Hidden Markov Models Modified by Winfried Just

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Hidden Markov ModelsModified by Winfried Just
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Outline

• CG-islands
• The Fair Bet Casino
• Hidden Markov Model
• Decoding Algorithm
• Forward-Backward Algorithm
• HMM Parameter Estimation
• Profile HMM Alignment

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CG-Islands

• Given 4 nucleotides probability of occurrence is
  1/4. Thus, probability of occurrence of a
  dinucleotide is 1/16.
• However, the frequencies of dinucleotides in DNA
  sequences vary widely.
• In particular, CG is typically underrepresented
• CG often mutates to TG. Thus, prob. of CG
  occurrence is typically lt (1/16)

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Why CG-Islands?

• CG is the least frequent dinucleotide because C
  in CG is easily methylated, then has the tendency
  to mutate into T
• However, the methylation is suppressed around
  genes in a genome. So, CG appears at relatively
  high frequency within these CG islands
• So, finding the CG islands in a genome is an
  important problem

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CG Islands and the Fair Bet Casino

• The CG islands problem can be modeled after a
  problem named The Fair Bet Casino

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TheFair Bet Casino

• The game is to flip coins, which results in only
  two possible outcomes Head or Tail.
• Suppose that the dealer uses both Fair and Biased
  coins.
• The Fair coin will give Heads and Tails with the
  same probability of ½.
• The Biased coin will give Heads with a
  probability of ¾.

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The Fair Bet Casino (contd)

• Thus, we define the probabilities
• P(HF) P(TF) ½
• P(HB) ¾, P(TB) ¼
• The crooked dealer changes between Fair and
  Biased coins with probability 10

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The Fair Bet Casino Problem

• Input A sequence of x x1x2x3xn of coin tosses
  made by two possible coins (F or B).
• Output A sequence p p1 p2 p3 pn, with each pi
  being either F or B indicating that xi is the
  result of tossing the Fair or Biased coin
  respectively.

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Problem
Fair Bet Casino Problem Any observed outcome
could have been generated by any sequence of coin
tosses!
Need to incorporate a way to grade different
sequences differently.
Decoding Problem
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P(xfair coin) vs. P(xbiased coin)

• Some definitions
• P(xfair coin) probability of generating the
  outcome x if the dealer uses the F coin.
• P(xbiased coin) probability of generating the
  outcome x if the dealer uses the B coin.
• k the number of Heads in x.

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P(xfair coin) vs. P(xbiased coin)

• P(xfair coin) 1/2n
• P(xbiased coin) 3k/4n
• P(xfair coin) P(xbiased coin)
• when k n / log23
• k 0.67n

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Log-odds Ratio

• We define the log-odds ratio as follows
• log2(P(xfair coin) / P(xbiased coin))
• Ski1 log2(p(xi) / p-(xi))
• n k log23

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Computing Log-odds Ratio in Sliding Windows
x1x2x3x4x5x6x7x8xn Consider a sliding window
of the outcome sequence. Find the log-odds for
this short window.
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Hidden Markov Model (HMM)

• Can be viewed as an abstract machine with k
  hidden states.
• Each state has its own probability distribution,
  and the machine switches between states according
  to this probability distribution.
• At each step, the machine makes 2 decisions
• What state should it move to next?
• What symbol from its alphabet should it emit?

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Why Hidden?

• Observers can see the emitted symbols of an HMM
  but have no ability to know which state the HMM
  is currently in.
• Thus, the goal is to infer the most likely states
  of an HMM basing on some given sequence of
  emitted symbols.

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HMM Parameters

• S set of all possible emission characters.
• Ex. S H, T for coin tossing
• S 1, 2, 3, 4, 5, 6 for dice
  tossing
• S a, c, g, t for nucleotide
  sequences
• Q set of hidden states, each emitting symbols
• from S.
• Ex. Fair or Biased coin
• CP island or not CP island
• coding region or non-coding region

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HMM Parameters (contd)

• A (akl) a Q x Q matrix of probabilities of
  changing from state k to state l.
• E (ek(b)) a Q x S matrix of probabilities
  of emitting symbol b during a step in which the
  HMM is in state k.

