An EM likelihood with two strict local maxima

Since $0\le ({\theta }^2-1{)}^2\le 9$ on $[-2,2]$, $1\mathrm{/}4\le r(\theta )\le 1\mathrm{/}2$. Thus $p_{\theta }$ is nonnegative and sums to one over $(x,z)\in \{0,1{\}}^2$ for every $\theta$. The observed density at $x=1$ is $m_{\theta }(1)=r(\theta )$.

The squared term $({\theta }^2-1{)}^2$ vanishes exactly at $\theta =\pm 1$ and is positive elsewhere. Therefore $r(\theta )$, and hence the observed likelihood and log likelihood, have strict local maxima at those two distinct parameter values; both are global maximizers with equal value, so neither is a strict global maximum.

For every current value ${\theta }_t=\pm 1$, the latent conditional law is uniform on $Z$. For any candidate $\theta$, $Q(\theta \mid {\theta }_t)=\log r(\theta )-\log 2$, so its exact maximizer set is again $\{-1,1\}$. Starting from ${\theta }_0=-1$, choosing ${\theta }_{t+1}=-{\theta }_t$ at each step is an attained exact M-step; all fibers have positive finite marginals and identical latent support, with finite Q and KL. The algorithm therefore emits the infinite alternating sequence, which has no limit while its log likelihood is constant. This verifies the limitation described in [F5] without contradicting the one-step ascent result [F4].