Maximum likelihood inference by estimating the parameters of the probability distribution

Question

I'm wondering if the following two formulations of maximum likelihood inference yield the same result.

Let $Z$ be a 0-or-1 latent random variable and $X$ a random variable that depends on $Z$ according to some known conditional probability distribution $p(x|z)$. If we observe a value $x$ but not $z$, then the task of maximum likelihood inference is to find $z^*$ that maximizes $p(x|z^*)$.

Now consider the following alternative formulation. Let $Z'$ be Bernoulli$(\theta)$, where $\theta$ is a parameter, and $X'$ depend on $Z'$ in the same way. In other words, $p(x|z)$ and $p(x'|z')$ are the same conditional distribution. This induces a distribution $p_\theta(x')$ on $x$ given by $p_{\theta}(x') = \sum_z p(x'|z') p_\theta(z')$ where $p_\theta(z')$ is the probability distribution of Bernoulli$(\theta)$, i.e., $p_\theta(1)=\theta$ and $p_\theta(0)=1-\theta$. Now if we observe a value $x$, we can imagine finding $\theta^*$ that maximizes $p_{\theta^*}(x)$.

My question: Is it true that if $z^*=1$ then $\theta^* \ge 1/2$, and if $\theta^* > 1/2$ then $z^*=1$?

In other words, if we use the alternative formulation to form a maximum likelihood estimate of $\theta$, and then use the most likely value of Bernoulli$(\theta)$ as our estimate of $Z$, does this yield the same result as ordinary maximum likelihood inference (the first formulation)? To put it another way, can we do maximum likelihood inference by parametrizing a model for the latent variable we want to infer, forming a maximum likelihood estimate for those parameters, and then using that to find the maximum likelihood value of the latent variable?

Answer 1

For this particular case, yes, it works.

If $p(x|0)<p(x|1)$, then $z^*=1$, and $$p_{\theta^*}(x) = p(x|0) (1-\theta^*) + p(x|1) \theta^* = p(x|0) + (p(x|1)-p(x|0))\theta^*,$$ which is maximized at $\theta^*=1$.

Similarly, if $p(x|0)>p(x|1)$, then $z^*=0$ and $\theta^*=0$.