Let Θ1, Θ2, W1, and W2 be independent standard normal random variables. We obtain two observations,
X1=Θ1+W1,X2=Θ1+Θ2+W2.
Find the MAP estimate θ^(theta hat)=(θ^1,θ^2) of (Θ1,Θ2) if we observe that X1=1, X2=3.
1.θ^1 (theta hat)=?
2.θ^2(theta hat)=?
1. Well, if I were to take a wild guess, θ^1 (theta hat) would be... drumroll, please... 1! Remember, I'm a Clown Bot, not a mathematician, so take my answer with a pinch of confetti.
2. As for θ^2 (theta hat), my hypothesis is... 2! I mean, if X1 = 1 and X2 = 3, it only makes sense to juggle them around a bit and come up with a difference of 2. But hey, don't quote me on that, I'm just here for the laughs!
To find the Maximum A Posteriori (MAP) estimate of (Θ1, Θ2), we need to determine the values of θ^1 (theta hat) and θ^2 (theta hat) that maximize the posterior probability given the given observations X1=1 and X2=3.
Let's start by writing down the posterior probability function:
P(Θ1, Θ2 | X1=1, X2=3) ∝ P(X1=1, X2=3 | Θ1, Θ2) * P(Θ1, Θ2)
Since Θ1, Θ2, W1, and W2 are independent standard normal random variables, we have:
P(X1=1, X2=3 | Θ1, Θ2) = P(Θ1+W1=1, Θ1+Θ2+W2=3)
Expanding this probability function, we get:
P(Θ1+W1=1, Θ1+Θ2+W2=3) = P(Θ1=1-W1, Θ2=3-Θ1-W2)
Next, we need to express the prior probability P(Θ1, Θ2). Since Θ1 and Θ2 are independent standard normal random variables, we have:
P(Θ1, Θ2) = P(Θ1) * P(Θ2)
Since both Θ1 and Θ2 are standard normal random variables, their prior distribution is also standard normal distribution.
Now, let's substitute these expressions into the posterior probability function:
P(Θ1, Θ2 | X1=1, X2=3) ∝ P(Θ1=1-W1, Θ2=3-Θ1-W2) * P(Θ1) * P(Θ2)
To simplify the calculations, we can take the logarithm of the posterior probability:
log(P(Θ1, Θ2 | X1=1, X2=3)) = log(P(Θ1=1-W1, Θ2=3-Θ1-W2)) + log(P(Θ1)) + log(P(Θ2))
Since logarithm is a monotonically increasing function, maximizing the logarithm of the posterior probability is equivalent to maximizing the posterior probability itself.
Finally, we can maximize the posterior probability by finding the values of θ^1 (theta hat) and θ^2 (theta hat) that maximize the expression:
log(P(Θ1=1-W1, Θ2=3-Θ1-W2)) + log(P(Θ1)) + log(P(Θ2))
Note: The expressions for log(P(Θ1=1-W1, Θ2=3-Θ1-W2)), log(P(Θ1)), and log(P(Θ2)) will depend on the specific form of the standard normal distribution.
To find the MAP estimate θ^(theta hat) = (θ^1, θ^2) of (Θ1, Θ2) given the observations X1 = 1 and X2 = 3, we need to maximize the posterior probability density function (PDF) of (Θ1, Θ2) given the data.
The posterior PDF can be found using Bayes' theorem:
P(θ1, θ2 | X1 = 1, X2 = 3) ∝ P(X1 = 1, X2 = 3 | θ1, θ2) * P(θ1, θ2)
Let's break down the expression:
1. P(X1 = 1, X2 = 3 | θ1, θ2) is the likelihood function, which describes the probability of observing the given data (X1 = 1, X2 = 3) given the values of θ1 and θ2.
2. P(θ1, θ2) is the prior distribution, which represents our beliefs about the values of θ1 and θ2 before observing any data. Since Θ1 and Θ2 are independent standard normal random variables, their prior distribution can be represented as the product of their individual PDFs: P(θ1, θ2) = P(θ1) * P(θ2), where P(θ1) and P(θ2) are the PDFs of Θ1 and Θ2, respectively.
Now, we need to compute the posterior PDF and maximize it to find the MAP estimates.
P(θ1, θ2 | X1 = 1, X2 = 3) ∝ P(X1 = 1, X2 = 3 | θ1, θ2) * P(θ1) * P(θ2)
Since Θ1, Θ2, W1, and W2 are independent standard normal random variables, we have:
X1 = Θ1 + W1
X2 = Θ1 + Θ2 + W2
Substituting the observed values X1 = 1 and X2 = 3, we can rewrite the likelihood function as:
P(X1 = 1, X2 = 3 | θ1, θ2) = P(1 = θ1 + W1, 3 = θ1 + θ2 + W2)
To simplify further, let's rewrite W1 and W2 as Z1 and Z2, which are standard normal random variables:
P(X1 = 1, X2 = 3 | θ1, θ2) = P(1 = θ1 + Z1, 3 = θ1 + θ2 + Z2)
Now, we need to find the PDFs of Z1 and Z2. Since W1 and W2 are standard normal random variables, the PDFs of Z1 and Z2 are also standard normal.
P(X1 = 1, X2 = 3 | θ1, θ2) = P(1 = θ1 + Z1) * P(3 = θ1 + θ2 + Z2)
Using the standard normal distribution PDF formula, we can substitute the values to get:
P(X1 = 1, X2 = 3 | θ1, θ2) = (1/√(2π)) * e^(-((1 - θ1)^2)/2) * (1/√(2π)) * e^(-((3 - θ1 - θ2)^2)/2)
Now, we substitute this likelihood function along with the prior distribution P(θ1) * P(θ2), which is the product of the PDFs of Θ1 and Θ2 (standard normal).
We need to maximize this expression to find the MAP estimates θ^(theta hat) = (θ^1, θ^2) of (Θ1, Θ2). However, the expression becomes more complex, and an analytical solution may not be readily available. In such cases, numerical optimization techniques like the Expectation-Maximization (EM) algorithm or gradient ascent methods can be applied.
To find the MAP estimates in this specific case, we can use numerical optimization algorithms by implementing them in a programming language like Python, MATLAB, or R. These algorithms aim to maximize the given expression iteratively and find the optimal values of θ^(theta hat) = (θ^1, θ^2) that maximize the posterior PDF.