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
## 1 0.125 0.1052632 0.002667683

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
## 1 0.3333333 0.3295455 0.002442002

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
## 1     prior     5   35 0.1250000 0.1052632 0.002667683
## 2 posterior    30   60 0.3333333 0.3295455 0.002442002