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+are+powerful+tools+for+modeling+sequential+data%252C+where+the+underlying+states+evolve+in+a+stochastic+manner+and+are+only+indirectly+observable.+Traditional+HMM+approaches+are+well-established+for+linear+sequences%252C+and+have+been+extended+to+other+structures+such+as+trees.+In+this+paper%252C+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%252C+decoding%252C+and+parameter+learning+problems+for+tree-based+HMMs+with+coupled+branches.+Our+approach+scales+polynomially+with+the+number+of+states+and+nodes%252C+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.&cdnurl=https%3a%2f%2fth.bing.com%2fth%2fid%2fR.6d6ea02e122e8c55ffcedcf988ff7440%3frik%3dh%252f3kQJXKOpmh4Q%26pid%3dImgRaw%26r%3d0&exph=630&expw=1200&q=Hidden+Markov+Model&FORM=IRPRST&ck=EA130DC2B6FC45E2FA5E2A5AEE649FF0&selectedIndex=34&itb=0)
