Consider a two-dimensional data set D of N points, where some points have only a single coordinate value observed. In particular, without loss of generality, assume that the points in D are ordered so that:
• the first n points have only their first coordinate observed, collected in data set ;
• the following m points have only their second coordinate observed, collected in data set
1,…,n + m;
• the rest of the data points have both coordinates observed, collected in data set , i = n + m + 1,…,N.
• D = D1∪D2∪D3. Obviously, n + m < N.
We would like to estimate a bi-variate Gaussian model of the data D. To make our life easier, assume that we already know the full covariance matrix C∈R2×2 and its inverse (precision matrix) P = C−1 ∈R2×2, but we need to estimate the mean µ = (µ1,µ2)T.
Formulate an E-M algorithm to estimate the mean µ by treating the unobserved coordinate values as latent variables.
• Clearly show the Initialisation
• Cleary demonstrate the E-Step
• Clearly show the M-step
Sample Solution