EM does not decrease the observed log likelihood

By [F2] and [F3], $P:=q_t\mu$ and $Q:=q_{\theta }^x\mu$ are probability measures. For $u=q_t$ and $u=q_{\theta }^x$, $n{\mathbf{1}}_{\{u=\mathrm{\infty }\}}\le u$ and $\int ud\mu =1$, so [F8] gives $n\mu \{u=\mathrm{\infty }\}\le 1$ for every positive integer $n$; [F9] forces both infinite-value sets to be $\mu$-null. Let $N$ be their union. By [F10], $P(N)=Q(N)=0$. Also $P\{q_{\theta }^x=0\}=0$, since $Q\{q_{\theta }^x=0\}=0$ and $P\ll Q$. Finally $P\{q_t=0\}=0$ by the definition of $P$. Thus $q_t$ and $q_{\theta }^x$ are finite and strictly positive on a $P$-full set.

By [A1] and [F4], choose a finite-valued Radon–Nikodym derivative $r=dP\mathrm{/}dQ$. Define $\tilde{r}=r{q}_{\theta }^x$ off $N$ and $\tilde{r}=0$ on $N$, and define ${\tilde{q}}_t=q_t$ off $N$ and ${\tilde{q}}_t=0$ on $N$. These are finite-valued measurable functions. For every measurable $A$, [F5] and [F10] give ${\int }_A\tilde{r}d\mu ={\int }_{A}r{q}_{\theta }^{x}d\mu ={\int }_{A}rdQ=P(A)={\int }_A{q}_{t}d\mu ={\int }_A{\tilde{q}}_{t}d\mu \mathrm{.}$ Since $\mu$ is sigma-finite and $P$ is finite, a sigma-finite exhaustion for $\mu$ is also finite for $P$. Uniqueness [F11] therefore gives $\tilde{r}={\tilde{q}}_t$ $\mu$-almost everywhere. On the $P$-full set from step 1.1, $r=q_t\mathrm{/}{q}_{\theta }^x$ and $\log r=\log p({\theta }_t,x,z)-\log m_{{\theta }_t}(x)-\log p(\theta ,x,z)+\log m_{\theta }(x)\mathrm{.}$ The stated absolute-integrability conditions make these log terms integrable. Integrating this equality with respect to $P=q_t\mu$ and using [F4] yields $D(P\mathrm{\parallel }Q)=Q({\theta }_t\mid {\theta }_t)-Q(\theta \mid {\theta }_t)+{\mathrm{\ell }}_x(\theta )-{\mathrm{\ell }}_x({\theta }_t),$ which rearranges to the claimed identity.

By [F6], the KL term in the identity is nonnegative, so the observed log-likelihood difference is at least the Q difference. A generalized admissible update has nonnegative Q difference by [F7]. An attained exact maximizer over a set containing ${\theta }_t$ has the same property. Hence either update does not decrease the observed log likelihood.