Analytic Gaussian Mixture

Overview

This benchmark consists of an analytically defined PDF \(\pi : \mathbb{R}^2 \rightarrow \mathbb{R}\) consisting of a Gaussian mixture.

Authors

• Linus Seelinger

Run

docker run -it -p 4243:4243 linusseelinger/benchmark-analytic-gaussian-mixture

Properties

Model	Description
posterior	Posterior density

posterior

Mapping	Dimensions	Description
input	[2]	2D coordinates \(x \in \mathbb{R}^2\)
output	[1]	Log PDF \(\pi\) evaluated at \(x\)

Feature	Supported
Evaluate	True
Gradient	True
ApplyJacobian	True
ApplyHessian	False

Config	Type	Default	Description
None

Mount directories

Mount directory	Purpose
None

Source code

Model sources here.

Description

Let \(X_1 \sim \mathcal{N}(\begin{pmatrix} -1.5 \\ -1.5 \end{pmatrix}, 0.8 I)\), \(X_2 \sim \mathcal{N}(\begin{pmatrix} 1.5 \\ 1.5 \end{pmatrix}, 0.8 I)\), \(X_3 \sim \mathcal{N}(\begin{pmatrix} -2 \\ 2 \end{pmatrix}, 0.5 I)\). Denote by \(f_{X_1}, f_{X_2}, f_{X_3}\) the corresponding PDFs.

The PDF \(\pi\) is then defined as

\[ \pi(x) := \sum_{i=1}^3 f_{X_i}(x), \]

and the benchmark outputs \(\log(\pi(x))\).

This distribution is inspired by Chi Feng’s excellent online mcmc-demo.