#Task 1	

y <- c(3,6,3,5,9,14,12,11,19,18,15,4,1,6,11,21,11,3,7,18)

#Task 2

alpha <- 0.1
beta <- 0.1

alpha_post_1 = alpha + sum(y[1:5])
alpha_post_2 = alpha + sum(y[6:10])
beta_post = beta + 5
alpha_post_1
alpha_post_2
beta_post

#Task 3
#Notice that there is a difference between the parameterisation of the Gamma density in #the practical, and that used in R;  type help(rgamma) to see the details.  Hence, for the #R functions, we need to set the scale parameter to 1/beta.  

n <- 10000
lambda_1 <- rgamma(n, shape=alpha_post_1, scale=1/beta_post)
lambda_2 <- rgamma(n, shape=alpha_post_2, scale=1/beta_post)

weights <- rep(NA, times=n)
for(i in 1:n){
  weights[i] <- prod(0.5 * dpois(y[11:20], lambda_1[i]) + 0.5 * dpois(y[11:20], lambda_2[i]))
}

#Task 4

post_mean_1_labelled = alpha_post_1 / beta_post
post_mean_2_labelled = alpha_post_2 / beta_post
post_mean_1_labelled
post_mean_2_labelled

post_mean_1_all = sum(lambda_1 * weights) / sum(weights)
post_mean_2_all = sum(lambda_2 * weights) / sum(weights)
post_mean_1_all
post_mean_2_all

#Task 5
#Note that the function density must take as argument a vector of weights that sum to 1.

lambda_1_density = density(lambda_1, weights=weights/sum(weights))
lambda_2_density = density(lambda_2, weights=weights/sum(weights))

t <- seq(0, 20, length.out = 1000)

par(mfrow=c(1,2))
plot(lambda_1_density, main="Posterior density of lambda1")
lines(t, dgamma(t, shape=alpha_post_1, scale=1/beta_post), lty=2)
legend("topright",lty=c(1,2), legend=c("All data","Labelled data"))
plot(lambda_2_density, main="Posterior density of lambda2")
lines(t, dgamma(t, shape=alpha_post_2, scale=1/beta_post), lty=2)
legend("topright",lty=c(1,2), legend=c("All data","Labelled data"))

#Task 6

allocations <- matrix(NA, nrow=n, ncol=10)
for(i in 1:n){
  allocations[i,] <- dpois(y[11:20], lambda_1[i]) / (dpois(y[11:20], lambda_1[i]) + dpois(y[11:20], lambda_2[i]))
}

prob_of_group_1 = rep(NA, times=10)
for(i in 1:10){
  prob_of_group_1[i] = sum(allocations[,i] * weights) / sum(weights)
}
prob_of_group_1