The Beta Binomial Model

ISI-BUDS 2023

Bike ownership

The transportation office at our school wants to know \(\pi\) the proportion of people who own bikes on campus so that they can build bike racks accordingly. The staff at the office think that the \(\pi\) is more likely to be somewhere 0.05 to 0.25. The plot below shows their prior distribution. Write out a reasonable \(f(\pi)\). Calculate the prior expected value, mode, and variance.

Plotting the prior

plot_beta(5, 35)

Summarizing the prior

summarize_beta(5, 35)

   mean      mode         var         sd
1 0.125 0.1052632 0.002667683 0.05164962

Posterior for the Beta-Binomial Model

Let \(\pi \sim \text{Beta}(\alpha, \beta)\) and \(Y|n \sim \text{Bin}(n,\pi)\).

\(f(\pi|y) \propto \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}\pi^{\alpha-1} (1-\pi)^{\beta-1} {n \choose y}\pi^y(1-\pi)^{n-y}\)

\(f(\pi|y) \propto \pi^{(\alpha+y)-1} (1-\pi)^{(\beta+n-y)-1}\)

\(\pi|y \sim \text{Beta}(\alpha +y, \beta+n-y)\)

\(f(\pi|y) = \frac{\Gamma(\alpha+\beta+n)}{\Gamma(\alpha+y)\Gamma(\beta+n-y)} \pi^{(\alpha+y)-1} (1-\pi)^{(\beta+n-y)-1}\)

Conjugate prior

We say that \(f(\pi)\) is a conjugate prior for \(L(\pi|y)\) if the posterior, \(f(\pi|y) \propto f(\pi)L(\pi|y)\), is from the same model family as the prior.

Thus, Beta distribution is a conjugate prior for the Binomial likelihood model since the posterior also follows a Beta distribution.

Bike ownership posterior

The transportation office decides to collect some data and samples 50 people on campus and asks them whether they own a bike or not. It turns out that 25 of them do! What is the posterior distribution of \(\pi\) after having observed this data?

\(\pi|y \sim \text{Beta}(\alpha +y, \beta+n-y)\)

\(\pi|y \sim \text{Beta}(5 +25, 35+50-25)\)

\(\pi|y \sim \text{Beta}(30, 60)\)

Plotting the posterior

plot_beta(30, 60)

Summarizing the posterior

summarize_beta(30,60)

       mean      mode         var         sd
1 0.3333333 0.3295455 0.002442002 0.04941662

Plot summary

plot_beta(30, 60, mean = TRUE, mode = TRUE)

Bike ownership: balancing act

plot_beta_binomial(alpha = 5, beta = 35,
                   y = 25, n = 50)

Posterior Descriptives

\(\pi|(Y=y) \sim \text{Beta}(\alpha+y, \beta+n-y)\)

\[E(\pi | (Y=y)) = \frac{\alpha + y}{\alpha + \beta + n}\] \[\text{Mode}(\pi | (Y=y)) = \frac{\alpha + y - 1}{\alpha + \beta + n - 2} \] \[\text{Var}(\pi | (Y=y)) = \frac{(\alpha + y)(\beta + n - y)}{(\alpha + \beta + n)^2(\alpha + \beta + n + 1)}\\\]

Bike ownership - descriptives of the posterior

What is the descriptive measures (expected value, mode, and variance) of the posterior distribution for the bike ownership example?

summarize_beta_binomial(5, 35, y = 25, n = 50)

      model alpha beta      mean      mode         var         sd
1     prior     5   35 0.1250000 0.1052632 0.002667683 0.05164962
2 posterior    30   60 0.3333333 0.3295455 0.002442002 0.04941662