Lartillot Gaussian model
The multidimensional Gaussian model from Lartillot et al '06 is useful for testing Bayesian applications.
The model is parameterised by a vector $\theta = (\theta_1, \theta_2, \ldots, \theta_d)$ of dimension $d$. The prior on $\theta$ is a product of independent normals with a null mean and unit variance. The likelihood function is given by $$\mathcal{L}(\theta, v) = {(2\pi v)^{-d/2}} \prod_{i = 1}^d \text{exp}\bigg[\frac{-\theta_i^2}{2v}\bigg],$$ where $v$ is the common variance for all $d$ dimensions.
The joint posterior is given by $d$ products of $\mathcal{N}(0, v/(1+v))$. Finally, the evidence is given by $$ \mathcal{Z} = (2\pi v)^{-d/2} \left(\frac{v}{1 + v} \right)^{d/2}=\left(2\pi(1+v)\right)^{-d/2}\ . $$
This codebase provides a simple implementation of the Lartillot Gaussian model in Python.
pip install lartillot_gaussian
from lartillot_gaussian import LartillotGaussianModel
import numpy as np
model = LartillotGaussianModel(d=10, v=1.0)
theta = np.array([[0]])
print(model.lnz)
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