Deriving posterior means for regression with horseshoe prior

Question

I'm reading the Horseshoe prior regression paper which formulates regression like so:

$(y|\beta) \sim N(\beta, \sigma^2I)$

$(\beta_i|\lambda_i,\tau) \sim N(0, \lambda_i^2 \tau^2)$

$\lambda_i \sim C^{+}(0,1)$

where $C^{+}(0,1)$ is the half-Cauchy distribution.

In section 2.1 of the paper it says:

$E[\beta_i | y_i, \lambda_i^2] = (\frac{\lambda_i^2}{1 + \lambda_i^2}) y_i + (\frac{1}{1 + \lambda_i})0$

Where does this expression come from?

Answer 1

By the usual conjugacy of the normal prior and likelihood, the posterior distribution (note also that $\tau$ and $\sigma$ are fixed to 1) is $\beta_i|y_i, \lambda_i\sim N(\lambda_i(\lambda_i+1)^{-1}y_i, \lambda_i(\lambda_i+1)^{-1})$. And so the expectation they show is simply the expectation of this posterior.