(5), Bayesian inference derives a posterior distribution (P( x | Y)) of variables (x), offsetting constants, based on a likelihood function
(P(Y | x)) and a prior distribution of the variables ([pi](x)).
Notations f(*): Probability density function E(*): Mean function Var(*): Variance function [gamma](y, x): Lower incomplete gamma function [GAMMA](y, x): Upper incomplete gamma function F(*): Cumulative distribution function S(*): Survival function h(*): Hazard rate function L(*): Likelihood function
l(*): Log-likelihood function [psi](*): Digamma function [psi]'(*): Trigamma function I(*): Expected Fisher information matrix [mu]: Positive parameter [phi]: Positive parameter [alpha]: Positive parameter [beta]: Real parameter [lambda]: Positive parameter Q(*): Quantile function [THETA]: Vector of parameters H(*: Observed Fisher information matrix [[mu].sub.k]: kth moment G(*: TTT-plot [D.sub.n]: Kolmogorov-Smirnov statistic [y.sup.*]: Predictive value.
The likelihood function
of the whole failure time data set could be obtained by multiplying the three individual contributions as
If we let [L.sub.i] = log f([y.sub.i]; [[theta].sub.i], [THETA]) as a contribution of [y.sub.i] to the likelihood function
L = [[summation].sub.i][L.sub.i], then
Obtaining maximum likelihood function
under population ([OMEGA]).
Assuming that [theta]' was a parameter value of [theta], the logarithmic likelihood function
concerning [theta]' was
(i) Individual speed follows a normal (Gaussian) distribution with mean [mu] and variance [[sigma].sup.2], since a normal distribution reduces likelihood function
One main challenge in these inferential tasks is attributed to the burden of computational load of minimizing/maximizing corresponding object functions, including likelihood functions
or the squared differences [8-11].
The local likelihood functions
are formulated using four of the 37 original datasets, one from each fault case.
The estimation for the parameters of DWD via maximum likelihood estimation technique presented the independent observations are x1,x2,....xn then the likelihood function
of the DWD shown:
Since the multidimensional likelihood function
is a nonlinear function of time delays and has many local maxima, the exact ML estimate needs multidimensional grid search.
Another approach  was to employ linear transform (LT) on the phase noise term and derive an approximate expression of the likelihood function
. By using the likelihood function
, a modified LLR metric was obtained.