So I have this prior: $p(\mathbb{w}|\alpha)$. And then I am given that the posterior is: $$p(\mathbb{w}|\mathbb{t},\alpha,\sigma^2) \propto p(\mathbb{w}|\alpha)p(\mathbb{t}|\mathbb{w},\sigma^2)$$
I always find hard to show something like this, since there are so many ways to factorise the conditional probability.
We can re-write the above as:
$$\frac{p(\mathbb{w},\mathbb{t},\alpha,\sigma^2)}{p(t,\alpha,\sigma^2)}$$ I have then tried to use probability chain rule on the above to no avail.
EDIT: Paper that this is coming from. Equation 14.
It is a useful trick when manipulating conditional density functions to just ignore some subset of the variables and then after the manipulation return the "ignored" variables to the resulting expression after the $"|"$, this is so because the conditional probabilities are themselves valid probability density functions - if you like you could write $q(\cdot) := p(\cdot | \alpha,\sigma^2)$ - and so we have that $$ \begin{align*} q(\mathbb{w}|\mathbb{t}) = \frac{q(t|\mathbb{w})q(\mathbb{w})}{q(\mathbb{t})} \\ \end{align*}, $$ becomes $$ p(\mathbb{w}|\mathbb{t},\sigma^2,\alpha) = \frac{ p(\mathbb{t}|\mathbb{w},\sigma^2,\alpha) p(\mathbb{w}|\sigma^2,\alpha)}{p(\mathbb{t}|\sigma^2,\alpha)}. $$ You then inspect all of these terms and apply any conditional independence properties of the generative model to remove any redundant conditioning, so for example looking at the paper you link to we have the hierarchical structure $$ \begin{align*} \alpha &\sim\pi(\alpha) \\ \mathbb{w}|\alpha &\sim p(\mathbb{w}|\alpha)\\ \mathbb{t}|\mathbb{w},\sigma^2 &\sim p(\mathbb{t}|\mathbb{w},\sigma^2) \end{align*} $$ similarily we see that $p(\mathbb{w}|\sigma^2,\alpha)=p(\mathbb{w}|\alpha)$, inserting these we have $$ p(\mathbb{w}|\mathbb{t},\sigma^2,\alpha)=\frac{p(\mathbb{t}|\mathbb{w},\sigma^2)p(\mathbb{w}|\alpha)}{p(\mathbb{t}|\alpha,\sigma^2)}. $$