class: title-slide <br> <br> .right-panel[ # The Beta-Binomial Model ## Dr. Mine Dogucu Examples from [bayesrulesbook.com](https://bayesrulesbook.com) ] --- ## 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. <img src="slide-02c-beta-binomial_files/figure-html/unnamed-chunk-3-1.png" style="display: block; margin: auto;" /> --- ## Plotting the prior ```r plot_beta(5, 35) ``` <img src="slide-02c-beta-binomial_files/figure-html/unnamed-chunk-5-1.png" style="display: block; margin: auto;" /> --- ## Summarizing the prior ```r 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 ```r plot_beta(30, 60) ``` <img src="slide-02c-beta-binomial_files/figure-html/unnamed-chunk-8-1.png" style="display: block; margin: auto;" /> --- ## Summarizing the posterior ```r summarize_beta(30,60) ``` ``` ## mean mode var ## 1 0.3333333 0.3295455 0.002442002 ``` --- ## Plot summary ```r plot_beta(30, 60, mean = TRUE, mode = TRUE) ``` <img src="slide-02c-beta-binomial_files/figure-html/unnamed-chunk-11-1.png" style="display: block; margin: auto;" /> --- ## Bike ownership: balancing act ```r plot_beta_binomial(alpha = 5, beta = 35, y = 25, n = 50) ``` <img src="slide-02c-beta-binomial_files/figure-html/unnamed-chunk-13-1.png" style="display: block; margin: auto;" /> --- ## 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? -- ```r 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 ```