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The existence of the latent variables z prevents to optimize a complete likelihood function defined using Eq. (4).
The spatial likelihood is a reduction of the complete likelihood applied to a forecast with rate value normalized to match the total observed number of targets (Zechar et al. 2010a).
For all magnitude classes, the complete likelihood counts the Poisson joint log-likelihood of the observed number ω x,t) given the forecast λ x,t): L ( t ) = ∑ ( − λ ( x, t ) + ω ( x, t ) log ( λ ( x, t ) ) − log ( ω ( x, t ) ! ).
In practice, however, determining the complete likelihood surface is both difficult and computationally expensive.
For model comparisons the Deviance Information Criterion (DIC) for mixed‐effects models based on the complete likelihood was used [ 18].
The complete likelihood model combines the prior probability with the sequencing data to derive the posterior probability of the alleles.
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Let L(Y o,Y m,θ μ,D,σ2) represent the complete data likelihood.
Given ({z}=left( z_{1},z_{2},ldots,z_{N}right) ), we first compute the complete log likelihood as begin{aligned} l_{c}left( theta mid Dright) =text {log},pleft( Y,mathbf {z}mid X,theta right) =sum _{n=1}^{N}text {log},pleft( y_{n},z_{nn}|x_{n},theta right).
The complete data likelihood is then where is a multinomial p.m.f.
This is done by taking the partial derivative of the expected complete log likelihood with respect to each parameter.
The joint posterior distribution was defined as the product of the prior probabilities and the complete data likelihood (cf. [ 6]).
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