jonas_homework_3.Rmd

---
title: "Chapter 3 Excecise Jonas"
output: html_notebook
---


```{r}
library(tidyverse)
library(rethinking)
p_grid <- seq( from=0 , to=1 , length.out=1000 ) 
prior <- rep( 1 , 1000 ) 
likelihood <- dbinom( 6 , size=9 , prob=p_grid ) 
posterior <- likelihood * prior 
posterior <- posterior / sum(posterior) 
set.seed(100) 
samples <- sample( p_grid , prob=posterior , size=1e4 , replace=TRUE)
```

```{r}
#E1
plot(p_grid, posterior)
dens(samples, add = FALSE)
abline( v=0.2 )
```

```{r}

dens(samples)
abline( v=0.2 )
length(samples[samples<0.2])/length(samples)#E1:  5e-04
length(samples[samples>0.8])/length(samples)#E2: 0.1117
length(samples[(samples>0.2) & (samples<0.8)])/length(samples)#E3: 0.8878
quantile(samples, seq(0,1,0.2))[2]#E4: 0.5195195 
quantile(samples, seq(0,1,0.2))[5]#E5: 0.7567568 
HPDI(samples, p=0.66)#E6: 0.5205205 0.7847848
PI(samples, 0.66)#E7: 0.5005005 0.7687688
```

# Medium
```{r}
library(rethinking)
p_grid <- seq(from=0, to=1.0, length.out=1000)
prior <- rep(1, length(p_grid))
likelihood <- dbinom(8, size=15, prob=p_grid) #TRY with rbinom, avg over results?
posterior.unstandardized  <- likelihood*prior
posterior <- posterior.unstandardized / sum(posterior.unstandardized)
#plot(p_grid, posterior)

samples <- sample( p_grid, 5000, replace=TRUE,prob=posterior) #M2
HPDI(samples,prob = 0.9) #M2: 0.3313313 0.7197197 

#posterior check
dsamples <- rbinom(n=length(samples), size=15, prob = samples)
hist(dsamples)
length(dsamples[dsamples==8])/length(samples) #M3: 0.1436

#Using the posterior distribution constructed from the new (8/15) data, now calculate the probability of observing 6 water in 9 tosses.
#what's the new data? the dsamples
#use old posterior as prior:
newprior <- posterior
newlikelihood <- dbinom(6, 9, prob=p_grid)
newposterior <- newprior * newlikelihood 
newposterior <- newposterior / sum(newposterior)
plot(p_grid, newposterior)
newsamples <- sample(p_grid, 5000, replace = TRUE, prob = newposterior)
hist(newsamples)


```

```{r}
#non flat prior
p_grid <- seq(from=0, to=1.0, length.out=500)
### lolololol
prior <- c(p_grid[p_grid<0.5]*0, p_grid[p_grid>=0.5]/p_grid[p_grid>=0.5] )
likelihood <- dbinom(8, size=15, prob=p_grid) #TRY with rbinom, avg over results?
posterior.unstandardized  <- likelihood*prior
posterior <- posterior.unstandardized / sum(posterior.unstandardized)

#plot(p_grid, posterior)

samples <- sample( p_grid, 5000, replace=TRUE,prob=posterior) #M2
HPDI(samples,prob = 0.9) #M2: 0.3313313 0.7197197 NOW: M4:0.5010020 0.7114228

#posterior check
dsamples <- rbinom(n=length(samples), size=15, prob = samples)
hist(dsamples) ## NOW closer to the data
length(dsamples[dsamples==8])/length(samples) #M3: 0.1436 NOW M$: 0.1482

#Using the posterior distribution constructed from the new (8/15) data, now calculate the probability of observing 6 water in 9 tosses.
#what's the new data? the dsamples
#use old posterior as prior:
newprior <- posterior
newlikelihood <- dbinom(6, 9, prob=p_grid)
newposterior <- newprior * newlikelihood 
newposterior <- newposterior / sum(newposterior)
plot(p_grid, newposterior)
newsamples <- sample(p_grid, 5000, replace = TRUE, prob = newposterior)
hist(newsamples)


```
# HARD


```{r}
library(rethinking)
data(homeworkch3)
#H1
p_grid <- seq(from=0, to=1, length.out = 1000)
prior <- rep(1, length(p_grid))
#
all_births <- c(birth1, birth2)
cboys <- sum(all_births)
size <- length(all_births)


likelihood <- dbinom(x=cboys, size=size, prob = p_grid)#this is mind twisting: we resample from x thus shaping pgrid
posterior.unstandardized <- prior*likelihood
posterior <- posterior.unstandardized / sum(posterior.unstandardized)
plot(p_grid, posterior)

#H2
samples <- sample(p_grid, replace = TRUE, size = 10000, prob = posterior)
hist(samples)
HPDI(samples, prob = .5)
HPDI(samples, prob = .89)
HPDI(samples, prob = .97)

```
```{r}
#H3
sims <- rbinom(n = 10000, size = 200, prob = samples) # samples: values for p drawn from posterior
dens(sims)
abline(v = 111) # looks spot on
```

```{r}
#H4
nboysb1 <- sum(birth1) # 51
# what's the probability to draw nboysb1 from 100?
sims100 <- rbinom(10000, size = 100, prob = samples) #samples: p vals from posterior on 200 data
dens(sims100)
abline(v = nboysb1) # looks spot on


p_grid <- seq(from=0, to=1, length.out = 1000)
prior <- rep(1, length(p_grid))
likelihoodb1 <- dbinom(x=nboysb1, size = length(birth1), prob = p_grid)
posteriorb1.unstandardized <- prior*likelihoodb1 
posteriorb1 <- posteriorb1.unstandardized / sum(posteriorb1.unstandardized)
samplesb1 <- sample(p_grid, replace = TRUE, size = 10000, prob = posteriorb1)

simsb1 <- rbinom(n = 10000, size = 200, prob = samplesb1) # samples: values for p drawn from posterior
dens(sims)
dens(simsb1, add=TRUE, type = 'o',)

abline(v = 111) # looks spot on
```
```{r}
firstgirls <- sum(birth1==0)
firstwasgirl <- birth2[birth1==0]
firstgirls
boyfirstwasgirl <- sum(firstwasgirl)

simboys <- rbinom(n = 10000, size = firstgirls, prob = samples) #10k samples from posterior if idependent
dens(simboys, adj=1,)
abline(v=boyfirstwasgirl,  ) #what an outlier this number of boys following a girl is! probs engineering of designer babies.
```











```{r}
p_grid <- seq( from=0 , to=1 , length.out=1000 ) 
d <- c(birth1, birth2)
boys <- sum(d)
size <- length(d)
prior <- rep( 1 , length( p_grid ) )
likelihood <- dbinom( x=boys, size=size , prob=p_grid )
posterior.unweighted <- likelihood*prior
posterior <- posterior.unweighted / sum(posterior.unweighted)
plot( p_grid , posterior , type='l' )

#H2
samples <- sample(posterior,10000,  replace = TRUE)
hist( samples)
HPDI(samples, 0.97)

#H5
#data(homeworkch3)


#likelihood <- rbinom(p_grid, size=200, prob=p_grid)
#posterior.unweighted <- likelihood*prior
#posterior <- posterior.unweighted / sum(posterior.unweighted)
#hist(posterior)
```