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Non-Identifiable Pedigrees and a Bayesian Solution

Some methods aim to correct or test for relationships or to reconstruct the pedigree, or family tree. We show that these methods cannot resolve ties for correct relationships due to identifiability of the pedigree likelihood which is the probability of inheriting the data under the pedigree model. This means that no likelihood-based method can produce a correct pedigree inference with high probability. This lack of reliability is critical both for health and forensics applications. In this paper we present the first discussion of multiple typed individuals in non-isomorphic pedigrees, $\mathcal{P}$ and $\mathcal{Q}$, where the likelihoods are non-identifiable, $Pr[G~|~\mathcal{P},\theta] = Pr[G~|~\mathcal{Q},\theta]$, for all input data $G$ and all recombination rate parameters $\theta$. While there were previously known non-identifiable pairs, we give an example having data for multiple individuals. Additionally, deeper understanding of the general discrete structures driving these non-identifiability examples has been provided, as well as results to guide algorithms that wish to examine only identifiable pedigrees. This paper introduces a general criteria for establishing whether a pair of pedigrees is non-identifiable and two easy-to-compute criteria guaranteeing identifiability. Finally, we suggest a method for dealing with non-identifiable likelihoods: use Bayes rule to obtain the posterior from the likelihood and prior. We propose a prior guaranteeing that the posterior distinguishes all pairs of pedigrees. Shortened version published as: B. Kirkpatrick. Non-identifiable pedigrees and a Bayesian solution. Int. Symp. on Bioinformatics Res. and Appl. (ISBRA), 7292:139-152 2012.

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