Welcome to my blog. I will review papers, discuss ideas and write tutorials on topics at the interface of applied probability, computational statistics and machine learning. Most of my research to date has been in Markov chain Monte Carlo sampling algorithms. The idea is that to compute an expectation \[ \mathbb{E}_\pi[f] = \int f(x)\pi(dx), \] where \(f \in L^2(\pi)\) is some function of interest and \(\pi\) is some probability distribution, then we can simulate a single realisation of a Markov chain with transition kernel \(P\) that has limiting distribution \(\pi\) and compute ergodic averages.
Markov chains can be easily simulated, see below for an example.
# generate storage variables
nits <- 1000
x <- rep(NA, nits)
y <- rep(NA, nits)
x[1] <- rnorm(1)
y[1] <- rnorm(1)
a <- 0.9
# simulate chains
for (i in 2:nits) {
x[i] <- a * x[i-1] + sqrt(1-a^2) * rnorm(1)
y[i] <- a * y[i-1] + sqrt(1-a^2) * rnorm(1)
}
# plot results
plot(x, type = 'l', col = "red")
lines(y, col = "darkgreen")
I’m looking forward to sharing my thoughts on the blog!