Last lecture we were trying to understand \(\pi\) the acceptance rate of a graduate program in a specific department. Let’s make a fresh start to the same problem. This time we will let \(\pi \in [0,1]\).
Continuous probability models
Let \(\pi\) be a continuous random variable with pdf \(f(\pi)\). Then \(f(\pi)\) has the following properties:
\(\int_\pi f(\pi)d\pi = 1\), ie. the area under \(f(\pi)\) is 1
\(f(\pi) \ge 0\)
\(P(a < \pi < b) = \int_a^b f(\pi) d\pi\) when \(a \le b\)
Interpreting \(f(\pi)\):
\(f(\pi)\) can be used to compare the plausibility of two different values of \(\pi\).
Plotting the continuous prior
For each of the student’s prior ideas for \(\pi\) plot the pdf of the prior. Your plot will not be exact since no exact values are given.
Morteza thinks that it is extremely difficult to get into this program.
Jared thinks that it is difficult to get into this program.
Erin does not have any strong opinions whether it is difficult or easy to get into this program.
Xuan thinks that it is easy to get into this program.
Beyoncé thinks that it is extremely easy to get into this program.
Morteza’s prior
Jared’s prior
Erin’s prior
Xuan’s prior
Beyoncé’s prior
Beta Prior model
Let \(\pi\) be a random variable which can take any value between 0 and 1, ie. \(\pi \in [0,1]\). Then the variability in \(\pi\) might be well modeled by a Beta model with shape parameters\(\alpha > 0\) and \(\beta > 0\):
\[\pi \sim \text{Beta}(\alpha, \beta)\]
The Beta model is specified by continuous pdf \[\begin{equation}
f(\pi) = \frac{\Gamma(\alpha + \beta)}{\Gamma(\alpha)\Gamma(\beta)} \pi^{\alpha-1} (1-\pi)^{\beta-1} \;\; \text{ for } \pi \in [0,1]
\end{equation}\] where \(\Gamma(z) = \int_0^\infty y^{z-1}e^{-y}dy\) and \(\Gamma(z + 1) = z \Gamma(z)\). Fun fact: when \(z\) is a positive integer, then \(\Gamma(z)\) simplifies to \(\Gamma(z) = (z-1)!\).
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.
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 \(Y|n \sim \text{Bin}(n,\pi)\).
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)
