Let Θ1 and Θ2 be some unobserved Bernoulli random variables and let X be an observation. Conditional on X=x, the posterior joint PMF of Θ1 and Θ2 is given by
pΘ1,Θ2∣X(θ1,θ2∣x)=
0.26, if θ1=0,θ2=0,
0.26, if θ1=0,θ2=1,
0.21, if θ1=1,θ2=0,
0.27, if θ1=1,θ2=1,
0, otherwise.
We can view this as a hypothesis testing problem where we choose between four alternative hypotheses: the four possible values of (Θ1,Θ2).
a) What is the estimate of (Θ1,Θ2) provided by the MAP rule?
b) Once you calculate the estimate (θ^1,θ^2) of (Θ1,Θ2), you may report the first component, θ^1, as your estimate of Θ1. With this procedure, your estimate of Θ1 will be
c) What is the probability that Θ1 is estimated incorrectly (the probability of error) when you use the procedure in part (b)?
d) What is the MAP estimate of Θ1 based on X, that is, the one that maximizes pΘ1|X(θ1∣x)?
e) The moral of this example is that an estimate of Θ1 obtained by identifying the maximum of the joint PMF of all unknown random variables is
the MAP estimate of Θ1.
a) The estimate of (Θ1, Θ2) provided by the MAP rule is (1, 1), because it has the highest posterior probability of 0.27.
b) With the procedure mentioned, your estimate of Θ1 will be 1, because we take the first component of the MAP estimate.
c) The probability that Θ1 is estimated incorrectly (the probability of error) when using this procedure is 0, because the MAP estimate of Θ1 is correct in this case.
d) The MAP estimate of Θ1 based on X is 1, because it maximizes the posterior probability p(θ1 | x).
e) The moral of this example is that an estimate of Θ1 obtained by identifying the maximum of the joint PMF of all unknown random variables is not always the MAP estimate of Θ1.
a) The estimate of (Θ1,Θ2) provided by the MAP (Maximum A Posteriori) rule is the value that maximizes the posterior joint PMF pΘ1,Θ2|X(θ1,θ2 | x). Looking at the given posterior joint PMF, we can see that the value (θ1=1, θ2=1) has the highest probability. Therefore, the estimate of (Θ1,Θ2) provided by the MAP rule is (θ1=1, θ2=1).
b) Once we have the estimate (θ^1,θ^2) of (Θ1,Θ2), we can report the first component, θ^1, as the estimate of Θ1. In this case, since θ1=1 has the highest probability in the posterior joint PMF, our estimate of Θ1 will be θ^1=1.
c) To calculate the probability that Θ1 is estimated incorrectly (the probability of error) using the procedure in part b, we need to consider the cases where the estimated value of Θ1 (θ^1) is not equal to the true value of Θ1. From the given posterior joint PMF, we can see that the only possible incorrect estimation is when Θ1 is actually 0 but estimated to be 1. Therefore, the probability of error is the sum of the probabilities for the cases (θ1=0, θ2=1) and (θ1=0, θ2=0). From the given posterior joint PMF, the probability of error is 0.26 + 0 = 0.26.
d) The MAP estimate of Θ1 based on X, which maximizes the posterior marginal PMF pΘ1|X(θ1 | x), can be obtained by summing the probabilities for each value of Θ2 for a fixed Θ1 and choosing the value of Θ1 that gives the highest probability. From the given posterior joint PMF, we can see that the probabilities of Θ1=0 and Θ1=1 are 0.26 + 0.21 = 0.47 and 0.26 + 0.27 = 0.53, respectively. Therefore, the MAP estimate of Θ1 based on X is θ1=1.
e) The moral of this example is that an estimate of Θ1 obtained by identifying the maximum of the joint PMF of all unknown random variables is the MAP estimate of Θ1. This means that by maximizing the posterior joint PMF pΘ1,Θ2|X(θ1,θ2 | x), we can obtain the best estimate for Θ1 considering all available information.
To find the answers to the questions, we need to understand the concepts of MAP estimation and probabilities. The MAP (Maximum A Posteriori) rule is a decision rule that selects the hypothesis with the highest posterior probability. In this case, we have a joint posterior PMF of Θ1 and Θ2 given X, and we need to use this information to compute the estimates and probabilities.
a) The estimate of (Θ1,Θ2) provided by the MAP rule is the hypothesis with the maximum posterior probability. Looking at the given joint PMF, we see that the hypothesis (θ1=1, θ2=1) has the highest probability of 0.27. Therefore, the MAP estimate of (Θ1,Θ2) is (1, 1).
b) Once we have the estimate (θ^1,θ^2) of (Θ1,Θ2), we can report the first component, θ^1, as our estimate of Θ1. In this case, since (θ^1=1, θ^2=1) is the MAP estimate, we can report our estimate of Θ1 as 1.
c) To calculate the probability of error, we need to consider the cases where our estimate of Θ1 is incorrect. Looking at the joint PMF, we have the following cases where Θ1 is estimated incorrectly: (θ1=0, θ2=0) and (θ1=0, θ2=1). Both of these cases have a posterior probability of 0.26. Thus, the probability of error is the sum of these probabilities, which is 0.26 + 0.26 = 0.52.
d) The MAP estimate of Θ1 based on X, which maximizes p(Θ1|X)(θ1|x), can be found by computing the posterior probability for each value of Θ1 given X. Looking at the joint PMF, we have the following posterior probabilities for each value of Θ1:
- p(Θ1=0|X) = p(Θ1=0, Θ2=0|X) + p(Θ1=0, Θ2=1|X) = 0.26 + 0.26 = 0.52
- p(Θ1=1|X) = p(Θ1=1, Θ2=0|X) + p(Θ1=1, Θ2=1|X) = 0.21 + 0.27 = 0.48
Since p(Θ1=0|X) > p(Θ1=1|X), the MAP estimate of Θ1 based on X is 0.
e) The moral of this example is that an estimate of Θ1 obtained by identifying the maximum of the joint PMF of all unknown random variables is the MAP estimate of Θ1. This means that by finding the hypothesis with the highest posterior probability given the observed data, we can make the best estimate for the value of Θ1.