Two moons

sbisandbox.benchmarks.twomoons.TwoMoonsBenchmark

Bases: UniformPriorMixin, Benchmark

Two moons benchmark task.

The parameters $\mathit{\theta }\in {\mathbb{R}}^2$ are sampled from

$${\theta }_1\sim \mathcal{U}([-1,1])$$

$${\theta }_2\sim \mathcal{U}([-1,1])$$

The data $\mathit{x}\in {\mathbb{R}}^2$ are generated as follows:

• Sample the latent variables

$$\alpha \sim \mathcal{U}([-\pi /2,\pi /2])$$

$$r\sim \mathcal{N}(\mu =0.1,\sigma =0.01)$$

• Define

$$p=(r\cos \alpha +0.25,r\sin \alpha )$$

• Output

$$\mathit{x}=p-(\frac{|{\theta }_1+{\theta }_2|}{\sqrt{2}},\frac{{\theta }_1-{\theta }_2}{\sqrt{2}})$$

The name of the model comes from the shape of the posterior distribution $p(\mathit{\theta }|\mathit{x})$: bimodal with a banana-shaped mass around the modes (a-la a crescent moon).