Now assume we do not know which coin is tossed in each experiment, and we decide to use the EM algorithm to estimate θA,θB . Suppose we initialize the parameters as θ0A=0.6 and θ0B=0.4 . Also, we initially think coin A and coin B are selected with equal probability and independently in each experiment.
Experiment | Coin | Outcome
1 | ? | H T T H
2 | ? | H H H T
Perform the first iteration of E-step, find the value of Q1(z(1)=A) , i.e. the conditional probability that coin A is selected in Experiment 1, and Q2(z(2)=A) , i.e. the conditional probability that coin A is selected in Experiment 2.
(Enter numerical answers accurate to at least 2 decimal places. )
Q1(z(1)=A)=
unanswered
Q2(z(2)=A)=
unanswered
After performing the first iteration of E-step, perform the first iteration of M-step, find θ1A,θ1B , the value of θ after the first iteration of the EM algorithm.
(Enter numerical answers accurate to at least 2 decimal places. )
θ1A=
unanswered
θ1B=
In the EM algorithm, the E-step involves estimating the values of the hidden variables (in this case, the coin that is tossed in each experiment), given the observed data and the current parameter estimates. The M-step involves updating the parameter estimates based on the estimated values of the hidden variables.
To find the conditional probability that coin A is selected in Experiment 1 (Q1(z(1)=A)), we calculate the probability that coin A produced the observed outcome in Experiment 1, given the current parameter estimates.
Q1(z(1)=A) = P(Z1=A | Outcome1) = (P(Z1=A) * P(Outcome1 | Z1=A)) / P(Outcome1)
Given that we initially think coin A and coin B are selected with equal probability and independently in each experiment, we can assume P(Z1=A) = P(Z1=B) = 0.5.
To calculate P(Outcome1 | Z1=A), we first need to estimate the probability of obtaining each outcome (H or T) when using coin A. Let's denote θA as the parameter representing the probability of getting a head when tossing coin A. From the given initial parameter values, θ0A=0.6.
P(H | Z1=A) = θA = 0.6
P(T | Z1=A) = 1 - θA = 1 - 0.6 = 0.4
Now we can calculate P(Outcome1 | Z1=A) by multiplying the probabilities of getting each outcome in the given sequence of heads and tails:
P(Outcome1 | Z1=A) = P(H T T H | Z1=A) = P(H | Z1=A) * P(T | Z1=A) * P(T | Z1=A) * P(H | Z1=A) = 0.6 * 0.4 * 0.4 * 0.6 = 0.0576
To calculate P(Outcome1), we need to consider the joint probabilities of each possible hidden variable value (coin A or coin B) and the observed outcome. Since we initially think coin A and coin B are selected with equal probability and independently in each experiment, we can assume P(Z1=A) = P(Z1=B) = 0.5. Similarly, θ0B=0.4.
P(Outcome1) = P(Outcome1 | Z1=A) * P(Z1=A) + P(Outcome1 | Z1=B) * P(Z1=B) = 0.0576 * 0.5 + P(Outcome1 | Z1=B) * 0.5
Now, we can find Q1(z(1)=A) as follows:
Q1(z(1)=A) = P(Z1=A | Outcome1) = (P(Z1=A) * P(Outcome1 | Z1=A)) / P(Outcome1) = (0.5 * 0.0576) / (0.0576 * 0.5 + P(Outcome1 | Z1=B) * 0.5)
To find the conditional probability that coin A is selected in Experiment 2 (Q2(z(2)=A)), we follow the same procedure. We use the observed outcome in Experiment 2 and the current parameter estimates to calculate the probabilities and joint probabilities.
After performing the first iteration of the E-step and finding Q1(z(1)=A) and Q2(z(2)=A), we move on to the M-step.
In the M-step, we update the parameter estimates based on the estimated values of the hidden variables. We calculate the new values of θA and θB by taking the average of the probabilities of getting a head for each experiment, weighted by the corresponding conditional probabilities.
