BayesianTools.jl
is a Julia package with methods useful for Monte Carlo Markov Chain simulations. The package has two submodules:
ProductDistributions
: defines aProductDistribution
type and related methods useful for defining and evaluating independent priorsLink
: useful to rescale MC proposals to live in the support of the prior densities
The package is registered
(v1.x) pkg> add BayesianTools
The following code shows how a product distribution resulting from multiplying a normal and a Beta can be obtained
using BayesianTools.ProductDistributions
p = ProductDistribution(Normal(0,1), Beta(1.,1.))
n = length(p) ## 2 > Number of distributions in the product
To check whether an Array{Float64}
is in the support of p
insupport(p, [.1,2.]) ## false
insupport(p, [.1,1.]) ## true
The logpdf
and the pdf
at a point x::Array{Float64}(n)
are
logpdf(p, [.1,.5]) # = logpdf(Normal(0,1), .1) + logpdf(Beta(1.,1.), .5)
pdf(p, [.1,.5]) # = pdf(Normal(0,1), .1) * pdf(Beta(1.,1.), .5)
It is also possible to draw a sample from p
rand!(p, Array{Float64}(2,100))
invlink
and link
are useful to transform and backtransform the parameters of a parametric statistical model according to the support of its distribution. logjacobian
provides the log absolute Jacobian of the inverse transformation applied by invlink
.
The typical use case of the methods in the Links
is best understood by an example. Suppose interest lies in sampling from a Gamma(2,1) distribution
This is a simple distribution and there are many straightforward ways to draw from it. However, we will consider employing a random walk MetropolisHastings (MH) sampler with a standard Gaussian proposal.
The support of this distribution is x > 0 and there are four options regarding the proposal distribution:

Use a
Normal(0,1)
and proceed as you normally would if the support of the density was (Inf, +Inf). 
Use a truncated normal distribution

Sample from a Normal(0,1) until the draw is positive

Reparametrise the distribution in terms of and draw samples from
The first strategy will work just fine as long as the density evaluates to 0 for values outside its support. This is the case for the pdf
of a Gamma
in the Distributions
package.
The second and the third strategy is going to work as long as the acceptance ratio includes the normalizing constant (see Darren Wilkinson's post).
The last strategy also needs an adjustment to the acceptance ratio to incorporate the Jacobian of the transformation.
The code below use invlink
, link
, and logjacobian
to carry out the r.v. transformation and the Jacobian adjustment.
Notice that the Improper
distribution is a subtype of ContinuousUnivariateDistribution
. Links
defines methods for Improper
that allow the transformations to go through automatically. (Improper
can also be used as a component of the ProductDistribution
which is useful if an improper prior was elicited for some components of the parameter.)
using BayesianTools.Links
function mcmc_wrong(iters)
chain = Array{Float64}(iters)
gamma = Gamma(2, 1)
d = Improper(0, +Inf)
lx = 1.0
for i in 1:iters
xs = link(d, lx) + randn()
lxs = invlink(d, xs)
a = logpdf(gamma, lxs)logpdf(gamma, lx)
(rand() < exp(a)) && (lx = lxs)
chain[i] = lx
end
return chain
end
function mcmc_right(iters)
chain = Array{Float64}(iters)
gamma = Gamma(2, 1)
d = Improper(0, +Inf)
lx = 1.0
for i in 1:iters
xs = link(d, lx) + randn()
lxs = invlink(d, xs)
a = logpdf(gamma, lxs)logpdf(gamma, lx)
## Log absolute jacobian adjustment
a = a  logjacobian(d, lxs) + logjacobian(d, lx)
(rand() < exp(a)) && (lx = lxs)
chain[i] = lx
end
return chain
end
The results is
mc0 = mcmc_wrong(1_000_000)
mc1 = mcmc_right(1_000_000)
using Plots
Plots.histogram([mc0, mc1], normalize=true, bins = 100, fill=:slategray, layout = (1,2), lab = "draws")
title!("Incorrect sampler", subplot = 1)
title!("Correct sampler", subplot = 2)
plot!(x>pdf(Gamma(2,1),x), w = 2.6, color = :darkred, subplot = 1, lab = "Gamma(2,1) density")
plot!(x>pdf(Gamma(2,1),x), w = 2.6, color = :darkred, subplot = 2, lab = "Gamma(2,1) density"))
png("sampler")