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A joint problem formulation, where quantities that are not of particular interest are marginalized out from a joint posterior distribution, or an intrinsically marginal formulation, which is based on an integrated likelihood function, can be chosen according to the inferential objective and computational convenience.
These unobserved or latent variables are marginalized out instead of being explicitly calculated.
Evaluation of the full posterior turns out to be not only computationally prohibitive but useless if states cannot be marginalized out analytically.
In state-based marginalization, each state output PDF is a function of the reliable components only, while the unreliable components are marginalized out, i.e., each unreliable component is integrated over the range of values it can assume.
In this case the total likelihood function will be a weighted sum of the likelihoods for all possible waiting times, so that in effect the w i are marginalized out in favor of the actually observed data.
For the likelihood function itself, p 0 and p t are nuisance (unobserved, latent state) parameters that need to be marginalized out by summing over all possible combinations of p 0 and p t.
Similar(40)
First, the approach breaks up the interdependence of the hidden states and the static parameters by marginalizing out the states using a particle filter.
Similarly, the a posteriori pdf of the channel vector is obtained as a Gaussian mixture by marginalizing out all possible ISI state transitions (41).
Under the minimum mean square error (MMSE) criterion, the forward filtered channel vector estimated at instant is obtained by marginalizing out the ISI state variable (42).
Similarly, the MMSE smoothed estimate of the channel vector at instant is obtained by marginalizing out all possible ISI state transitions (43).
The marginal posterior pdf required in (7) is computed from the joint posterior pdf of the variables in the probabilistic model of the system by marginalizing out all variables but the bit of interest.
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Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com