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HMM for Fair Bet Casino

• The Fair Bet Casino can be defined in HMM terms
  as follows
• S 0, 1 (0 for Tails and 1 Heads)
• Q F,B F for Fair B for Biased coin.
• aFF aBB 0.9
• aFB aBF 0.1
• eF(0) ½ eF(1) ½
• eB(0) ¼ eB(1) ¾

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HMM for Fair Bet Casino (contd)

• Visualization of the Transition Probabilities A

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HMM for Fair Bet Casino (contd)

• Visualization of the Emission Probabilities E

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HMM for Fair Bet Casino (contd)
HMM model for the Fair Bet Casino Problem
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Hidden Paths

• A path p p1 pn in the HMM is defined as a
  sequence of states.
• Consider path p FFFBBBBBFFF and sequence x
  01011101001

x 0 1 0 1 1 1
0 1 0 0 1 p F F F B
B B B B F F F P(xipi) ½ ½
½ ¾ ¾ ¾ ¾ ¾ ½ ½ ½ P(pi-1 ? pi) ½
9/10 9/10 1/10 9/10 9/10 9/10 9/10
1/10 9/10 9/10
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P(xp) Calculation

• P(xp) Probability that sequence x was generated
  and the path p was followed, according to the
  model M.
• n
• P(xp) P(p0? p1) . ? P(xi pi).P(pi ? pi1)
• i1
• n
• a p0, p1 . ? e pi (xi) . a pi, pi1
• i1

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Decoding Problem

• Goal Find an optimal hidden path of states given
  observations.
• Input Sequence of observations x x1..xn
  generated by an HMM M(S, Q, A, E)
• Output A path that maximizes P(xp) (and thus
  P(px) ) over all possible paths p.

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Building Manhattan for Decoding Problem

• Andrew Viterbi used the Manhattan grid model to
  solve our Decoding Problem.
• Every choice of p p1 pn corresponds to a path
  in the graph.
• The only valid direction in the graph is
  eastward.
• This graph has Q2(n-1) edges.

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Edit Graph for Decoding Problem
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Decoding Problem vs. Alignment Problem
Valid directions in the alignment problem.
Valid directions in the decoding problem.
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Decoding Problem as Finding a Longest Path in a
DAG

• The Decoding Problem is reduced to finding a
  longest path in the directed acyclic graph (DAG)
  above.
• Notes the length of the path is defined as the
  product of its edges weights, not the sum.

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Decoding Problem (contd)

• Every path in the graph has weight P(xp).
• The Viterbi algorithm finds the path that
  maximizes P(xp) among all possible paths.
• The Viterbi algorithm runs in O(nQ2) time.

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Decoding Problem (contd)
w
(k, i)
(l, i1)
The weight w is given by w el(xi1). akl
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Decoding Problem (contd)

• Initialization
• sbegin,0 1
• sk,0 0 for k ? begin.
• Final result
• Let p be the optimal path. Then,
• P(xp) maxk ? Q sk,n . ak,end

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Viterbi Algorithm

• The value of the product can become extremely
  small, which leads to overflowing.
• To avoid overflowing, use log value instead. So,
• sk,i1 logel(xi1) max k ? Q sk,i
  log(akl)

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Forward-Backward Problem

• Given a sequence of coin tosses generated by
  an HMM.
• Goal find the probability that the dealer was
  using a biased coin at a particular time.

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Forward Algorithm

• Defined fk,i (forward probability) as the
  probability of emitting the prefix x1xi and
  reaching the state pi k.
• The recurrence for the forward algorithm is
• fk,i ek(xi) . S fk,i-1 . alk
• l ?
  Q

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Backward Algorithm

• However, forward probability is not the only
  factor affecting P(pi kx).
• The sequence of transitions and emissions that
  the HMM undergoes between pi and pn also affect
  P(pi kx).

