Debiasing illustration

Illustration of debiasing for the posterior mean of a 2D Gaussian with unknown mean $\mu$ and fixed covariance $\Sigma$. Data is $\mathcal{D}=\{\mathbf{x}_i\sim\mathcal{N}(\mathbf{x}_i|\mu=\mathbf{2},\Sigma)\}_{i=1}^{100}$ with $\Sigma=[(-1,3)^\top,(3,1)^\top]$, prior is $p(\mathbf{\mu})=\mathcal{N}(\mathbf{\mu}|\mathbf{0}, I)$. We aim to compute the posterior mean $\int \mu p(\mu|\mathcal{D})d\mu$. Debiasing computes multiple posterior paths (coloured solid lines), which are randomly truncated (solid line stops), and then plugged into the debiasing estimator to estimate the posterior mean of $\mu_1$ and $\mu_2$ (coloured round dots, dotted lines connect path end-point to estimate). The procedure is averaged $R=1000$ times (gray dots), after which the empirical mean matches the full posterior mean. A kernel density estimate of the gray dots is shown in the background.

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