Question

Expectation Maximization (EM) Assume we have a dataset of two points {x(1),x(2)} : x(1)=−1,x(2)=1. Assume x(i) is drawn i.i.d. from a simple mixture distribution of two Gaussian components: f(x∣μ1​,μ2​)=21​ϕ(x∣μ1​,1)+21​ϕ(x∣μ2​,1), where ϕ(⋅∣μi​,1) denotes the probability density function of Gaussian distribution N(μi​,1) with mean μi​ and unit variance. We want to estimate the unknown parameters μ1​ and μ2​. Assume we run EM starting from an initialization of μ1​=−2 and μ2​=2.

Answer 1

The Expectation Maximization (EM) algorithm is used to estimate the unknown means μ1 and μ2 of a mixture of two Gaussian components in this problem.

Step-by-step explanation:

The Expectation Maximization (EM) algorithm is used to estimate unknown parameters in a mixture model. In this case, we are trying to estimate the unknown means μ1 and μ2 of a mixture of two Gaussian components. The data set consists of two points: x(1) = -1 and x(2) = 1.

1. Initialize the means with μ1 = -2 and μ2 = 2.
2. Expectation Step: Calculate the responsibilities of each component for each data point using the current means and the Gaussian density function.
3. Maximization Step: Update the means by calculating the weighted average of the data points based on the responsibilities.
4. Repeat steps 2 and 3 until convergence.

By running the EM algorithm iteratively, we will converge to estimates of μ1 and μ2 that best fit the given data set.