Correlated Gaussian

This example will explore a highly-correlated Gaussian using Models.CorrelatedGaussian. This model uses a conjuage Gaussian prior, see the docstring for the mathematical definition.

Setup

For this example, you'll need to add the following packages

julia>]add Distributions MCMCChains Measurements NestedSamplers StatsBase StatsPlots

Define model

using NestedSamplers

# set up a 4-dimensional Gaussian
D = 4
model, logz = Models.CorrelatedGaussian(D)

let's take a look at a couple of parameters to see what the likelihood surface looks like

using StatsPlots

θ1 = range(-1, 1, length=1000)
θ2 = range(-1, 1, length=1000)
loglike = model.prior_transform_and_loglikelihood.loglikelihood
logf = [loglike([t1, t2, 0, 0]) for t2 in θ2, t1 in θ1]
heatmap(
    θ1, θ2, exp.(logf),
    aspect_ratio=1,
    xlims=extrema(θ1),
    ylims=extrema(θ2),
    xlabel="θ1",
    ylabel="θ2"
)

Sample

using MCMCChains
using StatsBase
# using single Ellipsoid for bounds
# using Gibbs-style slicing for proposing new points
sampler = Nested(D, 50D;
    bounds=Bounds.Ellipsoid,
    proposal=Proposals.Slice()
)
names = ["θ_$i" for i in 1:D]
chain, state = sample(model, sampler; dlogz=0.01, param_names=names)
# resample chain using statistical weights
chain_resampled = sample(chain, Weights(vec(chain[:weights])), length(chain));

Results

chain_resampled

Chains MCMC chain (2386×5×1 Array{Float64, 3}):

Iterations        = 1:2386
Number of chains  = 1
Samples per chain = 2386
parameters        = θ_1, θ_2, θ_3, θ_4
internals         = weights

Summary Statistics
  parameters      mean       std      mcse    ess_bulk    ess_tail      rhat   ⋯
      Symbol   Float64   Float64   Float64     Float64     Float64   Float64   ⋯

         θ_1    1.6108    0.4664    0.0097   2328.6362   1936.1660    0.9996   ⋯
         θ_2    1.6247    0.4596    0.0096   2268.8746   1958.8120    1.0001   ⋯
         θ_3    1.5988    0.4543    0.0095   2307.4521   2194.8809    1.0001   ⋯
         θ_4    1.6134    0.4677    0.0097   2331.0871   2434.4555    0.9998   ⋯
                                                                1 column omitted

Quantiles
  parameters      2.5%     25.0%     50.0%     75.0%     97.5%
      Symbol   Float64   Float64   Float64   Float64   Float64

         θ_1    0.7428    1.2820    1.5899    1.9481    2.4679
         θ_2    0.8042    1.3181    1.5984    1.9191    2.5305
         θ_3    0.7181    1.2721    1.5976    1.8958    2.4717
         θ_4    0.7550    1.3243    1.5942    1.9329    2.5527

corner(chain_resampled)