Uniform choice for Prior Distribution

Uniform choice for Prior Distribution

My prior function is
$\Phi\left(\mathbf{k}_\ell,W_\ell\right)=\frac{1}{N}\log
p\left(\mathbf{k}_\ell,W_\ell\right)$
which is determined once I choose the Bayesian prior parameter likelihood
$p(\mathbf{k}_\ell,W_\ell)$. A natural choice is to take this proportional
to the effective number of graphs
$\mathcal{N}(\mathbf{k}_\ell,W_\ell)=\exp[N\cdot
S(\mathbf{k}_\ell,W_\ell)]$.
where $S(\mathbf{k}_\ell,W_\ell)]$ is the Shanon entropy per node
However, much further down the line I realise that my choice was 'not so
good'. Instead I wish to use a uniform choice, so the prior is just a
constant function of its variables.
How would I go about this?
EDIT FYI: $\mathbf{k}_\ell$ is a degree sequence for species $\ell$ and
$W_\ell$ is the degree correlation for connected pairs (of nodes) for
species $\ell$