Assume a linear model for the data : $y = x^T\beta + \epsilon$, with $\epsilon \sim \mathcal{N}(0,\sigma^2)$ and an a priori on the parameter: $\beta \sim \mathcal{N}(0,\Sigma^2 _p)$. Let $\mathbf{y}, \mathbf{X}$ be the training set. It is widely known that the a posteriori distribution of $\beta$ is $$ p(\beta |\mathbf{y},\mathbf{X}) = \mathcal{N}(\sigma^{-2}(\sigma^{-2}XX^T + \Sigma_p)^{-1}Xy, \sigma^{-2} XX^T + \Sigma_p^{-1})$$

now what if we didn't know how to derive this result, how can we 'observe' this a posteriori by sampling?


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