# Estimating sparse precision matrices - penalised likelihood method

In order to estimate sparse precision matrices, there is a method called "penalised likelihood" which leads to this formula.

Can someone write down the demonstration ? I do not understand how we arrive to this equation... Thanks !

The equation is: $$\hat\Theta = argmin_{Theta} \{tr{S\Theta}-log det{\Theta} + \lambda\sum_{i\neq j}{|\Theta_{ij}|}\}$$
The purpose is to find the best estimate of $$\Theta$$ the inverse of covariance matrix, called precision matrix. We have to maximize the likelihood of a Gaussian distribution, with $$\mu_x = 0$$. $$f(x)=\frac{1}{\sqrt{(2π)^n|\Sigma|}} exp\{−\frac{1}{2} x^T \Sigma^{-1} x \}$$ constrained by sparsity of precision matrix. We can substitute $$\Sigma^{-1}$$ with $$\Theta$$.
Usually for log-linear probabilities we solve them by getting logarithm of the distribution. Now that we want to find $$\Theta$$ instead of $$\Sigma$$, we can minimize the negate of logarithm of Gaussian likelihood.
The third term is the penalty term, for an optimization problem with Lagrange multiplier $$\lambda$$. Why is it penalized? Because we have the constraint of sparsity on the precision matrix. It is applied with limiting the norm $$l_1$$ (sum of absolute values of elements) of matrix $$\Theta$$.