Marjolijn Elsinga & Elze de Groot1 Markov Chains and Hidden Markov Models Marjolijn Elsinga & Elze...

Marjolijn Elsinga amp Elze de Groot 1

Markov Chains Markov Chains and and

Hidden Markov ModelsHidden Markov Models

Marjolijn Elsinga

amp

Elze de Groot


Andrei A MarkovAndrei A Markov

Born 14 June 1856 in Ryazan RussiaDied 20 July 1922 in Petrograd Russia

Graduate of Saint Petersburg University (1878)

Work number theory and analysis continued fractions limits of integrals approximation theory and the convergence of series


Todays topicsTodays topics

Markov chains

Hidden Markov models- Viterbi Algorithm- Forward Algorithm- Backward Algorithm- Posterior Probabilities


Markov Chains (1)Markov Chains (1)

Emitting states



Transition probabilities

Probability of the sequence


Key property of Markov ChainsKey property of Markov Chains

The probability of a symbol xi depends only on the value of the preceding symbol xi-1


Begin and End statesBegin and End states

Silent states


Example CpG IslandsExample CpG Islands

CpG = Cytosine ndash phosphodiester bond ndash Guanine

100 ndash 1000 bases long Cytosine is modified by methylation Methylation is suppressed in short stretches

of the genome (start regions of genes)High chance of mutation into a thymine (T)


Two questionsTwo questions

How would we decide if a short strech of genomic sequence comes from a CpG island or not

How would we find given a long piece of sequence the CpG islands in it if there are any


DiscriminationDiscrimination

48 putative CpG islands are extractedDerive 2 models

- regions labelled as CpG island (lsquo+rsquo model)

- regions from the remainder (lsquo-rsquo model)

Transition probabilities are set- Where Cst+ is number of times letter t follows letter s


Maximum Likelihood EstimatorsMaximum Likelihood Estimators

Each row sums to 1Tables are asymmetric


Log-odds ratioLog-odds ratio


Discrimination shownDiscrimination shown


Simulation lsquo+rsquo modelSimulation lsquo+rsquo model


Simulation lsquo-rsquo modelSimulation lsquo-rsquo model



Markov chains



Hidden Markov Models (HMM) (1)Hidden Markov Models (HMM) (1)

No one-to-one correspondence between states and symbols

No longer possible to say what state the model is in when in xi

Transition probability from state k to l

πi is the ith state in the path (state sequence)



Begin state a0k

End state a0k

In CpG islands example



We need new set of parameters because we decoupled symbols from states

Probability that symbol b is seen when in state k


Example dishonest casino (1)Example dishonest casino (1)

Fair die and loaded dieLoaded die probability 05 of a 6 and

probability 01 for 1-5Switch from fair to loaded probability

005Switch back probability 01


Dishonest casino (2)Dishonest casino (2)

Emission probabilities HMM model that generate or emit sequences



Hidden you donrsquot know if die is fair or loaded

Joint probability of observed sequence x and state sequence π


Three algorithmsThree algorithmsWhat is the most probable path for generating a

given sequence

Viterbi AlgorithmHow likely is a given sequence

Forward AlgorithmHow can we learn the HMM parameters given a

set of sequences

Forward-Backward (Baum-Welch) Algorithm


Viterbi AlgorithmViterbi Algorithm

CGCG can be generated on different ways and with different probabilities

Choose path with highest probability

Most probable path can be found recursively


Viterbi Algorithm (2)Viterbi Algorithm (2)

vk(i) = probability of most probable path ending in state k with observation i





Most probable path for CGCG


Viterbi AlgorithmViterbi AlgorithmResult with casino example



given sequence



set of sequences



Forward Algorithm (1)Forward Algorithm (1)Probability over all possible paths

Number of possible paths increases exponentonial with length of sequence

Forward algorithm enables us to compute this efficiently


Forward Algorithm (2) Forward Algorithm (2)

Replacing maximisation steps for sums in viterbi algorithm

Probability of observed sequence up to and including xi requiring πi = k


Forward Algorithm (3)Forward Algorithm (3)



given sequence



set of sequences



Backward Algorithm (1)Backward Algorithm (1)Probability of observed sequence from xi to the

end of the sequence requiring πi = k


Disadvantage AlgorithmsDisadvantage Algorithms

Multiplying many probabilities gives very small numbers which can lead to underflow errors on the computer

can be solved by doing the algorithms in log space calculating log(vl(i))


Backward AlgorithmBackward Algorithm


Posterior State Probability (1)Posterior State Probability (1)

Probability that observation xi came from state k given the observed sequence

Posterior probability of state k at time i when the emitted sequence is known

P(πi = k | x)


Posterior State Probability (2)Posterior State Probability (2)First calculate probability of producing entire

observed sequence with the ith symbol being produced by state k

P(x πi = k) = fk (i) bk (i)


