# When does this prior dominate likelihood?

This is a simple Bayesian inference problem, where we are trying to infer some weight parameter $$w$$. Our posterior distribution is $$P\propto \exp\left(-\frac{1}{\sigma^2} w^Tw\right) \exp\left(-f(w)\right)$$ $$= \exp\left(-\frac{1}{\sigma^2} w^Tw-f(w)\right)$$ where the first term expresses Guassian prior (let's say $$g(w)=\frac{1}{\sigma^2}w^Tw$$). The second term expresses likelihood.

The first term scales with the number of elements ($$N$$) in vector $$w$$, and let's assume that the second term does not scale as fast as the first term and its scaling can only be estimated through numerical experiments. I believe it's logical to say that at very large $$N$$, the first term will dominate, and the posterior will be basically Gaussian. However, I am struggling to come up with a numerical analysis that will determine the threshold for $$N$$. What can I do?

I first thought about finding a maximum a posteriori estimation of $$w$$ (call it $$w^*$$), and see at which $$N$$, $$g(w^*)$$ is much greater than $$f(w^*)$$. But soon realized this doesn't make sense, because if the Gaussian term dominates, $$g(w^*) \rightarrow 0$$. Is this a correct argument?

The second thing I tried was using $$w$$ whose elements are i.i.d. $$\mathcal{N}(0,\sigma^2)$$, and see at which $$N$$, $$g(w)$$ is much greater than $$f(w)$$. I think this is a way to go, but I cannot clearly explain why this is right. Could anyone help?