To find the Maximum A Posteriori (MAP) estimate of Θ, we need to compute the conditional probability of Θ given the observed values X_1 = -1 and X_2 = 1, denoted as P(Θ | X_1, X_2).
By Bayes' theorem, we have:
P(Θ | X_1, X_2) = (P(X_1, X_2 | Θ) * P(Θ)) / P(X_1, X_2)
Since Θ, W1, and W2 are independent standard normal random variables, we can assume that X1 and X2 are also normally distributed with means Θ and 2Θ, respectively, and unit variances:
P(X_1, X_2 | Θ) = P(X_1 | Θ) * P(X_2 | Θ) = (1/√(2π)) * exp(-0.5 * ((-1 - Θ)^2)) * (1/√(2π)) * exp(-0.5 * ((1 - 2Θ)^2))
Substituting these values into the Bayes' theorem equation, we get:
P(Θ | X_1, X_2) = ((1/√(2π)) * exp(-0.5 * ((-1 - Θ)^2)) * (1/√(2π)) * exp(-0.5 * ((1 - 2Θ)^2)) * P(Θ)) / P(X_1, X_2)
We want to maximize this probability, which is equivalent to maximizing its logarithm:
log(P(Θ | X_1, X_2)) = log(((1/√(2π)) * exp(-0.5 * ((-1 - Θ)^2)) * (1/√(2π)) * exp(-0.5 * ((1 - 2Θ)^2)) * P(Θ)) / P(X_1, X_2))
Simplifying further:
log(P(Θ | X_1, X_2)) = log(P(Θ)) - 0.5 * ((-1 - Θ)^2) - 0.5 * ((1 - 2Θ)^2) - log(P(X_1, X_2))
Since Θ, W1, and W2 are independent standard normal random variables, the prior probability P(Θ) follows a standard normal distribution with mean 0 and variance 1. Therefore, log(P(Θ)) = -0.5 * Θ^2 + constant.
Regarding the term log(P(X_1, X_2)), since we are given the values X1 = -1 and X2 = 1, we can compute this term as well.
Finally, to find the MAP estimate of Θ, we need to find the value of Θ that maximizes the expression log(P(Θ | X_1, X_2)).
It's important to note that this is a general outline of the process, and the specific calculations involved may require further details or numerical methods depending on the values of X1 and X2.
Suppose that X_1=\Theta +W_1 and X_2=2\Theta +W_2, where \Theta ,W_1,W_2 are independent standard normal random variables. If the values that we observe happen to be X_1=-1 and X_2=1, then the MAP estimate of \Theta is
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