Ecco qui - tre esempi. Ho reso il codice molto meno efficiente di quanto sarebbe in una vera applicazione per rendere la logica più chiara (spero).
# 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)
Per il campionatore di vaniglia, otteniamo:
> 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
che è una bassa probabilità di accettazione, ma comunque ... mettere a punto la proposta sarebbe di aiuto o adottarne una diversa. Ecco i risultati della proposta di camminata casuale:
> 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
Risultati simili, come si potrebbe sperare, e una migliore probabilità di accettazione (puntando a ~ 50% con un parametro).
E, per completezza, il campionatore di indipendenza:
> 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
Poiché non si "adatta" alla forma del posteriore, tende ad avere la più bassa probabilità di accettazione ed è più difficile sintonizzarsi bene per questo problema.
Nota che in generale preferiremmo proposte con code più grasse, ma questo è un altro argomento.