A scheme for constructing an irreducible Markov chain for pedigree data

Area of Science:

• Computational Biology
• Genetics
• Statistical Modeling

Background:

• Exact computation of probabilities in genetic pedigrees can be computationally infeasible.
• Markov chain Monte Carlo (MCMC) methods offer an alternative but require specific properties like irreducibility for guaranteed convergence.
• Reducibility in Markov chains poses challenges for genetic pedigree analysis.

Purpose of the Study:

• To propose a novel scheme for constructing an irreducible Markov chain specifically for genetic pedigree data.
• To address the limitations of reducible Markov chains in genetic analyses.
• To enhance the efficiency of probability estimation in complex pedigrees.

Main Methods:

• Development of a scheme to construct an irreducible Markov chain for pedigree data.
• Utilizing a Metropolis jumping kernel to facilitate transitions between explicitly identified communicating classes.
• Application of the ergodic theorem for convergence guarantees.

Main Results:

• Successfully constructed an irreducible Markov chain applicable to genetic pedigrees.
• Demonstrated significantly improved efficiency compared to existing probability estimation methods.
• The proposed method ensures convergence of estimates to true probabilities.

Conclusions:

• The developed method provides an efficient and reliable approach for estimating probabilities in genetic pedigrees.
• The use of irreducible Markov chains with Metropolis kernels overcomes limitations of previous methods.
• This technique is valuable for complex genetic analyses where exact computations are intractable.