= Solution
{c}
An unknown <variance> requires a joint <prior distribution> for $(\alpha,\beta,\sigma^2)$. \b[The analysis is not determined until a prior for the unknown variance, and its relation to the coefficients, is specified.] A <maximum-likelihood estimate> substituted for $\sigma^2$ does not by itself account for its posterior uncertainty.
For example, retain the independent coefficient <prior distributions> from part (b) and independently assign $s=\sigma^2$ an <inverse-gamma distribution> with shape $a_0>0$ and scale $b_0>0$, whose <probability density function> is proportional to $s^{-a_0-1}e^{-b_0/s}$. Then the joint <posterior density> is proportional to
$$
s^{-a_0-1-n/2}\exp\left\{-\frac{b_0+\frac12\sum_i(y_i-\alpha-\beta x_i)^2}{s}
-\frac{(\alpha-\alpha_0)^2}{2\tau_0^2}
-\frac{(\beta-\beta_0)^2}{2\tau_1^2}\right\}.
$$
Conditional on $s$, the two coefficient <posterior distributions> are still the independent <normal distributions> in part (b), with $s$ substituted for $\sigma^2$. The other <full conditional distribution> is
$$
s\mid\alpha,\beta,\boldsymbol y\sim
\operatorname{InvGamma}\left(a_0+\frac n2,
b_0+\frac12\sum_i(y_i-\alpha-\beta x_i)^2\right).
$$
The marginal coefficient <posterior distribution> requires <marginalization> over $s$; it is generally nonnormal and the two coefficients need not remain independent. In particular, the posterior <cumulative distribution function> of the requested <mean> becomes
$$
\mathbb P(M_0\leq t\mid\boldsymbol y)
=\int_0^\infty
\Phi\left(\frac{t-m_\alpha(s)-x_0m_\beta(s)}
{\sqrt{v_\alpha(s)+x_0^2v_\beta(s)}}\right)
\pi(s\mid\boldsymbol y)\,ds.
$$
Its shortest 95% <credible interval> must be found from this marginal <posterior distribution>, for example by numerical <integration> or posterior simulation. This is <normal regression with independent coefficient and variance priors>.
A different, convenient choice is a <normal-inverse-gamma prior>, in which the conditional coefficient prior <covariance matrix> is proportional to $\sigma^2$. That choice gives a multivariate <Student's t-distribution> after <marginalization>, so the interval uses its univariate <Student's t-distribution> <quantiles>. It changes the coefficient prior specification; a Student interval does not follow automatically from retaining the fixed-$\tau_0,\tau_1$ priors in part (b).
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