Simulating the Beta-Binomial

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# Simulating the Beta-Binomial
## Dr. Mine Dogucu
Examples from [bayesrulesbook.com](https://bayesrulesbook.com)

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## Prior

```r
plot_beta(45, 55)
```

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## Data and the Posterior

```r
summarize_beta_binomial(45, 55, y = 30, n = 50)
```

```
##       model alpha beta mean      mode         var         sd
## 1     prior    45   55 0.45 0.4489796 0.002450495 0.04950248
## 2 posterior    75   75 0.50 0.5000000 0.001655629 0.04068942
```

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```r
plot_beta_binomial(45, 55, y = 30, n = 50)
```

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```r
library(tidyverse)
set.seed(84735)
```

```r
michelle_sim <- data.frame(pi = rbeta(10000, 45, 55)) 
```

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```
##          pi
## 1 0.4884391
## 2 0.4430527
## 3 0.4953429
## 4 0.4633678
## 5 0.4963052
## 6 0.5495968
```

```
##              pi
## 9995  0.4741776
## 9996  0.3756744
## 9997  0.4660905
## 9998  0.4791408
## 9999  0.4394663
## 10000 0.4229081
```

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```r
set.seed(84735)
michelle_sim <- data.frame(pi = rbeta(10000, 45, 55)) %>% 
  mutate(y = rbinom(10000, size = 50, prob = pi))
```

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```
##          pi  y
## 1 0.4884391 23
## 2 0.4430527 14
## 3 0.4953429 35
## 4 0.4633678 29
## 5 0.4963052 36
## 6 0.5495968 26
```

```
##              pi  y
## 9995  0.4741776 24
## 9996  0.3756744 13
## 9997  0.4660905 25
## 9998  0.4791408 21
## 9999  0.4394663 24
## 10000 0.4229081 21
```

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```r
michelle_posterior <- michelle_sim %>% 
  filter(y == 30)
```

```r
nrow(michelle_posterior)
```

```
## [1] 211
```

`\(f(\pi|y =30)\)`
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```r
ggplot(michelle_posterior, aes(x = pi)) + 
  geom_histogram(color = "white", binwidth = 0.025)
```

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```r
ggplot(michelle_posterior, aes(x = pi)) + 
  geom_density() +
  xlim(0,1)
```

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