Online Importance Sampling
Hamza El-Saawy Stanford Stats 362 Final Project
This package supports online (batched) Importance Sampling (IS) with or without control variates
The package provides a MixtureDistribution{F, S} <: Distribution{F, S}
for mixture importance sampling.
Note: Even if f! (or g!) is a scalar function, it must write its output to a vector.
Moreover, w should only return a scalar.
Also, even if q <: UnivariateDistribution, x will be a Vector of length 1.
The exception is with respect to p, it should follow the convention of Distributions.jl,
which, unfortunetely, wants logpdf(p::UnivariateDistrbution, x::Real) but
logpdf(p::MultivariateDistrbution, x::Vector).
So, for the univariate case, w accepts a vector of length 1, but logpdf(p, x)
takes a scalar.
Note: when passing in external data (update!(is, X=X, F=F, W=W,...)), F and G
will be modified in place (F .*= W' and G ./= Q))
IS
Basic IS
ImportanceSampler(f!, lengthf::Int, q::Distribution ; p=nothing, w=nothing)f!(r::AbstractVector, x::AbstractVector) is a function (or anything callable) that
modifies r, its first argument. Note that x will always have the same length
as q, the sampling distribution. r will always have the same length as lengthf, the
output dimension of f.
Either p or w should be provided. p should have
logpdf(p, x::AbstractVector) defined, for x = rand(q).
w(x) should compute p(x)/q(x), the ratio of their pdfs.
IS with Control Variates:
CvImportanceSampler(f!, lengthf::Int, q::Distribution;
g!s::Union{AbstractVector{<:Tuple{Any, Vector{Float64}}}, Void}=nothing,
p=nothing, w=nothing,
use_q::Bool=false)Here, g!s is a vector of tuples, (g!, θ), where
g!(r::AbstractVector, x::AbstractVector) takes a vector always of size length(q)
and writes the result in r, which always has the size length(θ). θ is the integral
of g over the support of q. use_q uses q, or its components if
q <: MixtureDistribution, as control variates as well, with a θ of [1.0].
Running IS
The general syntax to run an ImportanceSampler is:
update!(is; X [, F, W])
update!(is; F, W)
update!(is; niters, nbatches, batchsize)All arguments except for is::ImportanceSampler use keywords. X is optional if
F and W are provided.
If the date is not provided, update will generate random nbatches batches of data
sized batchsize, for a total number of iterations: niters == nbatches * batchsize.
Only two of niters, nbatches, batchsize, all ::Int, should be provided.
For a CvImportanceSampler, it is:
update!(is; X [, F, G, W, Q])
update!(is; F, G, W, Q)
update!(is; niters, nbatches, batchsize)Here G is the value of all functions in g!s, concatenated together vertically.
Q is q(x) at each point, which is used to generate p(x) = w(x)*q(x).
Similar to above, if X is omitted, F, G, W, and Q must be provided.
There are also the keywords updateμ::Bool=true and updateβ::Bool=false that update the estimates
of the mean and regression coefficients, respectively.
The functions updateμ!(...) and updateβ!(...) with similar syntax to above can also be used
as well to update only one of the two. (Updating both together introduces a bias.)