Parameter estimation was then performed in an M-step returning maximum likelihood point estimates.
It should be noted as a caveat that there are potential errors in estimation for maximum likelihood point estimates.
Maximum likelihood point and interval estimates were obtained using the module AMFIT from the computer package EPICURE (Preston et al, 1993).
Using maximum likelihood point estimates of these parameters, we compute the posterior probabilities of each genotype given the read data and classify an individual's genotype as the one with the largest posterior probability.
Multiple sets of 170 SNPs were sampled with replacement from the data set, which yielded different maximum-likelihood points { θ ^ i * } i = 1 B in the parameter space.
The Jacobian is obtained by orthogonalising the Taylor-expansion of around the maximum-likelihood point,.
Unfortunately, although PAML provides maximum-likelihood point estimates of parameters, it does not provide any confidence intervals for these parameters.
Bayesian inference can be used to assess the uncertainty due to such confounding factors, unlike the maximum-likelihood point estimates provided by an EM algorithm.
While in the model-averaging approach, inferences are conditioned on Maximum-Likelihood point estimates of nuisance parameters, the Bayesian approach integrates out nuisance parameters, giving marginal probabilities of species limits.
To search for the maximum-likelihood point over the parameter space, we applied the ECM (Expectation/Conditional Maximization) method (Meng and Rubin 1993), which was previously used in the demographic inference study by Excoffier et al. (2013).
As a second way to quantify parameter estimation performance, we used the Hessian matrix, which is the matrix of second-order partial derivatives of a function with respect to its parameters, to construct 95% confidence intervals (CI) around the maximum-likelihood point estimates for the parameters of our models.
