Suppose we have two coins (coin A and coin B ), and we conduct two independent experiments in which a single coin is tossed four times. The outcomes of the experiments are presented in the table below. We use H to denote a Heads and T to denote a Tails. Denote θA:=P(H|A) , i.e. the probability of getting H when tossing coin A , and θB:=P(H|B) , i.e. the probability of getting H when tossing coin B , as the parameters we want to estimate, x(i) as the outcome of each Experiment i , z(i) as the index ( A or B ) of the coin tossed in Experiment i .
Experiment Coin Outcome
1 B H T T H
2 A H H H T
First assume we know from the table in each experiment which coin is tossed. Hence there is no "hidden" information and we can do the maximum likelihood estimation (MLE) directly.
As all the experiments are independent, the expression of P(x(1),x(2)|θA,θB) should be in the form of θaA(1−θA)bθcB(1−θB)d . Find a,b,c,d based on the outcomes in the table.
a=
unanswered
b=
unanswered
c=
unanswered
d=
To find the values of a, b, c, and d based on the outcomes in the table, we need to count the number of heads (H) and tails (T) for each experiment and for each coin.
For Experiment 1 with Coin B, the outcome is H T T H. So, there are 3 tails (b=3) and 1 head (a=1).
For Experiment 2 with Coin A, the outcome is H H H T. So, there are 3 heads (c=3) and 1 tail (d=1).
Therefore:
a=1
b=3
c=3
d=1
To find values of a, b, c, and d, we need to count the number of heads and tails for each coin in the given outcomes.
For Coin A:
In Experiment 2, there are 3 heads and 1 tail. Therefore, a=3 (the number of heads).
For Coin B:
In Experiment 1, there is 1 head and 3 tails. Therefore, b=1 (the number of heads).
Now let's calculate the number of tails for each coin:
For Coin A:
In Experiment 2, there is 1 tail. Therefore, d=1 (the number of tails).
For Coin B:
In Experiment 1, there are 3 tails. Therefore, c=3 (the number of tails).
So, the values of a, b, c, and d are:
a=3, b=1, c=3, d=1.