This approach mirrors the classic parametric maximum likelihood estimation given right censored data in terms of the likelihood containing a density function component and a survival function component whose relative contributions depends upon whether or not an observation is censored.
Their covariances are determined by maximising the likelihood of the density function via expectation-maximisation (EM).
Note that in cases where the likelihood (the probability density function, pdf, viewed as a function of the parameters values) is known except for a normalizing constant, MCMC has been the main option for numerical Bayesian inference since the 1990s [ 1].
Maximum likelihood estimation/estimator. Probability density function.
The EM algorithm [11 14] tries to estimate the parameter set Θ = { A, H, Q, R } of the system (4), (5) by maximizing the likelihood of its probability density function P ( x, Θ ).
In the likelihood expression above, the density function f depends on the fitness coefficient s via the mean of this normal distribution, which is (1 + s) z t /(1 + sz t ).
We exploit the fact that y has a multivariate normal distribution, y ∼ N (W β, σ u 2 Z A AZ T + K ν, h, σ K + σ e 2 I ), and estimate the parameters β, σu, σe, ν, h and σ K by maximizing the log likelihood of the corresponding density function.
Therefore, we maximize the log likelihood J of the density function f y, u, g ; i.e., we maximize (3) J = log (c ) − 1 2 ⋅ [ 1 σ e 2 | | y − W β − Zu u − g (X ) | | 2 + 1 σ u 2 u T A − 1 u + g (X ) T K ν, h, σ K − 1 g (X ) ], with respect to β, u, and g(X).
