Let ${}_1$ and ${}_2$ be some unobserved Bernoulli random variables and let $x$ be an observation. Conditional on $x=x,$ the posterior joint PMF of ${}_1$ and ${}_2$ is given by

$|)(x|)=\{\begin{array}{ccc}0\mathrm{.}26,& if{}_1=0,& {}_2=0\\ 0\mathrm{.}26,& if{}_1=0,& {}_2=1\\ 0\mathrm{.}21,& if{}_1=1,& {}_2=0\\ 0\mathrm{.}27,& if{}_1=1,& {}_2=1\\ 0,& \text{o}\text{t}\text{h}\text{e}\text{r}\text{w}\text{i}\text{s}\text{e}\text{.}& \end{array}$

We can view this as a hypothesis testing problem where we choose between four alternative hypotheses: the four possible values of $({}_1,{}_2).$

a$)$ What is the estimate of $({}_1,{}_2)$ provided by the MAP rule?

Select an option $v$

b$)$ Once you calculate the estimate $(hat({)}_1,$hat$({)}_2)$ of $({}_1,{}_2),$ you may report the first component, hat$({)}_1,$ as your estimate of ${}_1.$ With this procedure, your estimate of ${}_1$ will be

c$)$ What is the probability that ${}_1$ is estimated incorrectly $($the probability of error$)$ when you use the procedure in part $($b$)?$

d$)$ What is the MAP estimate of ${}_1$ based on $x,$ that is$,$ the one that maximizes $|)(x|)?$

e$)$ The moral of this example is that an estimate of ${}_1$ obtained by identifying the maximum of the joint PMF of all unknown random variables

$$ the MAP estimate of ${}_1.$