Coordinate-ascent variational inference

Inputs: A finite number $m\ge 1$ of sigma-finite factor reference spaces $({\mathrm{\Omega }}_j,{\mathcal{A}}_j,{\mu }_j)$, their product reference measure $\mu$, a measurable target $\gamma$ with finite positive normalizer, an admissible initial mean-field density $q={\prod }_{j=1}^m{q}_j$, and a finite sweep count $N\ge 0$.

Output: The current admissible product density after at most $N$ full sweeps, together with a status indicating whether the requested sweeps completed or a coordinate update was unavailable.

For each sweep $t=1,\dots ,N$, visit coordinates in the fixed order $j=1,\dots ,m$. At coordinate $j$, keep $q_{-j}:={\prod }_{k\mathrm{\ne }j}{q}_k$ fixed; for $m=1$, the empty product reference space is a singleton with unit measure ${\mu }_{-j}$ and its density is $1$. Form the conditional log expectation $h_j(z_j):={\int }_{{\prod }_{k\mathrm{\ne }j}{\mathrm{\Omega }}_k}\log \gamma (z_j,z_{-j})q_{-j}(z_{-j})d{\mu }_{-j}(z_{-j})\mathrm{.}$ Use this signed integral only when it is defined without an $\mathrm{\infty }-\mathrm{\infty }$ ambiguity and has a measurable extended-real version; require it to be finite for current $q_j$-almost every $z_j$. Set $c_j:={\int }_{{\mathrm{\Omega }}_j}{e}^{h_j(z_j)}{\mu }_j(d{z}_j)\mathrm{.}$ If $0<c_j<\mathrm{\infty }$ and the normalized density $q_j^{\ast }(z_j):=e^{h_j(z_j)}\mathrm{/}{c}_j$ makes $q_j^{\ast }{q}_{-j}$ an admissible member of the mean-field family, replace $q_j$ by $q_j^{\ast }$. Otherwise stop and return the current density with status “coordinate update unavailable.”

The algorithm performs only the requested finite sweeps. It makes no convergence, stationary-point, global-optimum, calibration, or approximation-error claim. A separate theorem on this page proves exact coordinate monotonicity under these conditions.