Sampling from a Box of Two Types of Colored Balls

In TeachBayes: Teaching Bayesian Inference

knitr::opts_chunk$set(fig.width=6, fig.height=4, fig.path='Figs/',
                      echo=TRUE, warning=FALSE, message=FALSE)

Here is a standard sampling-without-replacement problem. Suppose we have $N$ balls in a box, $B$ that are black and the remaining $W = N - B$ balls are white.

You sample $n$ balls from the box and $b$ turn out to be black. What have you learned about the number $B$ that are black in the box?

This is a Bayes' rule problem. We'll illustrate it for the case where $N = 50$.

1. (Prior) First the number of black balls in the box $B$ could be any value from 0 to 50. We place a uniform prior on the values 0, 1, 2, ..., 50.

library(TeachBayes)
bayes_df <- data.frame(B=0:50, Prior=rep(1/51, 51))

1. (Data) We take a sample of $n = 10$ balls without replacement and observe that the number of black balls is $b = 3$. The likelihood is the probability of this outcome, expressed as a function of $B$. I use the special function dsampling().

sample_b <- 3
pop_N <- 50
sample_n <- 10
bayes_df$Likelihood <- dsampling(sample_b, pop_N, 
                                 bayes_df$B, sample_n)

1. Last we turn the Bayesian crank using the bayesian_crank() function and obtain the posterior probabilities for $B$.

bayes_df <- bayesian_crank(bayes_df)

I compare the prior and posterior probabilities for $B$ graphically.

prior_post_plot(bayes_df)

Here is a 90 percent probability interval for $B$:

library(dplyr)
discint(select(bayes_df, B, Posterior), 0.90)

TeachBayes documentation built on May 1, 2019, 9:17 p.m.