Here we will introduce a random walk on in which the next step is determined by the beta distribution with parameter and . Later we use this random walk as a proposal Markov chain on . A smaller keeps a sample path closer to either of the boundary. The larger the value is the smaller the move of each step becomes Thus, will change the shape of stationary distribution of the random walk, and will influence the speed of convergence of random walk.
Explore it. Download bwalk.r, and see how a sample path of the beta random walk looks for a different choice of and .
> source("bwalk.r") > sample.path = rwalk(move=bmove, trajectory=T, delta=0.8, theta=20) > plot(sample.path, type="l", xlab="time", ylab="state", main="Beta random walk")A long run behavior can be observed from the histogram of from the end of runs repeatedly. Change the running time, and see if the distribution of is different. Also obtain the histogram of for a different choice of and .
> sample.data = rwalk(move=bmove, run.time=100, delta=0.8, theta=20, sample.size=500) > hist(sample.data, freq=F, breaks=seq(0,1,by=0.05), col='red')
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