HMMs Hidden Markov Models

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HMMs Hidden Markov Models
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Definitions Uses

• A probabilistic model which deals with sequences
  of symbols.
• Originally used in speech recognition (the
  symbols being phonemes)
• Useful in biology the sequence of symbols being
  the DNA.

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Questions where HMMs are used

• Does this sequence belong to a particular family?
• Can we identify regions in a sequence (for
  instance alpha helices, beta sheets)?
• and less directly
• Pairwise/multiple sequence alignment
• Searching databases for protein families
  (building profiles).

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Markov Chains (reminder)

• A sequence of random variables X1,X2, where
  each present state depends only on the previous
  state.
• A?C?G?G?T?A(vertical or horizontal!)
• These conditional probabilities can be
  illustrated as follows (in DNA)

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Markov chain probabilities

• Each arrow has a transition probability PCA
  P(xiAXi-1C)
• Thus the probability of a sequence x will be

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Some terminology

• We differentiate between states and
  symbolsSymbols are what we see, our
  observations the sequence we see.States are
  what we talk of in theory. They are part of the
  probabilistic model (Pst the probability of
  the transition between 2 theoretic states, s and
  t. Here s and t are A , C, G or T).We will see
  the importance of this difference later on.

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CpG islands

• In the human genome, for biological reasons, the
  occurrence of the dinucleotide CG is lower than
  would be expected from C G independent
  probabilities.

• On the other hand, around promoters or the
  start regions of many genes, we see many more
  CG couplets (and in fact more C and G generally).

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Markov chains in use

• Lets answer one of the previous questions, using
  Markov Chains. For example Given a sequence, is
  it a CpG island?
• Lets assume that we have the 2 sets of
  transition probabilities in standard DNA, and
  transition probabilities in CpG DNA.

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Markov chains in use - continued
CpG islands ( model)
normal DNA (- model)
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Markov chains in use - continued

• Thus, we have two models which we can compare
  statistically (likelihood ratio test)

Note Px0x1P(x1) the probability of beginning
with x1.
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And now to HMMs

• HMMs can answer a question we posed before that
  Markov chains cannot how do we find CpG islands
  in a long un-annotated sequence?

• Instead of two Markov chain models (like before)
  we need one model that incorporates both the
  models from before. How do we do that in the
  previous example?

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The solution

• We relable the states. In order to integrate, we
  will define A , G , C , T (CpG areas) and A-,
  G-, C-, T- (normal DNA). Both A and A- emit
  the symbol A.

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• Many state sequences emit a sequence of
  symbolsAGCTG-C-C-T- A-G-CTGCCT- A
  G C T G C C T

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Most probable state path - Viterbi algorithm

• We can compute the likelihood of each one of
  these state paths.
• Look for the maximum likelihood state sequence.
• This way, given a long unnotated sequence, I get
  the ML state at the endA-G-C-G-T-T-T-C-GCAGA
  -C-G-T-CGT
• Viterbi algorithm does this with dynamic
  programming algorithm (an example later).

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HMM Formalism
A
A
A
A
Si1
Si
Si-1
S1
Sn
. . .
. . .
B
B
B
Ki1
Ki
Ki-1
K1
Kn
. . .
. . .

• S, K, P, A, B
• S s1sN are the values for the hidden states
• K k1kM are the values for the observations
• The hidden states emit/generate the symbols
  (observations)
• ? ?i are the initial state probabilities
• P Pij are the state transition probabilities
• B bik are the emission probabilities (which
  in our case are 0 or 1 P(A/A)P(A/A-)1,
  P(A/C)0)

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Markov chain vs. HMM

• The state sequence itself follows a simple Markov
  chain. But-
• The essential difference between a Markov chain
  and an HMM is it is no longer possible to know
  the state by looking at the symbols the state
  is hidden.

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Another example the dishonest casino

• In a casino, they use a fair die most of the
  time, but occasionally switch to an unfair die.
  The switch between dice can be represented by an
  HMM

0.9
0.95
1 1/6 2 1/6 3 1/6 4 1/6 5 1/6 6 1/6
1 1/10 2 1/10 3 1/10 4 1/10 5 1/10 6 1/2
0.05
0.1
FAIR
UNFAIR
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Dishonest casino - continued

• The symbols (observations) are the sequence of
  rolls3 5 6 2 1 4 6 3 6
• What is hidden?If the die is fair or unfairf f
  f f u u u f fThis is a Markov chain. Except for
  that, we have
• Emission probabilitiesGiven a state, we have 6
  possible matching symbols, each with an emission
  probability.

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Exposing the casino

• Once again we can estimate which is the most
  probable state path, and estimate when the casino
  was cheating

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Rate Variation Among Sites

• Many methods assume equal rate of evolution in
  all sites unrealistic
• Rate4site a method for deducing the rate at
  each sites. Assumes independence of the rate
  between sites.
• Rate variation estimation shouldAllow some
  correlation between the rates of evolution at
  adjacent sites.

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A Hidden Markov Model Approach to Variation Among
Sites in Rate of Evolution (Churchill and
Felsenstein 1996)
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Motivation

• To find the path of rates that fits the data in
  the best way.
• We will represent a general path as
  follows(c1,c2,,cn)where ci 10.0 or 2.0 or
  0.3 (in the example above).ci corresponds to the
  rate at site i.n is the length of the sequence.

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Paths ? HMM

• Each path corresponds to a state path.The real
  path is hidden.
• Before we had ACG ? A C G
• Now we have c1,c2,c3 corresponding to a multiple
  sequence alignment

x1 y1 y3 x2 y2 y3 s1 s2 s3
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Motivation - notes

• Finding the path that best fits the datahow?The
  path that makes the maximum contribution to the
  likelihood of the data.
• Instead of going over all possible paths (lots of
  computation) ? dynamic programming.

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Outline of the algorithm

• Assume we have k categories of rater1, r2, ,
  rk.We know ?ri for i1kWe know Pri rj for
  i,j1k (correlation values)

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Outline - continued

• 2. The likelihood of a given phylogeny T
• n the number of sites in sequence.
• The combination (c1,,cn) denotes a combination
  of rates for each site

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Outline - continued

• 3. Look for the combination which makes the
  largest contribution to the likelihood

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the gory details1. Choosing rate categories

• How do we choose the rates?- estimate the values
  of ri and fi by ML, using the EM algorithm-
  examine a few sets of rates and choose the one
  with ML- equal-probability categories from gamma
  distribution (Yang, 1995)

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Computing the likelihood

• Computing the likelihoodVia the up algorithm.

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The most probable rates combination

• The most probable rates combinationa version of
  the Viterbi algorithm.Definewhere for kn

• D(k) data of sites k till n
• Dk data at site k

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The most probable rates combination - continued

• This is the likelihood contribution of sites k
  thru n, where site k has rate ck, and the rest
  (k1 thru n) have the combination of rates that
  maximizes the contribution.

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The most probable rates combination - continued

• We thus get

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Proof (the essentials)

• Because the rate categories are the outcome of a
  Markov chain and P(c1,cn)fc1Pc1,c2Pc2c3Pcn-1,cn
  
• Due to the assumption that once the rate is set,
  the sites evolve independently

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The most probable rates combination - continued

• Thus, we compute the above equation from sites n,
  n-1, n-2 up till 1, for each one of the rate
  categories.
• At the end, we take the largest of the quantities
  
• This is the quantity that maximizes the
  contribution to the likelihood.

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Backtracking the rate combination

• We still dont know which combination c1,cn
  brought us here, so we backtrack

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THE END!