Gaussian linear uniform

sbisandbox.benchmarks.gaussian_linear_uniform.GaussianLinearUniformBenchmark

Bases: UniformPriorMixin, Benchmark

Gaussian linear benchmark task.

The parameters \(\boldsymbol{\theta} \in \mathbb{R}^n\) are sampled independently from a uniform distribution,

\[ \theta_i \sim \mathcal{U}([-1, 1]),\]

for \(i \in \{1, \ldots, n\}\). The data \(\boldsymbol{x} \in \mathbb{R}^n\) are generated as follows:

\[ \boldsymbol{x} \sim \mathcal{N}(\mu=\boldsymbol{\theta}, \Sigma_2=\sigma \boldsymbol{I}_n) \]

The posterior is analytical and equal to the likelihood, i.e

\[ \log p(\boldsymbol{\theta} | \boldsymbol{x}) \propto -\frac{1}{2\sigma^2}(\boldsymbol{x} - \boldsymbol{\theta})^T(\boldsymbol{x} - \boldsymbol{\theta}) \]