Particle swarm optimization in constrained maximum likelihood estimation a case study

Parametric statistical models are commonly used in many sub-fields of bioinformatics [1], [2]. For simplicity and computational concerns, bioinformatic scientists prefer to use differentiable and unconstrained statistical models than non-differentiable and constrained ones. For example, in pseudotime analysis (see section 3), in [3], the authors propose to regress gene expression on pseudotime using cubic B-spline so that an analytical solution is available. Other authors suggest to replace B-spline with a generalized linear model and a gradient-based method is applied to find maximum likelihood estimation [4]. In zero imputation problem, the authors construct a Gamma-Normal mixture model so that parameters can be estimated analytically [5]. In [6], the authors propose an unconstrained LASSO-type objective function and optimize it with a convex optimization algorithm. However, in real applications, it is common to impose constraints on parameters for interpretability. Besides, analytically solutions are not always available and the likelihood function is not differentiable or convex if discrete parameters are contained. Thus, constrained models without desirable mathematical properties can be more realistic and interpretable in many cases.