Proof-of-concept parallel tempering

[richfitz]

Merge after #104, contains those commits

Closes #24

A basic parallel tempering sampler. There's much more to add (particularly embedding other samplers and sorting out sampling from the hot chain), but this is a start at least

Marc, a simple example to get you going. This actually looks pretty good now I think:

set.seed(1) # so we see the same thing
pkgload::load_all()
likelihood <- ex_mixture(5)
prior <- monty_dsl({
  x ~ Normal(0, 10)
})
posterior <- likelihood + prior
s <- monty_sampler_parallel_tempering(n_rungs = 10, vcv = matrix(0.1))
res <- monty_sample(posterior, s, 10000, n_chains = 4)

hist(c(res$pars), freq = FALSE)
x <- seq(min(res$pars), max(res$pars), length.out = 1001)
y <- exp(posterior$density(rbind(x)))
lines(x, y / sum(y) / diff(x)[[1]], col = "red")

(takes 17s on my machine for 40,000 steps, so 2k steps/s which is ok, especially as this is really 11 times that many samples).

I've run out a much longer chain and this is what I see:

set.seed(42) # so we see the same thing
pkgload::load_all()
likelihood <- ex_mixture(5)
prior <- monty_dsl({
  x ~ Normal(0, 10)
})
posterior <- likelihood + prior
s <- monty_sampler_parallel_tempering(n_rungs = 10, vcv = matrix(0.1))
r <- monty_runner_callr(dust2::dust_openmp_threads(10, "fix"))
res <- monty_sample(posterior, s, 50000, n_chains = 10, runner = r)

hist(c(res$pars), freq = FALSE)
x <- seq(min(res$pars), max(res$pars), length.out = 1001)
y <- exp(posterior$density(rbind(x)))
lines(x, y / sum(y) / diff(x)[[1]], col = "red")

(27s for 500,000 points, I can live with that)

I suspect we're out by a factor related to the beta swaps at the moment?

[codecov]

Codecov Report

Attention: Patch coverage is 99.13043% with 1 line in your changes missing coverage. Please review.

Project coverage is 99.77%. Comparing base (774cbd0) to head (7fb063e).

Files with missing lines	Patch %	Lines
R/sampler-parallel-tempering.R	99.13%	1 Missing ⚠️

Additional details and impacted files

@@            Coverage Diff             @@
##             main     #106      +/-   ##
==========================================
- Coverage   99.78%   99.77%   -0.02%     
==========================================
  Files          66       67       +1     
  Lines        5226     5341     +115     
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- Misses         11       12       +1     

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