Posterior Prediction

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# Posterior Prediction
## Dr. Mine Dogucu
Examples from [bayesrulesbook.com](https://bayesrulesbook.com)

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### Posterior prediction of new outcomes

Suppose we get our hands on 20 more artworks.
What number would you _predict_ are done by artists that are Gen X or younger?

- __sampling variability__

- __posterior variability__ in `$\pi$`.\index{sampling variability}

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First, let `$Y' = y'$` be the (yet unknown) number of the 20 new artworks that are done by Gen X or younger artists.
Thus conditioned on `$\pi$`, the randomness in `$Y'$` can be modeled by `$Y'|\pi \sim \text{Bin}(20,\pi)$` with pdf

`$$f(y'|\pi) = P(Y' = y' | \pi) = \binom{20}{ y'} \pi^{y'}(1-\pi)^{20 - y'}$$`
--

`$$f(y'|\pi) f(\pi|(y=14)) \; ,$$`
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---
class: middle

`$f(y'|(y=14)) = \int_0^1 f(y'|\pi) f(\pi|(y=14)) d\pi$`

`$f(y'|(y=14)) = {20 \choose y'} \frac{\Gamma(110)}{\Gamma(18)\Gamma(92)}\frac{\Gamma(18+y')\Gamma(112-y')} {\Gamma(130)}  \text{ for } y' \in \{0,1,\ldots,20\}$`

`$f((y'=3)|(y=14)) = {20 \choose 3}\frac{\Gamma(110)}{\Gamma(18)\Gamma(92)}\frac{\Gamma(18+3)\Gamma(112-3)} {\Gamma(130)} = 0.2217$`

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Let `$Y'$` denote a new outcome of variable `$Y$`.  Further, let pdf `$f(y'|\pi)$` denote the dependence of `$Y'$` on `$\pi$` and posterior pdf `$f(\pi|y)$` denote the posterior plausibility of `$\pi$` given the original data `$Y = y$`.  Then the posterior predictive model for `$Y'$` has pdf

`$$f(y'|y) = \int f(y'|\pi) f(\pi|y) d\pi \; .$$`

In words, the overall chance of observing `$Y' = y'$` weights the chance of observing this outcome under _any_ possible `$\pi$` ( `$f(y'|\pi)$` ) by the posterior plausibility of `$\pi$` ( `$f(\pi|y)$` ).

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```r
head(art_chains_df, 3)
```

```
##          pi
## 1 0.1300708
## 2 0.1755003
## 3 0.2214110
```

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```r
set.seed(84735)
# Predict a value of Y' for each pi value in the chain
art_chains_df <- art_chains_df %>% 
  mutate(y_predict = rbinom(length(pi), size = 20, prob = pi))
```

```r
head(art_chains_df, 3)
```

```
##          pi y_predict
## 1 0.1300708         4
## 2 0.1755003         4
## 3 0.2214110         4
```

---

```r
ggplot(art_chains_df, aes(x = y_predict)) + 
  stat_count()
```