Stationary Points and Likelihood Geometry

Introduction

A Gaussian-mixture likelihood is generally not a single smooth hill with one peak. Its surface can contain local maxima, saddle points, and directions along which the likelihood does not change. These distinctions matter because an iterative method can settle at a point where the likelihood cannot be improved by a small move in some directions, even though other directions lead to a better fit.

This tutorial focuses on the geometry of regular interior points: points where the mixture weights are positive, covariance matrices are positive definite, and the likelihood is differentiable in the chosen parameterization. The examples use fixed, known variances to make the calculations transparent. The central lesson is that convergence of EM is evidence about the path taken by an algorithm, not a certificate that the best possible likelihood has been found.

In a mixture model, \(\theta\) typically includes component parameters and coordinates for the weights. For \(K\) components, the observed-data density for observation \(x_i\) has the form \(f(x_i;\theta)=\sum_{k=1}^K\pi_k f_k(x_i;\eta_k)\), with \(\pi_k>0\) and \(\sum_k\pi_k=1\). The log-likelihood is the sum of the logarithms of these mixture densities. Because it is a sum of component densities inside a logarithm, its curvature is not simply the curvature of any one component likelihood.

What the Hessian Can Tell Us

At a stationary point, the Hessian matrix \(H(\theta_*)=\nabla^2\ell(\theta_*)\) describes the leading change in log-likelihood for small parameter perturbations. A negative-definite Hessian gives a strict local maximum. An indefinite Hessian gives a saddle: there are directions in which the likelihood rises and others in which it falls. If the Hessian has zero eigenvalues, its second-order information is insufficient by itself.

To see why, write a nearby point as \(\theta_*+h\). The second-order Taylor expansion at a stationary point is

If the Hessian is negative definite, there is a constant \(c>0\) such that \(h^\mathsf{T}H(\theta_*)h\leq-c\lVert h\rVert^2\). The remainder divided by \(\lVert h\rVert^2\) tends to zero, so for sufficiently small nonzero \(h\), the negative quadratic term dominates and the likelihood decreases. If the Hessian has positive and negative eigenvalues, choose unit eigenvectors \(v_+\) and \(v_-\) for those eigenvalues. Along \(h=t v_+\), the leading change is positive for sufficiently small nonzero \(t\); along \(h=t v_-\), it is negative. Thus every neighborhood contains points on both sides of the stationary likelihood value. \(\square\)

This test is decisive when the Hessian has the specified curvature, but it is not a complete classification rule. A zero eigenvalue means that the quadratic term vanishes in at least one direction. Higher-order terms—or an exact comparison of likelihood values—are then needed.

Mixture Curvature and Responsibilities

Responsibilities make the structure of the Hessian explicit. For one observation, write \(q_k(\theta)=\log\{\pi_k f_k(x_i;\eta_k)\}\) in a smooth coordinate system for the parameters. Then \(\log f(x_i;\theta)=\log\sum_k\exp(q_k(\theta))\), and the posterior responsibility is \(r_{ik}=\exp(q_k)/\sum_j\exp(q_j)\). Differentiating this log-sum yields a Hessian with two contributions.

For a derivation, differentiating \(\log\sum_k\exp(q_k)\) gives \(\sum_k r_{ik}u_k=\bar u\). Differentiating the responsibility gives \(\nabla r_{ik}=r_{ik}(u_k-\bar u)\). Therefore

The first sum equals \(\sum_k r_{ik}(u_k-\bar u)(u_k-\bar u)^\mathsf{T}\), since \(\sum_k r_{ik}(u_k-\bar u)=0\). This proves the stated identity. Summing the identity over observations gives the observed log-likelihood Hessian. The second term is a weighted covariance matrix of component score vectors and is positive semidefinite; the first term contributes the component log-kernel curvature. Their combination, rather than either part alone, determines the observed likelihood geometry. \(\square\)

Flat Directions and Redundant Representations

A flat direction is a direction along which the likelihood remains unchanged, not merely one where its first derivative happens to be zero. Mixture models can have such directions when different parameter values describe the same density. In particular, if two component densities coincide, changing their relative weights does not change their combined density. Label switching creates another familiar form of non-uniqueness: permuting component labels leaves the mixture distribution unchanged, although that symmetry generally gives distinct equivalent points rather than a continuous flat path.

A zero Hessian eigenvalue alone does not establish that an entire flat path exists. For example, the first nonzero change in a direction could be cubic or quartic. Classification in such cases requires examining higher-order terms or the likelihood itself. It is also important to distinguish a flat direction from label switching: label switching gives equivalent likelihood values at permuted parameter points, while a flat direction continuously connects parameter values with identical likelihood.

What EM Convergence Does—and Does Not—Establish

For exact EM, the E-step computes posterior responsibilities at the current parameter, and the M-step maximizes the corresponding expected complete-data log-likelihood. Under the usual conditions for these steps, the observed log-likelihood does not decrease. This monotonicity is useful, but it compares consecutive values along one algorithmic path; it does not compare the endpoint with every other parameter value.

There is a precise connection between an interior EM fixed point and stationarity. Suppose the EM update is well-defined, the likelihood and expected complete-data log-likelihood are differentiable near \(\theta_*\), and \(\theta_*\) is an interior fixed point that maximizes the M-step objective. At the fixed point, the posterior distribution used in the E-step is the posterior under \(\theta_*\). The gradient of the M-step objective with respect to its proposed parameter, evaluated at \(\theta_*\), equals the observed likelihood score there: this follows by differentiating the finite sum or integral defining the conditional expectation of the complete-data log-likelihood, under conditions permitting differentiation under the integral. Since an interior maximizer has zero gradient, \(\nabla\ell(\theta_*)=0\). Thus such an EM fixed point is stationary. Stationarity, however, does not specify whether the point is a local maximum, a saddle, or part of a flat set.

In practice, EM’s nondecreasing likelihood and a small change in parameters or likelihood can be useful convergence diagnostics. They answer whether a particular run has nearly stopped changing. They do not, on their own, reveal the Hessian’s curvature, rule out flat directions, or compare the result to other stationary points. A careful assessment separates three questions: has the algorithm stabilized, what is the local geometry at its endpoint, and how does its likelihood compare with those from other starts or candidate solutions?

Common Pitfall

• Wrong reasoning: “The EM iterates stopped changing and the likelihood stopped increasing, so this must be the global maximum.”
• Correction: A fixed point can be stationary without being a maximum, as the saddle example demonstrates. Even a strict local maximum is only locally optimal. Use curvature to assess the local geometry, and do not treat convergence from one initialization as a global-optimality result.
• Another warning: A zero Hessian eigenvalue does not by itself prove either a flat region or a maximum. Inspect higher-order behavior or compare likelihood values along the relevant direction.