Question 16: There are two coins from which one is randomly chosen an…

There are two coins from which one is randomly chosen and tossed 5 times, and the sequence of heads or tails is recorded. Consider the table below, which contains sample data from the experiment described above.

Based on the above data, answer the given subquestions.

(BONUS QUESTION)

Now assume that the coin identity $Z$ is hidden from the given table, but the outcomes $X$ are observed. Two coins have unknown head probabilities $\theta_1$ and $\theta_2$. One of the coins is chosen at random with equal probability ($P(Z = 1) = P(Z = 2) = 0.5$), and then tossed 5 times.

Suppose the initial parameter guesses are:

$\theta_1^{(0)} = 0.4, \quad \theta_2^{(0)} = 0.8.$

Perform one iteration of the EM algorithm:

E-step: Compute the posterior probabilities

$r_{i1} = P(Z_i = 1 \mid X_i, \theta^{(0)}), \quad r_{i2} = 1 - r_{i1}$

for each trial $i = 1, \ldots, 6$.

M-step: Use these posterior weights to update

$\theta_1^{(1)} = \frac{\sum_{i=1}^{6} r_{i1} \cdot (\text{heads in trial } i)}{\sum_{i=1}^{6} r_{i1} \cdot 5},$

$\theta_2^{(1)} = \frac{\sum_{i=1}^{6} r_{i2} \cdot (\text{heads in trial } i)}{\sum_{i=1}^{6} r_{i2} \cdot 5}.$

Using the updated parameters $\theta_1^{(1)}, \theta_2^{(1)}$, compute the posterior probability that trial 5 ($HTHHH$) was generated by coin 1:

$P(Z_5 = 1 \mid X_5, \theta_1^{(1)}, \theta_2^{(1)}).$