ARMS MH

ARMS stands for Adaptive Rejection Metropolis Sampling, and it is the generalization of ARS algorithm.

We can implement this algorithm with the following Julia program:

include("../GenRV/ars.jl")
using Main.corears
# ARMS
function arms(T, yfixed::Array)
x = ones(T+1)
for t = 1:T
# generate Y
while true
global y
y = gplus_sample(yfixed)
u = rand()
u <= exp(h(y)-hplus(y, yfixed)) && break
end
# accept or not
v = rand()
r = exp(h(y))*phi(x[t], yfixed)/(exp(h(x[t]))*phi(y, yfixed))
println(r)
if r >= 1
x[t+1] = y
else
if v <= r
x[t+1] = y
else
x[t+1] = x[t]
end
end
end
return(x)
end

Now let's apply ARMS to poisson logistic model:

# data
x = rand(10)
y = rand(10)
# parameters
tau = 1
# function of h
function h(b)
return(b * sum(x .* y) - sum(log.(1 .+ exp.(b*x))) - sum(1 .- 1 ./ (1 .+ exp.(b*x))) - b^2/(2*tau^2))
end
# derivative of h
function dh(b)
return(sum(x .* y) - sum(x .* (1 .- 1 ./ (1 .+ exp.(b*x)) )) + sum(x .* exp.(b*x) ./ (1 .+ exp.(b*x)).^2 ) - b/tau^2)
end
function phi(x, yfixed::Array)
if exp(h(x)) <= exp(hplus(x, yfixed))
return(exp(h(x)))
else
return(exp(hplus(x, yfixed)))
end
end
## example
x = arms(10000, [-1.5, -0.7, 0.7, 1.5])