Maximum likelihood degree of the two-dimensional linear Gaussian covariance model

Abstract

In algebraic statistics, the maximum likelihood degree of a statistical model is the number of complex critical points of its log-likelihood function. A priori knowledge of this number is useful for applying techniques of numerical algebraic geometry to the maximum likelihood estimation problem. We compute the maximum likelihood degree of a generic two-dimensional subspace of the space of n × n Gaussian covariance matrices. We use the intersection theory of plane curves to show that this number is 2n − 3.

Estimating linear covariance models with numerical nonlinear algebra

“…However, the polynomiality along the rows of Table 1 seems to persist. For m = 2 the ML degree equals 2n − 3, as shown recently by Coons, Marigliano and Ruddy [7]. For m ≥ 3 we propose the following conjecture.…”

“…Outside a Zariski closed set ∆ ⊂ Q, every system in F Q has the same number of solutions. If p ∈ Q\∆ then F p is such a generic instance of the family F Q , and the following is a suitable homotopy [25]: (7) H(x, t) = F (1−t)p+tq (x) . Now, to compute V (F q ), it suffices to find all solutions of a generic instance F p and then track these along the homotopy (7).…”

“…If p ∈ Q\∆ then F p is such a generic instance of the family F Q , and the following is a suitable homotopy [25]: (7) H(x, t) = F (1−t)p+tq (x) . Now, to compute V (F q ), it suffices to find all solutions of a generic instance F p and then track these along the homotopy (7). Obtaining all solutions of a generic instance can be a challenge, but this has to be done only once!…”

Rational partition models under iterative proportional scaling

“…For example in [2], the authors connect invariant theory and algorithms to compute the MLE including a form of the IPS algorithm. Many recent works have also explored the number of complex critical points of the log-likelihood function, also known as the ML-degree (see [1,8,11,19,29,30,39]).…”

Proof of a conjecture of Sturmfels, Timme and Zwiernik

“…For m ≤ 4, the ML-degree M L m of a generic linear concentration model of dimension m is:M L 2 = 2n − 3,(2)M L 3 = 3n 2 − 9n + 7was proved in[CMR20]. Statements (2) and (3) were conjectured in [STZ20, Conjecture 4.2].…”