= Solution
Let $\mathcal D=(\{\widehat m_s\},\{z_s\})$. <Bayes' theorem> and the stated prior factorization give the <posterior density>
$$
\boxed{p(M_0,v,H_0,w,\Omega_M\mid\mathcal D)\propto L(M_0,v,H_0,w,\Omega_M)\,p(M_0)p(v)p(H_0)p(w,\Omega_M)}.
$$
The normalizing constant, when finite and positive, is the integral of this product over its admissible parameter domain. An <improper prior> requires checking <posterior propriety> before interpreting this expression as a probability density.
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