 Savage-Dickey density ratio

Compare an unrestricted model $H_1$ with parameters $(\psi ,\lambda )$ to $H_0:\psi ={\psi }_0$. Suppose the likelihoods agree under that restriction, the priors are proper, and the nuisance prior under $H_0$ equals the conditional prior ${\pi }_1(\lambda \mid {\psi }_0)$ under $H_1$. Using compatible density versions with $0<{\pi }_1({\psi }_0)<\infty$, Bayes' theorem givesIndeed, the numerator of the marginal posterior density at ${\psi }_0$ is ${\pi }_1({\psi }_0)\int L(y\mid {\psi }_0,\lambda ){\pi }_1(\lambda \mid {\psi }_0)d\lambda ={\pi }_1({\psi }_0)m_0(y)$.

This ratio concerns densities in the unrestricted model, not a posterior atom at the null. A separately chosen nuisance prior under the null can invalidate the simple ratio; matching marginal nuisance priors is insufficient when $\psi$ and $\lambda$ are prior-dependent. Density values on a measure-zero restriction also require an agreed continuous or otherwise specified version.^[ref]