VolumeEstimation.jl

Estimate the volume of arbitrary high-dimensional regions in Julia.

Given a boolean membership function that defines a region, VolumeEstimation.jl estimates its volume using parallel tempering and stepping-stone sampling via Pigeons.jl.

Quick Start

using VolumeEstimation, CommonSolve

# Estimate the volume of a 5D unit hypercube (true volume = 1)
membership(x) = all(abs.(x) .<= 0.5)
prob = VolumeProblem(membership, 5)
sol = solve(prob)

sol.log_volume  # ≈ 0.0 (log of 1)
sol.volume      # ≈ 1.0

# Estimate the volume of a 3D simplex (true volume = 1/6)
membership(x) = all(x .>= 0) && sum(x) <= 1
prob = VolumeProblem(membership, 3; x0=[1/4, 1/4, 1/4])
sol = solve(prob; n_rounds=15, n_chains=10)

sol.volume  # ≈ 0.1667

How It Works

The volume of a region S is the normalizing constant of the uniform distribution on S. VolumeEstimation.jl computes this by constructing a tempered path between:

• Target: uniform distribution on S (log-density 0 inside, -Inf outside)
• Reference: Gaussian N(x0, sigma^2 I) truncated to S, whose partition function is known analytically. The scale sigma is chosen automatically (see below).

Pigeons.jl runs parallel tempering across this path, and stepping-stone sampling gives the log normalizing constant ratio. The volume is then:

log(Volume) = log(Z_target / Z_ref) + log(Z_ref)

where Z_ref = (2*pi*sigma^2)^(dim/2) * p_acc and p_acc is the fraction of the Gaussian that falls inside S.

Automatic Scale Selection

VolumeEstimation.jl automatically finds the right Gaussian scale by searching for kmax -- the maximum spring constant k such that ~95% of samples from N(x0, (1/k)I) land inside the region. The reference standard deviation is then sigma = 1/sqrt(kmax).

API

VolumeProblem(membership, dim; x0=nothing)

• membership: function f(x) -> Bool, returns true if x is inside the region
• dim: number of dimensions
• x0: a point known to be inside the region (defaults to the origin)

solve(prob; n_rounds=10, n_chains=10, kwargs...)

Runs parallel tempering and returns a VolumeSolution with fields:

• log_volume: log of the estimated volume (use this in high dimensions to avoid underflow)
• volume: exp(log_volume)
• pt: the Pigeons PT object for diagnostics (swap rates, round trips, etc.)

Additional keyword arguments are forwarded to Pigeons.pigeons().