The Beta-Binomial Model

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# The Beta-Binomial Model
### Dr. Dogucu

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<div class="my-footer"> From Bayes Rules! book
 Copyright &copy; Drs. Alicia Johnson, Miles Ott & Mine Dogucu. <a href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a></div>

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## Posterior for the Beta-Binomial Model

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

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

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

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

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

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## Conjugate prior

We say that `$f(\pi)$` is a conjugate prior for `$L(\pi|x)$` if the posterior, `$f(\pi|x) \propto f(\pi)L(\pi|x)$`, 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.

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## 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.

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## Plotting the prior

```r
plot_beta(5, 35) 
```

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## Summarizing the prior

```r
summarize_beta(5, 35)
```

```
##    mean      mode         var
## 1 0.125 0.1052632 0.002667683
```

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## 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?

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

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

`$\pi|x \sim \text{Beta}(30, 60)$`

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## Plotting the posterior

```r
plot_beta(30, 60) 
```

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## 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) 
```

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## Bike ownership: balancing act

```r
plot_beta_binomial(alpha = 5, beta = 35,
                   x = 25, n = 50)
```

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## Posterior Descriptives

`$\pi|(X=x) \sim \text{Beta}(\alpha+x, \beta+n-x)$`

`$$E(\pi | (X=x)) = \frac{\alpha + x}{\alpha + \beta + n}$$` 
$$\text{Mode}(\pi | (X=x))  = \frac{\alpha + x - 1}{\alpha + \beta + n - 2} $$
`$$\text{Var}(\pi | (X=x))   = \frac{(\alpha + x)(\beta + n - x)}{(\alpha + \beta + n)^2(\alpha + \beta + n + 1)}\\$$`
---

## 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, x = 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
```