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I am trying to solve the following problem.

If $y | \beta \sim N(X \beta, I_n)$ and $\beta \sim N(0, g^{-1}(X^t X)^{-1})$ for $g>0$. Find $ \pi(\beta|y)$ and show that $E(\beta|y)$ is a function of MLE of $\beta$.

My approach,

I proved that to maximize the likelihood, it is equivalent to minimizing the SSE, so the OLS is equal to the MLE.

I know normal is conjugate prior. So, $$\pi(\beta|y)= \frac{f(y|\beta)\pi(\beta)}{m(y)} \propto f(y|\beta)\pi(\beta) $$

$$ \pi(\beta|y) \propto \exp\{ - \frac{1}{2}(y-X\beta)^t (y-X\beta) - \frac{1}{2}\beta^t(g^{-1}(X^tX)^{-1})^{-1}\beta \} $$ $$ = \exp\{ - \frac{1}{2}( y^t y -2y^tX \beta+(1+g)\beta^t(X^tX)\beta ) \}$$ and I get stuck. I read about g-priors and I think the a posterior is $N(\frac{1}{1+g}\hat{\beta}, \frac{1}{1+g} (X^t X)^{-1})$ but I'm not sure.

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You may consult any textbook on Bayesian econometrics (e.g.) to find the conditional posterior distribution for $\beta$ given $\sigma^2$, given "conjugate priors" for $\beta$ and $\sigma^2$, i.e., $$\pi \left( \beta ,\sigma ^{2}\right) =\pi \left( \beta |\sigma ^{2}\right)\pi \left( \sigma ^{2}\right) $$ with \begin{eqnarray} \beta |\sigma ^{2} &\sim &N\left( \beta _{0},\sigma ^{2}B_{0}\right) \\ \sigma ^{2} &\sim &IG\left( \alpha _{0}/2,\delta _{0}/2\right), \end{eqnarray} as \begin{eqnarray*} \beta |\left( \sigma^{2},y\right) &\sim& N\left( \beta _{1},\sigma^{2}B_{1}\right) \end{eqnarray*} where \begin{eqnarray*} B_{1} &=&\left( X^{\prime }X+B_{0}^{-1}\right) ^{-1} \\ \beta _{1} &=&B_{1}\left( X^{\prime }y+B_{0}^{-1}\beta _{0}\right) \end{eqnarray*} Here, it seems to be the case that $\sigma^2$ is known to be one, so will disappear from these expressions. Also, your prior mean is zero and $B_0=g^{-1}(X'X)^{-1}$.

Thus, \begin{eqnarray*} B_1 &=& (X'X + ((g\cdot X'X)^{-1})^{-1})^{-1}\\ & = &(X'X + g\cdot X'X)^{-1} = ((1+g)X'X)^{-1}\\ &=& 1/(1+g)(X'X)^{-1} \end{eqnarray*} and \begin{eqnarray*} \beta_1 &=& B_1(X'y+g\cdot X'X\beta_0)\\ & = & B_1 X'y = ((1+g)X'X)^{-1}X'y\\ & = & 1/(1+g)(X'X)^{-1}X'y = 1/(1+g)\hat\beta \end{eqnarray*} The part regarding the posterior mean should follow immediately.

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