Suppose that fÈ and fX|È are described by simple closed-form formulas. Suppose that È is one-dimensional but X is high-dimensional.
a) Suppose that a specific value x of the random variable X has been observed. Is it true that the calculation of the LMS estimate will always involve only ordinary integrals (integrals with respect to only one variable)?
Status: unsubmitted
b) Is it true that the calculation of the mean squared error of the LMS estimator will always involve only ordinary integrals (integrals with respect to only one variable)?
Status: unsubmitted
(a) Yes
(b) No
a) Yes. The denominator in Bayes' rule involves an integral with respect to È. Once the conditional PDF is available, the LMS estimate is calculated by integrating again over the one-dimensional variable È.
b) No. In this case, we need to average the conditional variance over all possible values of x, and this will involve a multiple integral.
a) Well, if X is high-dimensional and È is one-dimensional, it's like trying to fit an elephant into a clown car. It might be a tight squeeze! So, the calculation of the LMS estimate might not be as simple as just one ordinary integral. You may need to integrate over multiple variables to capture all the dimensions of X. It's like juggling multiple balls at once!
b) Calculating the mean squared error of the LMS estimator? Well, that sounds like a real circus act! Since X is high-dimensional, you might need to integrate over all those dimensions to get a comprehensive estimate. So, sorry to disappoint, but it seems like you'll need more than just one ordinary integral. You'll be juggling integrals like a skilled clown!
a) No, it is not necessarily true that the calculation of the LMS estimate will always involve only ordinary integrals when a specific value x of the random variable X has been observed. The LMS estimate involves finding the value of È that minimizes the mean squared error (MSE) between the observed value x and the predicted value of X given È. The calculation of this estimator depends on the specific formulas for fÈ and fX|È. If the formulas involve joint distributions or dependencies between the dimensions of X, then the calculation of the LMS estimate may involve multiple integrals.
b) Similarly, it is not necessarily true that the calculation of the mean squared error of the LMS estimator will always involve only ordinary integrals. The mean squared error is given by the expected value of the squared difference between the observed value x and the predicted value of X given È, averaged over all possible values of È. The calculation of this expectation may involve multiple integrals if the formulas for fÈ and fX|È involve joint distributions or dependencies.
a) The calculation of the LMS (Least Mean Squares) estimate involves finding the conditional expectation of X given È, denoted as E[X|È]. In this case, È is a one-dimensional random variable, while X is a high-dimensional random variable.
To calculate E[X|È], we need to evaluate the integral of X with respect to its high-dimensional domain, and this integration can involve multiple variables. Therefore, the calculation of the LMS estimate can potentially involve multiple integrals, not just ordinary integrals with respect to one variable.
b) The mean squared error (MSE) of the LMS estimator measures the average squared difference between the estimated value and the true value of X. It is given by E[(X - ȳ)^2], where ȳ is the LMS estimate.
To calculate the MSE, we need to evaluate the expected value of the squared difference (X - ȳ)^2. This involves taking the expectation over the joint distribution of X and È, which can be high-dimensional.
In general, when È is one-dimensional and X is high-dimensional, calculating the MSE of the LMS estimator may involve integrals over the high-dimensional domain of X and È. These integrals can potentially involve multiple variables, not just ordinary integrals with respect to one variable.