8.3 Exact HMM computations

Using the inheritance vectors or meiosis indicators, the structure of the problem is that of a hidden Markov model (HMM) with the Markov latent state being the S.j, Markov over markers j. When the pedigree is small, so that each S.j takes only a practical number of values, standard exact HMM computational methods apply. Likelihoods and lod scores can be computed exactly. Alternatively, a single forwards computation followed by (repeated) backwards sampling provides (multiple independent) realizations from the joint distribution of all the Sij given the marker data, or given the marker and trait data, if the latter is included in the set of loci j.

From MORGAN 2.8, exact computation is performed on small pedigree components. Further, these HMM computations are also a component of MCMC sampling on larger pedigree components (see next section).

Note that in fact Sij are independent over meioses i, so that the structure is that of a factored HMM. In MORGAN V2.8.2, forward HMM computation for multiple meioses has been replaced by a factored version (FHMM), enabling much faster exact computation on small pedigree components and multiple-meiosis sampling for larger numbers of meioses.

Exact computation of lodscores on small pedigree components has been implemented for lm_markers and lm_multiple: computation uses the FHMM version of the Baum algorithm.

In MORGAN V2.8.2, Gold standards for exact computation are added in the Lodscore/Gold2 subdirectory.