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 are powerful tools for modeling sequential data%2C where the underlying states evolve in a stochastic manner and are only indirectly observable. Traditional HMM approaches are well-established for linear sequences%2C and have been extended to other structures such as trees. In this paper%2C we extend the framework of HMMs on trees to address scenarios where the tree-like structure of the data includes coupled branches -- a common feature in biological systems where entities within the same lineage exhibit dependent characteristics. We develop a dynamic programming algorithm that efficiently solves the likelihood%2C decoding%2C and parameter learning problems for tree-based HMMs with coupled branches. Our approach scales polynomially with the number of states and nodes%2C making it computationally feasible for a wide range of applications and does not suffer from the underflow problem. We demonstrate our algorithm by applying it to simulated data and propose self-consistency checks for validating the assumptions of the model used for inference. This work not only advances the theoretical understanding of HMMs on trees but also provides a practical tool for analyzing complex biological data where dependencies between branches cannot be ignored.?quality=80&w=800)
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