Section Computation of likelihood density function discusses the calculation of the likelihood function.
A detailed discussion of how we compute this likelihood can be found in Section Computation of likelihood density function.
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].
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 ).
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.
For the subset of individuals recovered that died of senescence (i.e., uncensored, w = 1), the likelihood function simplifies to the density function, f(t).
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).
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.
The probability density function of the likelihood ratio is constructed using the statistics of multiple response quantities and Monte Carlo simulation.
