EMgLASSO

Simulate a precision matrix using graphical LASSO and EM

Consider a d-dimensional mixture distribution with k mixture components associated with the parameter vector Θ = (τ, µ, Σ), representing respectively the mixture probability, the mean vector, and the covariance matrix for each component in the MMN distribution. For a random sample X 0 = (X 0 1 , ..., X 0 n ) of size n drawn independently from the population, the unobserved data are the random variables Z 0 = (Z1, ..., Zn) that determine with probability τj from which mixture the observation originates: Xi |(Zi = j) ∼ Nd(µj , Σj), P(Zi = j) = τj , j = 1, ..., k, subject to X k j=1 τj = 1. These variables constitute a random sample on the multinomial random variable Z. Therefore Z1, ..., Zn are independently and identically distributed multinomial random variables with probabilities τ. Let x 0 = (x 0 1 , ..., x 0 n ) be the observed values of X given by the sample. And z 0 = (z1, ...,zn) are the latent values associated with the realization of the random sample. The observed likelihood function is L(Θ|x) = Qn i=1 Pk j=1 τj f(xi ; µjΣj), where where f Date: September 2018. 1 2 HARRISON WATTS is the pdf of the jth multivariate normal distribution in the mixture and is given by f(x; µ, Σ) = exp(− 1 2 (x − µ) 0Σ −1 (x − µ)) p (2π) d |Σ| , where |Σ| is the determinant of the covariance matrix Σ. The complete likelihood function is the product L c (Θ; x, z) = Yn i=1 Y k j=1 τj f(xn; µj , Σj) 1(z i=j) , where 1 is the indicator function. The product is used with an indicator function in the exponent because the latent values are given in the complete likelihood function, which is unknown in practice.