Metropolis algorithm

ggplot2 is used for plotting, tidyr for manipulating data frames

library(ggplot2)
theme_set(theme_minimal())
library(tidyr)
library(gganimate)
library(ggforce)
library(MASS)
library(rprojroot)
library(rstan)
root<-has_file(".BDA_R_demos_root")$make_fix_file()

Parameters of a normal distribution used as a toy target distribution

y1 <- 0
y2 <- 0
r <- 0.8
S <- diag(2)
S[1, 2] <- r
S[2, 1] <- r

Metropolis proposal distribution scale

sp <- 0.3

Sample from the toy distribution to visualize 90% HPD interval with ggplot's stat_ellipse()

dft <- data.frame(mvrnorm(100000, c(0, 0), S))

see BDA3 p. 85 for how to compute HPD for multivariate normal in 2d-case contour for 90% HPD is an ellipse, whose semimajor axes can be computed from the eigenvalues of the covariance matrix scaled by a value selected to get ellipse match the density at the edge of 90% HPD. Angle of the ellipse could be computed from the eigenvectors, but since the marginals are same we know that angle is pi/4 Starting value of the chain

t1 <- -2.5
t2 <- 2.5

Number of iterations.

M <- 5000

Insert your own Metropolis sampling here

# Allocate memory for the sample
tt <- matrix(rep(0, 2*M), ncol = 2)
tt[1,] <- c(t1, t2)    # Save starting point
# For demonstration load pre-computed values
# Replace this with your algorithm!
# tt is a M x 2 array, with M draws of both theta_1 and theta_2
load(root("demos_ch11","demo11_2a.RData"))

The rest is for illustration Take the first 200 draws to illustrate how the sampler works

df100 <- data.frame(id=rep(1,100),
                    iter=1:100, 
                    th1 = tt[1:100, 1],
                    th2 = tt[1:100, 2],
                    th1l = c(tt[1, 1], tt[1:(100-1), 1]),
                    th2l = c(tt[1, 2], tt[1:(100-1), 2]))

Take the first 5000 observations after warmup of 50

s <- 5000
warm <- 500
dfs <- data.frame(th1 = tt[(warm+1):s, 1], th2 = tt[(warm+1):s, 2])

Remove warm-up period of 50 first draws later

# labels and frame indices for the plot
labs1 <- c('Draws', 'Steps of the sampler', '90% HPD')
p1 <- ggplot() +
  geom_jitter(data = df100, width=0.05, height=0.05,
              aes(th1, th2, group=id, color ='1'), alpha=0.3) +
  geom_segment(data = df100, aes(x = th1, xend = th1l, color = '2',
                                 y = th2, yend = th2l)) +
  stat_ellipse(data = dft, aes(x = X1, y = X2, color = '3'), level = 0.9) +
  coord_cartesian(xlim = c(-4, 4), ylim = c(-4, 4)) +
  labs(x = 'theta1', y = 'theta2') +
  scale_color_manual(values = c('red', 'forestgreen','blue'), labels = labs1) +
  guides(color = guide_legend(override.aes = list(
    shape = c(16, NA, NA), linetype = c(0, 1, 1)))) +
  theme(legend.position = 'bottom', legend.title = element_blank())

The following generates a gif animation of the steps of the sampler (might take 10 seconds).

animate(p1 +   
          transition_reveal(along=iter) + 
          shadow_trail(0.01))
## Rendering [--------------------------------------------] at 4.5 fps ~ eta: 22s
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