- MCMC toy examples for Lecture 05.

Author: estrumbelj

Diff for: Lecture05 - MCMC/Lecture05 - MCMC.r

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+### FROG
+
+# transition matrix
+q    <- matrix(rep(0, 9), nrow = 3)
+q[1,] <- c(0.0, 0.5, 0.5)
+q[2,] <- c(0.5, 0.0, 0.5)
+q[3,] <- c(0.5, 0.5, 0.0)
+
+m  <- 10000
+x  <- 1
+
+set.seed(12345)
+for (i in 1:m) {
+  x <- c(x, sample(1:3, 1, prob = q[x[i],])) # sample next state
+}
+
+table(x) / (m + 1)
+
+### FLY
+m  <- 10000
+x  <- 0.5
+
+set.seed(1010101)
+for (i in 1:m) {
+  # propose next state
+  x_new <- x[i] + runif(1, -0.5, +0.5)
+  # wrap-around
+  if (x_new < 0) x_new <- 0 - x_new
+  if (x_new > 1) x_new<- 2 - x_new
+  # add new state
+  x <- c(x, x_new)
+}
+
+hist(x, breaks = 20)
+
+
+### 0-1
+q    <- matrix(rep(0, 4), nrow = 2)
+q[1,] <- c(0.5, 0.5)
+q[2,] <- c(0.5, 0.5)
+
+m  <- 10000
+x  <- 1
+
+set.seed(12345)
+for (i in 1:m) {
+  x <- c(x, sample(1:2, 1, prob = q[x[i],])) # sample next state
+}
+
+x <- x - 1 # get 0-1 instead of 1-2
+table(x) / (m + 1)
+
+
+
+### Bayesian posterior
+
+# We're working with a simple Bernoulli-Beta model
+# Prior Beta(1,1), 12 data points, 9 ones, 3 zeroes
+# The posterior is then Beta(10, 4)
+# We're going to calculate posterior probability of parameter
+# being greater than 0.75
+library(mcmcse)
+library(ggplot2)
+a0 <- 1; b0 <- 1
+y <- c(rep(1, 9), rep(0, 3))
+n <- length(y)
+
+## 1. Computation based on deriving that the posterior is Beta(10, 4)
+x  <- seq(0, 1, 0.01)
+xx <- data.frame(x = x, y = dbeta(x, a0 + sum(y), b0 + n - sum(y)), type = "posterior") 
+ggplot(xx, aes(x = x, y = y, group = type, colour = type)) + geom_line() + xlab("theta") + ylab("density")
+1 - pbeta(0.75, a0 + sum(y), b0 + n - sum(y))
+
+## 2. Computation based on just integrating the non-normalized
+## posterior and normalizing
+prop_posterior <- function(theta, a0, b0, y) {
+  dbinom(sum(y), length(y), theta) * dbeta(theta, a0, b0)
+}
+p1 <- integrate(prop_posterior, lower = 0.75, upper = 1.00, a0, b0, y)$value
+p2 <- integrate(prop_posterior, lower = 0.00, upper = 0.75, a0, b0, y)$value
+p1 / (p1 + p2) # normalization
+
+## 3. Computation based on Monte Carlo approximation 
+## with actual distribution
+set.seed(0)
+m <- 1000
+x <- rbeta(m, a0 + sum(y), b0 + n - sum(y))
+mcse(x > 0.75)
+
+## 4. Computation based on Monte Carlo approximation 
+## with prop_posterior and Metropolis algorithm (MCMC)
+set.seed(0)
+m <- 10000
+x <- 0.5 # starting position
+for (i in 1:m) {
+  # proposing a new state (with wrap-around)
+  x_new <- x[i] + runif(1, -0.5, +0.5)
+  if (x_new < 0) x_new <- 0 - x_new
+  if (x_new > 1) x_new<- 2 - x_new
+  
+  # Metropolis correction
+  p_curr <- prop_posterior(x[i], a0, b0, y)
+  p_new  <- prop_posterior(x_new, a0, b0, y)
+  acceptance_prob <- min(p_new / p_curr, 1.0)
+  if (runif(1) > acceptance_prob) {
+    # reject
+    x_new <- x[i]
+  }
+  
+  # add state
+  x <- c(x, x_new)
+}
+hist(x)
+mcse(x > 0.75)
+
+## 5. Computation using Stan
+library(rstan)
+if (!file.exists("../Lecture03 - Probabilistic programming/Temp/bernoulli.compiled")) {
+  bernoulli_compiled <- stan_model("../Lecture03 - Probabilistic programming/bernoulli.stan")
+  saveRDS(bernoulli_compiled, file = "../Lecture03 - Probabilistic programming/Temp/bernoulli.compiled")
+} else {
+  bernoulli_compiled <- readRDS("../Lecture03 - Probabilistic programming/Temp/bernoulli.compiled")  
+}
+
+stan_data <- list(y = y,
+                  n = length(y))
+
+samples <- sampling(bernoulli_compiled, data = stan_data,  # we supply the model and data
+                    chains = 1, iter = 1200, warmup = 200)
+x <- extract(samples)$theta
+hist(x)
+mcse(x > 0.75)
+