Posterior State Probability (3) Posterior State Probability (3)

Posterior probabilities will then be

P(x) is result of forward or backward calculation


Posterior Probabilities (4)Posterior Probabilities (4)

For the casino example






Prediction of CpG islandsPrediction of CpG islands

First way Viterbi Algorithm

- Find most probable path through the model

- When this path goes through the lsquo+rsquo state a CpG island is predicted


Prediction of CpG islandsPrediction of CpG islandsSecond Way Posterior Decoding

- function

- g(k) = 1 for k Є A+ C+ G+ T+

- g(k) = 0 for k Є A- C- G- T-

- G(i|x) is posterior probability according to the model that base i is in a CpG island


Summary (1)Summary (1)

Markov chain is a collection of states where a state depends only on the state before

Hidden markov model is a model in which the states sequence is lsquohiddenrsquo



Most probable path viterbi algorithmHow likely is a given sequence forward

algorithmPosterior state probability forward and

backward algorithms (used for most probable state of an observation)


Andrei A MarkovAndrei A Markov

Born 14 June 1856 in Ryazan RussiaDied 20 July 1922 in Petrograd Russia

Graduate of Saint Petersburg University (1878)

Work number theory and analysis continued fractions limits of integrals approximation theory and the convergence of series



Markov chains




Emitting states










Silent states





























Markov chains










Begin state a0k

End state a0k




















given sequence



set of sequences



















given sequence



set of sequences














given sequence



set of sequences















P(πi = k | x)























- function

- g(k) = 1 for k Є A+ C+ G+ T+

- g(k) = 0 for k Є A- C- G- T-













Markov chains




Emitting states










Silent states





























Markov chains










Begin state a0k

End state a0k




















given sequence



set of sequences



















given sequence



set of sequences














given sequence



set of sequences















P(πi = k | x)























- function

- g(k) = 1 for k Є A+ C+ G+ T+

- g(k) = 0 for k Є A- C- G- T-













Emitting states










Silent states





























Markov chains










Begin state a0k

End state a0k




















given sequence



set of sequences



















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set of sequences














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set of sequences















P(πi = k | x)























- function

- g(k) = 1 for k Є A+ C+ G+ T+

- g(k) = 0 for k Є A- C- G- T-




















Silent states





























Markov chains










Begin state a0k

End state a0k




















given sequence



set of sequences



















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- g(k) = 1 for k Є A+ C+ G+ T+

- g(k) = 0 for k Є A- C- G- T-
















Silent states





























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End state a0k




















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- g(k) = 1 for k Є A+ C+ G+ T+

- g(k) = 0 for k Є A- C- G- T-













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End state a0k




















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- g(k) = 0 for k Є A- C- G- T-







































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End state a0k




















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- g(k) = 0 for k Є A- C- G- T-


































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End state a0k




















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- g(k) = 0 for k Є A- C- G- T-





















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End state a0k




















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set of sequences



















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- function

- g(k) = 1 for k Є A+ C+ G+ T+

- g(k) = 0 for k Є A- C- G- T-



















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End state a0k




















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set of sequences



















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P(πi = k | x)























- function

- g(k) = 1 for k Є A+ C+ G+ T+

- g(k) = 0 for k Є A- C- G- T-

















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End state a0k




















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P(πi = k | x)























- function

- g(k) = 1 for k Є A+ C+ G+ T+

- g(k) = 0 for k Є A- C- G- T-















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End state a0k




















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- g(k) = 0 for k Є A- C- G- T-













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- g(k) = 0 for k Є A- C- G- T-













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- g(k) = 0 for k Є A- C- G- T-




























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- g(k) = 0 for k Є A- C- G- T-























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- g(k) = 0 for k Є A- C- G- T-




















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- g(k) = 0 for k Є A- C- G- T-


















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- g(k) = 0 for k Є A- C- G- T-















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- g(k) = 0 for k Є A- C- G- T-













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set of sequences














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- g(k) = 1 for k Є A+ C+ G+ T+

- g(k) = 0 for k Є A- C- G- T-























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- g(k) = 1 for k Є A+ C+ G+ T+

- g(k) = 0 for k Є A- C- G- T-



















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- g(k) = 1 for k Є A+ C+ G+ T+

- g(k) = 0 for k Є A- C- G- T-















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- g(k) = 1 for k Є A+ C+ G+ T+

- g(k) = 0 for k Є A- C- G- T-













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- g(k) = 0 for k Є A- C- G- T-
























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- g(k) = 0 for k Є A- C- G- T-

































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- g(k) = 0 for k Є A- C- G- T-





























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- g(k) = 0 for k Є A- C- G- T-

























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- g(k) = 0 for k Є A- C- G- T-






















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- g(k) = 0 for k Є A- C- G- T-


















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- g(k) = 0 for k Є A- C- G- T-













- function

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- g(k) = 0 for k Є A- C- G- T-

























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