Aquí tienes tres ejemplos. He hecho que el código sea mucho menos eficiente de lo que sería en una aplicación real para aclarar la lógica (espero).
# We'll assume estimation of a Poisson mean as a function of x
x <- runif(100)
y <- rpois(100,5*x) # beta = 5 where mean(y[i]) = beta*x[i]
# Prior distribution on log(beta): t(5) with mean 2
# (Very spread out on original scale; median = 7.4, roughly)
log_prior <- function(log_beta) dt(log_beta-2, 5, log=TRUE)
# Log likelihood
log_lik <- function(log_beta, y, x) sum(dpois(y, exp(log_beta)*x, log=TRUE))
# Random Walk Metropolis-Hastings
# Proposal is centered at the current value of the parameter
rw_proposal <- function(current) rnorm(1, current, 0.25)
rw_p_proposal_given_current <- function(proposal, current) dnorm(proposal, current, 0.25, log=TRUE)
rw_p_current_given_proposal <- function(current, proposal) dnorm(current, proposal, 0.25, log=TRUE)
rw_alpha <- function(proposal, current) {
# Due to the structure of the rw proposal distribution, the rw_p_proposal_given_current and
# rw_p_current_given_proposal terms cancel out, so we don't need to include them - although
# logically they are still there: p(prop|curr) = p(curr|prop) for all curr, prop
exp(log_lik(proposal, y, x) + log_prior(proposal) - log_lik(current, y, x) - log_prior(current))
}
# Independent Metropolis-Hastings
# Note: the proposal is independent of the current value (hence the name), but I maintain the
# parameterization of the functions anyway. The proposal is not ignorable any more
# when calculation the acceptance probability, as p(curr|prop) != p(prop|curr) in general.
ind_proposal <- function(current) rnorm(1, 2, 1)
ind_p_proposal_given_current <- function(proposal, current) dnorm(proposal, 2, 1, log=TRUE)
ind_p_current_given_proposal <- function(current, proposal) dnorm(current, 2, 1, log=TRUE)
ind_alpha <- function(proposal, current) {
exp(log_lik(proposal, y, x) + log_prior(proposal) + ind_p_current_given_proposal(current, proposal)
- log_lik(current, y, x) - log_prior(current) - ind_p_proposal_given_current(proposal, current))
}
# Vanilla Metropolis-Hastings - the independence sampler would do here, but I'll add something
# else for the proposal distribution; a Normal(current, 0.1+abs(current)/5) - symmetric but with a different
# scale depending upon location, so can't ignore the proposal distribution when calculating alpha as
# p(prop|curr) != p(curr|prop) in general
van_proposal <- function(current) rnorm(1, current, 0.1+abs(current)/5)
van_p_proposal_given_current <- function(proposal, current) dnorm(proposal, current, 0.1+abs(current)/5, log=TRUE)
van_p_current_given_proposal <- function(current, proposal) dnorm(current, proposal, 0.1+abs(proposal)/5, log=TRUE)
van_alpha <- function(proposal, current) {
exp(log_lik(proposal, y, x) + log_prior(proposal) + ind_p_current_given_proposal(current, proposal)
- log_lik(current, y, x) - log_prior(current) - ind_p_proposal_given_current(proposal, current))
}
# Generate the chain
values <- rep(0, 10000)
u <- runif(length(values))
naccept <- 0
current <- 1 # Initial value
propfunc <- van_proposal # Substitute ind_proposal or rw_proposal here
alphafunc <- van_alpha # Substitute ind_alpha or rw_alpha here
for (i in 1:length(values)) {
proposal <- propfunc(current)
alpha <- alphafunc(proposal, current)
if (u[i] < alpha) {
values[i] <- exp(proposal)
current <- proposal
naccept <- naccept + 1
} else {
values[i] <- exp(current)
}
}
naccept / length(values)
summary(values)
Para la muestra de vainilla, obtenemos:
> naccept / length(values)
[1] 0.1737
> summary(values)
Min. 1st Qu. Median Mean 3rd Qu. Max.
2.843 5.153 5.388 5.378 5.594 6.628
lo cual es una baja probabilidad de aceptación, pero aún así ... ajustar la propuesta ayudaría aquí, o adoptar una diferente. Aquí están los resultados de la propuesta de caminata aleatoria:
> naccept / length(values)
[1] 0.2902
> summary(values)
Min. 1st Qu. Median Mean 3rd Qu. Max.
2.718 5.147 5.369 5.370 5.584 6.781
Resultados similares, como uno esperaría, y una mejor probabilidad de aceptación (con el objetivo de ~ 50% con un parámetro).
Y, para completar, la muestra de independencia:
> naccept / length(values)
[1] 0.0684
> summary(values)
Min. 1st Qu. Median Mean 3rd Qu. Max.
3.990 5.162 5.391 5.380 5.577 8.802
Debido a que no se "adapta" a la forma de la parte posterior, tiende a tener la menor probabilidad de aceptación y es más difícil sintonizar bien este problema.
Tenga en cuenta que, en términos generales, preferiríamos propuestas con colas más gruesas, pero ese es un tema completamente diferente.