Problem 13.3.Rmd

---
title: "R Notebook"
output: html_notebook
---
# 13.1.1
```{r}
mean(rbeta(n = 100, shape1 = 1, shape2 = 1))
```
```{r}
fPriorPredictive <- function(numSamples, a, b, N){
  lX <- vector(length = numSamples)
  for(i in 1:numSamples){
    theta <- rbeta(n = 1, shape1 = a, shape2 = b)
    lX[i] <- rbinom(n = 1, size = N, prob = theta )
    #cat("\n\n theta: ", theta)
    #cat("\n lX[i] : ", lX[i])
  }
  
  return(lX)
}

lX <- fPriorPredictive(1000, 1, 1, 100)
cat("\n Mean: ", mean(lX))
hist(lX)
```
#13.3.2
```{r}
curve(dbeta(x, shape1 = 7, shape2 = 102))
cat(7/102)
abline(v=7/102)
```

#13.3.3
```{r}
fPosteriorIndependent <- function(numSamples, a, b, N){
  lX <- vector(length = numSamples)
  for(i in 1:numSamples){
    lX[i] <- rbeta(n = 1,shape1 = a, shape2 = b)
  }
  
  return(lX)
}

lX <- fPosteriorIndependent(100, 7, 95, 100)
cat("\n Mean: ", mean(lX))
#hist(lX)
#cat("\nlX", lX)
plot(density(lX), col="blue", ylim=c(0,20))
curve(dbeta(x, shape1 = 7, shape2 = 95), add=T, col="orange")

```
#13.1.4
```{r}
all_means <- vector()
all_sds <- vector()
j = 1
for ( i in seq(from=1, to=1000, by = 1)){
  temp <- fPosteriorIndependent(i, 7, 95, 100)
  all_means[j] <- mean(temp)
  all_sds[j] <- sd(temp, na.rm = TRUE)/sqrt(i)
  j <- j + 1
}
```

```{r}
plot(all_means, ylim = c(0.05, 0.1))
abline(h=0.06862745, col="red")
mean(all_means)

#cat(all_sds)
plot(all_sds, ylim = c(0.000, 0.008))
lines(all_sds)
summary(all_sds)
```
#13.1.5
```{r}
all_means <- vector()
all_sds <- vector()
all_variance <- vector()
j = 1
for ( i in seq(from=1, to=1000, by = 1)){
  temp <- fPosteriorIndependent(i, 7, 95, 100)
  all_means[j] <- mean(temp)
  all_sds[j] <- sd(temp, na.rm = TRUE)/sqrt(i)
  all_variance[j] <- var(temp)
  j <- j + 1
}

```

```{r}
hist(all_variance)
```
#13.1.6
```{r}
fproposal <- function(theta, step_size){
  theta_proposed <- rnorm(n=1, mean=theta, sd=step_size) %% 1
  cat("\n Proposing theta: ", theta_proposed)
  return (theta_proposed)

}
```

```{r}
facceptReject <- function(theta_current, theta_proposed){
  r <- ((pbinom(q=6, size = 100, prob = theta_proposed) * pbeta(q = theta_proposed, shape1 = 1, shape2 = 1)))  / ((pbinom(q=6, size = 100, prob = theta_current) *  pbeta(q = theta_current, shape1 = 1, shape2 = 1)))

  cat("\nr : ", r)
  if (r > runif(1, min = 0, max = 1)){
    return(theta_proposed)
  }
  else{
    return(theta_current)
  }
  
  
  
  # r <- 0
  # likelihood_current <- pbinom(q=6, size = 100, prob = theta_current)
  # prior_current <- pbeta(q = theta_current, shape1 = 1, shape2 = 1)
  # 
  # likelihood_proposed <- pbinom(q=6, size = 100, prob = theta_proposed)
  # prior_proposed <- pbeta(q = theta_proposed, shape1 = 1, shape2 = 1)
  # 
  # if (likelihood_proposed * prior_proposed >= likelihood_current * prior_current){
  #   r <- 1
  # }
  # else{
  #   r <- (likelihood_proposed * prior_proposed) / (likelihood_current * prior_current)
  # }
  # cat ("\n\nr :",r)
  # if(r > runif(1, 0,1)){
  #   return(theta_proposed)
  # }
  # else{
  #   return(theta_current)
  # }
  # 
}


```


#13.1.8
```{r}
  
frwm <- function(theta_current, step_size = 0.1){
  cat("\n\n theta_current: ", theta_current)
  theta_proposed <- fproposal(theta_current,step_size)
  cat("\ntheta_proposed:", theta_proposed)
  theta_accepted <- facceptReject(theta_current,theta_proposed)
  cat("\nAccepted theta: ", theta_accepted)
  return(theta_accepted)
}
```

#13.1.9

```{r}
fsampler <- function(numSamples, step_size = 0.1){
  ltheta <- vector()
  theta <- runif(n = 1, min = 0, max = 1)
  for (t in seq(numSamples)){
    ltheta[t] <- frwm(theta, step_size)
  }
  return(ltheta)
}
step_size <- 0.1
s <- fsampler(100,step_size )
plot(density(s))
cat("\n\n\n Mean:", mean(s))
```