, se_prior), n=1000, add=TRUE)
text(0.588, 5, "Prior")
curve(dnorm(x, theta_hat_data, se_data), n=1000, add=TRUE)
text(0.420, 8, "Likelihood")
```
```{r eval=FALSE, include=FALSE}
if (savefigs) dev.off()
```
```{r eval=FALSE, include=FALSE}
if (savefigs) pdf(root("ElectionsEconomy/figs","prior_data_posterior_b.pdf", height=3, width=5.5))
```
```{r }
par(mar=c(3,1,1,1), mgp=c(1.5, 0.5, 0), tck=-.02)
plot(0, 0, xlim=c(0.37, 0.67), ylim=c(0, 20), xlab=expression(theta), xaxt="n", ylab="", yaxs="i", yaxt="n", bty="n", cex.lab=1.2)
axis(1, seq(0.3, 0.7, 0.1))
curve(dnorm(x, theta_hat_prior, se_prior), n=1000, add=TRUE, col="gray30")
text(0.588, 5, "Prior")
curve(dnorm(x, theta_hat_data, se_data), n=1000, add=TRUE, col="gray30")
text(0.420, 8, "Likelihood")
curve(dnorm(x, theta_hat_bayes, se_bayes), n=1000, add=TRUE)
text(0.525, 15, "Posterior")
```
```{r eval=FALSE, include=FALSE}
if (savefigs) dev.off()
```