23/09/13 compbio / Hidden Markov Model compbio.pbworks.com/w/page/16252896/Hidden Markov Model 1/12 compbio log in hel p Get a free wiki | Try our free business product Wiki Pages & Files VIEW EDIT Page history earch this workspace Hidden Markov Model last edited by PBworks 5 years ago Hidden Markov Model Lecture by Professor Xiaole Shirley Liu Wiki by Da Lin and Allen Cheng 1. Discrete Markov Chain i. Markov Chain - Example 1 ii. Markov Chain - Example 2 iii. Markov Chain - Example 3 2. Hidden Markov Model 3. Problems 4. Problem 1 5. Problem 2 6. Viterbi Algorithm 7. Problem 3 8. Additional Applications of Hidden Markov Models 9. Additional Resources The Markov chain was named in honor of Professor Andrei A M arkov, who first published hi s findings in 1906. Markov was born on June 14, 1856 in Ryazan, Russia, and studied mathematics at the University of St. Petersburg where he later taught as a professo r until his death in 1922. Alth ough his work in mathematic s ranges from differential equations to proba bility theory, he has become best known for this research on stochastic processes and the Marko v chain. On this page, we first give a brief introduction to Markov processes and the discrete Markov chain. We then present the Hidden Markov model, and solve it with three different methods: forward, backwards, and the Viterbi algorithm. Finally, we will examine and solve a Hidden Markov problem in a biological context. Discrete Markov Chain To join this workspace, request access. Already have an account? Log in! Navigator options Pages Files 2 easy ways About Us Association Studies and Haplotype Inference Basic Bioconductor Basic HTML Basic Python SideBar To use the template to start a new chapter, go to the template page; click edit, then copy everything in the edit box; go to your own page, click edit, and paste. You are ready to create your own chapter! Recent Activity Microarr ay Classification edited by Microarr ay Classification edited by Microarr ay Classification edited by Microarr ay Classification edited by Microarr ay Classification edited by Microarray Classification edited by Microarr ay Classification edited by More activity...
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Markov Chain - Example 1
Suppose Diana has a test tomorrow, and she
wants t o k now what is the probability of receiving a
passing grade. In the 6 previous exams this
semester, she has passed the first three and failed
the most recent three. Her genius roommate has
also calculated for her the transition probabilities
a(i,j) for her exam states:
What is the probability that Sam will pass her exam tomorrow?
Solution:
We can see this as a Markov chain problem with states S(1) - pass, and S(2) - fail. Each exam can be
associated with a time frame, denoted as q(n), where n is the exam number. We want to find the
probability that q(7) = P, given the transition probabilities and the observed exam states PPPFFF.However, if we remember that the Markov process has no memory, the problem simplifies to finding
the probability that q(7) = P given that q(6) = F. This is shown mathematically below:
Markov Chain - Example 2
Frank the Weatherman’s computer has malfunctioned,
so he needs to predict the 7-day weather forecast by
hand. To simplify his task, he decides to report only 3
different states, rain (R), cloudy (C), or sunny (S).
From college, he remembers that the transition
probabilities for the different weather states are:
If it is sunny today, what is the probability that the observed weather for the next 7 days is
S(today)SSRRSCS?
Solution:
This is again a Markov chain problem with states S(1) - rain, and S(2) - cloudy, and S(3) - sunny. Eachday can be associated with a time frame, denoted as q(n), where n is the number of days since today.
We want to find the probability of observing SSSRRSCS given that q(0)=3. This is simply
multiplication of transition probabilities. If today is sunny, the probability that q(1)=3 is just a(3,3). If
1 =3 the robabilit that 2 =3 is a ain a 3 3 and so forth. This is solved mathematicall below: