Metropolis Algorithm Matlab practice Matlab code taken from

Metropolis Algorithm Matlab practice • Matlab code taken from Professor Joo-Ho Choi • He applied it to with proposal distribution N(x, 10). • Matlab code for that give in the notes. • Here applied to the triangular distribution with U(x 0. 25, x+0. 25)

Metropolis-Hastings algorithm • Metropolis algorithm – A proposal distribution q(x*|x) is symmetric w. r. t x* and x. Then the ratio is simplified. Example: normal pdf at x* with mean at x equals to the vice versa. • Practice with matlab – Generate samples of this distribution using a proposal pdf – As the random walk progresses, the number of samples are increased, and the distribution converges to the target distribution. -2 -

Triangular distribution • p=2 x in [0, 1] • Matlab realization p=@(x) heaviside(x). *heaviside(1 -x)*2. *x xx=linspace(-1, 2, 301); fp=p(xx); plot (xx, fp)

Uniform proposal distribution • q(x)=U(x-0. 25, x+0. 25) • Sampling: clear; X(1)=0; N=1 e 4; delta=0. 25; for i=1: N-1; x=X(i); xs=x+2*(rand-0. 5)*delta u=rand if u<min(1, p(xs)/(p(x)+1. e-10)); X(1+i)=xs; else; X(1+i)=x; end • Plotting N 0=1; xx=linspace(0, 1, 26); dx=0. 04; %It often pays to have larger N 0 nb=histc(X(N 0+1: N), xx); bar(xx+dx/2, nb/(N-N 0)/dx);

Compare CDFs ecdf(X); hold on xx=linspace(0, 1, 101); xx 2=xx. ^2; plot(xx, xx 2, 'r') legend('ecdf', 'exact')

practice problems 1. Try triangular distribution with a normal proposal distribution with different mean and standard deviations. 2. Do Professor Choi’s example with different starting points. Source: Smithsonian Institution Number: 2004 -57325