I have a model where the noise is modelled as independent and identically distributed across the various data points. The noise $e$ is modelled as a 0 mean gaussian with $\phi$ as the precision (inverse variance). The likelihood for a single data point $i$ is defined as:
$$ P(y|w, \phi) = (\frac{\phi}{2\pi})^{0.5} \exp ^{-0.5 e_{i}\phi e_{i}} $$
The prior $P(w|y)$ is also modelled as a multivariate gaussian with zero mean and a covariance matrix. There is a scale parameter $\lambda$ which is also defined probabilistically and modelled using a gamma distribution. The noise precision $\phi$ also needs be inferred and is modelled using a gamma distribution
So ultimately the posterior will be given us
$$ P(w, \phi, \lambda|y) = \frac{P(w|\lambda) p(\phi) p(\lambda) \prod_{i} P(y|w, \phi)}{P(y)} $$
Now, this is probably a really silly question but since the priors are modelled using the Gamma or Gaussian distribution and the likelihood is also modelled with Gaussian noise, can we say that the posterior distribution will have also belong, say to the exponential family of distributions? I would like to know how the shape of priors and the likelihood affect the shape of the posterior distribution, if at all.