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Backward Algorithm (contd)

• Backward probability bk,i the probability of
  being in state pi k and emitting the suffix
  xi1xn.
• The backward algorithms recurrence
• bk,i S el(xi1) . bl,i1 . Akl
• l ? Q

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Backward-Forward Algorithm

• The probability that the dealer used a biased
  coin at any moment i is as follows
• P(x, pi k)
  fk(i) . bk(i)
• P(pi kx) _______________
  ______________
• P(x)
  P(x)

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HMM Parameter Estimation

• So far, we have assumed that the transition and
  emission probabilities are known.
• However, in most HMM applications, the
  probabilities are not known. Its very hard to
  estimate the probabilities.

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HMM Parameter Estimation (contd)

• Let T be a vector combining the unknown
  transition and emission probabilities.
• Given training sequences x (x1,, xm), let
  P(xT) be the maximum probability of x given the
  assignment of parameters T.
• Then our goal is to find
• m
• maxT ? P(xjT)
• j1

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Finding Distant Members of a Protein Family

• Motivation Distant cousins of functionally
  related biological sequences in a protein family
  may have weak similarities, and thus fail
  statistical tests, but may have weak similarities
  with many members of the family. So, the goal is
  to align a sequence to all members of the family
  at once.
• Families of related proteins can be represented
  by their multiple alignment and the corresponding
  profile.

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Profile Representation of Protein Families

• Aligned DNA sequences can be represented by a 4n
  profile matrix reflecting the frequencies of
  nucleotides.

Protein family can be represented by a 20n
profile representing frequencies of amino acids.
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Profiles and HMMs

• HMMs can also be used for aligning a sequence
  against a profile representing
• a protein family.
• A 20n profile P corresponds to n sequentially
  linked match states M1,,Mn in the profile HMM of
  P.

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Profile HMM
A profile HMM
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Insertion and Deletion States of Profile HMM

• States Ii insertion states
• States Di deletion states
• Assumption
• eIj(a) p(a)
• where p(a) is the frequency of the occurrence of
  the symbol a in all the sequences.

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Profile HMM Alignment

• Define vMj as the logarithmic likelihood score of
  the best path for matching x1, , xi to a
  profile HMM ending with xi emitted by the state
  Mj.
• vIj(i) and vDj(i) are defined similarly.

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Profile HMM Alignment Dynamic Programming

• 
  vMj-1(i-1) log(aMj-1, Mj)
• vMj(i) log (eMj(xi)/p(xi)) max
  vIj-1(i-1) log(aIj-1, Mj)
• 
  vDj-1(i-1) log(aDj-1, Mj)
• 
  vMj(i-1) log(aMj, Ij)
• vIj(i) log (eIj(xi)/p(xi)) max
  vIj(i-1) log(aIj, Ij)
• 
  vDj(i-1) log(aDj, Ij)

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Profile HMM Alignment Dynamic Programming

• 
  vMj-1(i-1) log(aMj-1, Dj)
• vMj(i) max
  vIj-1(i-1) log(aIj-1, Dj)
• 
  vDj-1(i-1) log(aDj-1, Dj)

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Paths in Edit Graph and Profile HMM

• A path through an edit graph and the
  corresponding path through a profile HMM

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Speech Recognition

• Create an HMM of the words in a language
• Each word is a state in Q.
• Each of the basic sounds in the language is a
  symbol in S.
• Input use speech as the input sequence.
• Goal find the most probable sequence of states.

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Speech Recognition Building the Model

• Analyze some large source of English sentences,
  such as a database of newspaper articles, to form
  probability matrixes.
• A0i the chance that word i begins a sentence.
• Aij the chance that word j follows word i.

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Building the Model (contd)

• Analyze English speakers to determine what sounds
  are emitted with what words.
• Ek(b) the chance that sound b is spoken in word
  k. Allows for alternate pronunciation of words.

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Speech Recognition Using the Model

• Use the same dynamic programming algorithm as
  before
• Weave the spoken sounds through the model the
  same way we wove the rolls of the die through the
  casino model.
• p represents the most likely set of words.

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Using the Model (contd)

• How well does it work?
• Common words, such as the, a, of make
  prediction less accurate, since there are so many
  words that follow normally.
• We can add more states to incorporate a little
  context into the decision.