θ1A = (Q1(z(1)=A) * P(H | Z1=A) + Q2(z(2)=A) * P(H | Z1=A)) / (Q1(z(1)=A) + Q2(z(2)=A))
θ1B = (Q1(z(1)=B) * P(H | Z1=B) + Q2(z(2)=B) * P(H | Z1=B)) / (Q1(z(1)=B) + Q2(z(2)=B))
Now you can use the above formulas to calculate the values of Q1(z(1)=A), Q2(z(2)=A), θ1A, and θ1B based on the given initial parameters and observed outcomes in the experiments.
To perform the E-step, we need to calculate the conditional probabilities of selecting coin A (θA) and coin B (θB) given the outcomes of the experiments.
For Experiment 1:
We need to calculate Q1(z(1)=A), the conditional probability that coin A is selected in Experiment 1.
First, let's calculate the probability of getting the given outcome (H T T H) if coin A is selected:
P(outcome|coin A) = θA * (1 - θA) * (1 - θA) * θA
= 0.6 * (1 - 0.6) * (1 - 0.6) * 0.6
= 0.6 * 0.4 * 0.4 * 0.6
= 0.0576
Next, let's calculate the probability of getting the given outcome if coin B is selected:
P(outcome|coin B) = θB * θB * θB * (1 - θB)
= 0.4 * 0.4 * 0.4 * (1 - 0.4)
= 0.4 * 0.4 * 0.4 * 0.6
= 0.0576
Now, let's calculate the total probability of getting the given outcome:
P(outcome) = P(outcome|coin A) + P(outcome|coin B)
= 0.0576 + 0.0576
= 0.1152
Finally, let's calculate the conditional probability of selecting coin A in Experiment 1 using Bayes' theorem:
Q1(z(1)=A) = P(coin A|outcome) = P(outcome|coin A) * P(coin A) / P(outcome)
= 0.0576 * (0.5) / 0.1152
= 0.25
Now, let's move on to Experiment 2.
For Experiment 2:
We need to calculate Q2(z(2)=A), the conditional probability that coin A is selected in Experiment 2.
Using the same process, we can calculate the probability of the given outcome (H H H T) if coin A is selected:
P(outcome|coin A) = θA * θA * θA * (1 - θA)
= 0.6 * 0.6 * 0.6 * (1 - 0.6)
= 0.6 * 0.6 * 0.6 * 0.4
= 0.0864
Similarly, the probability of the given outcome if coin B is selected:
P(outcome|coin B) = θB * (1 - θB) * (1 - θB) * θB
= 0.4 * (1 - 0.4) * (1 - 0.4) * 0.4
= 0.4 * 0.6 * 0.6 * 0.4
= 0.0576
Total probability of getting the given outcome:
P(outcome) = P(outcome|coin A) + P(outcome|coin B)
= 0.0864 + 0.0576
= 0.144
Conditional probability of selecting coin A in Experiment 2 using Bayes' theorem:
Q2(z(2)=A) = P(coin A|outcome) = P(outcome|coin A) * P(coin A) / P(outcome)
= 0.0864 * (0.5) / 0.144
≈ 0.3
Now, let's move on to the M-step and calculate θ1A and θ1B.
In the M-step, we update the values of θA and θB based on the conditional probabilities obtained in the E-step.
To calculate θ1A, we use the following formula:
θ1A = Summation(Qi(z(i)=A) * xi) / Summation(Qi(z(i)=A)), where xi represents the number of heads in experiment i.
θ1A = (Q1(z(1)=A) * 3 + Q2(z(2)=A) * 3) / (Q1(z(1)=A) + Q2(z(2)=A))
= (0.25 * 3 + 0.3 * 3) / (0.25 + 0.3)
= 0.475
To calculate θ1B, we use the following formula:
θ1B = 1 - θ1A
= 1 - 0.475
= 0.525
Therefore, the values obtained after the first iteration of the EM algorithm are:
θ1A = 0.475
θ1B = 0.525