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Bayesian linear regression with known residual covariance (simply put)

August 15, 2012

Here’s a topic that needs a simple synopsis.

Assume:
\beta \sim \mathcal{N}(\beta_0, \Sigma_\beta)
y | \beta \sim \mathcal{N}(X\beta, \Sigma_y)

Then:
\beta | y \sim \mathcal{N}(\hat{\beta}, V_\beta)

Where:
V_\beta = (X^\top\Sigma_y^{-1}X + \Sigma_\beta^{-1})^{-1}
\hat{\beta} = V_\beta (X^\top\Sigma_y^{-1}y + \Sigma_\beta^{-1}\beta_0)

References: here and here

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