Posterior distribution regression model

Question

Suppose we have the regression model $$y_t=x_t'\beta +\lambda_t\epsilon_t$$ with flat prior $p(\beta) \propto 1$ , i.i.d $\epsilon_t \sim N(0,1)$, $\lambda_t$ i.i.d and fixed regressors $x_t$.

Now suppose I am interested in the posterior $p(\beta| \lambda, y) $ where $y$ and $\lambda$ are the vectors with $y_t$ and $\lambda_t$ stacked over time. Because we are looking at the posterior of $\beta$ conditional on $\lambda$, it seems that we could look at usual the regression model with constant error variance $$\frac{y_t}{\lambda_t}=\frac{x_t}{\lambda_t}\beta+\epsilon_t$$ and make use of the known bayesian regression model result which then suggests that $p(\beta| \lambda, y)$ is normal with mean $(\sum_{t=1}^T\lambda^{-2}_tx_tx_t')^{-1}\sum_{t=1}^T\lambda^{-2}_tx_ty_t$ and covariance matrix $(\sum_{t=1}^T\lambda^{-2}_tx_tx_t')^{-1}$.

How can we rigorously justify this step?

Answer 1

Your model for the residuals, $\lambda_t\epsilon_t$ with $\epsilon_t$ i.i.d. $N(0,1)$ and the $\lambda_t$ known, corresponds to the following:

$$y_t = x'_t\beta + e_t$$

with $e_t$ having mean $0$ and covariance matrix $\Sigma$, with $\sigma_{ii}=\lambda_i^2$ and $\sigma_{ij} = 0, \space i\neq j$. The likelihood function corresponding to this formulation is that associated with a multivariate Normal distribution with known covariance matrix:

$$\mathcal{L}(\beta) \propto \exp\left(-{1\over 2}(y-x\beta)^T\Sigma^{-1}(y-x\beta)\right)$$

where $y-x\beta$ is the vector of errors.

Making use of the fact that the non-diagonal elements of $\Sigma$ are all $0$, we can rewrite the exponentiated term in the likelihood function as a sum:

$$\mathcal{L}(\beta) \propto \exp\left(-{1\over 2}\sum_{i=1}^n(y_i-x_i'\beta)\lambda_i^{-2}(y_i-x_i'\beta)\right)$$

which evidently can be rewritten in the form you conjecture is correct:

$$\mathcal{L}(\beta) \propto \exp\left(-{1\over 2}\sum_{i=1}^n\left({y_i \over \lambda_i}-{x_i\over \lambda_i}'\beta\right)^2\right)$$

As you've realized, this implies that the posterior is indeed Normal with the mean and covariance matrix given in your problem statement.