1. Hidden Markov Models in Bioinformatics
O1 O2 O3 O4 O5 O6 O7 O8 O9 O10 H1 H2 H3
Definition
Three Key Algorithms
Summing over Unknown States
Most Probable Unknown States
Marginalizing Unknown States
Key Bioinformatic Applications
Pedigree Analysis
Profile HMM Alignment
Fast/Slowly Evolving States
Statistical Alignment

2. Hidden Markov Models
O1 O2 O3 O4 O5 O6 O7 O8 O9 O10 H1 H2 H3
(O1,H1), (O2,H2),……. (On,Hn) is a sequence of stochastic variables with 2 components - one that is observed (Oi) and one that is hidden (Hi).
The marginal distribution of the Hi’s are described by a Homogenous Markov Chain:
pi,j = P(Hk=i,Hk+1=j)
Let pi =P{H1=i) - often pi is the equilibrium distribution of the Markov Chain.
Conditional on Hk (all k), the Ok are independent.
The distribution of Ok only depends on the value of Hi and is called the emit function

3. What is the probability of the data?
O1 O2 O3 O4 O5 O6 O7 O8 O9 O10 H1 H2 H3
The probability of the observed is , which could be hard to calculate. However, these calculations can be considerably accelerated. Let the probability of the observations (O1,..Ok) conditional on Hk=j. Following recursion will be obeyed:

4. Example - probability of the data
Observables {0, 1} at times 1, 2, 3. Hidden states {a, b}.
Emission probabilities: a b 1 .9 .1 .7 .3
Transition probabilities:
Equilibrium distribution, p, of a b is .5 .5
Example. Observation
P(aaa) =.5*.9 * P(011| aaa) = .7 *.3 *.3
P(aab) =.5*.9 * P(011| aab) = .7 *.3 *.7
…………………………………………….
Direct calculation:
Forward recursion:
Observations:
Hidden states:
.15
.35
pa*P(0|a) = .5 * .7
.099 .3 .9 .1 a b 1
pa = .5
pb = .5
.7(.35*.1+.15*.9)=.119
.3(.099*.9*+.119*.1)=0.0303
.7(.099* *.9)=0.0819
Hence P(O) up to the 3rd state is = 4

5. What is the most probable ”hidden” configuration?
This algorithm is also called Viterby.
Let be the sequences of hidden states in the most probably hidden path ie ArgMaxH[ ]. Let be the probability of the most probable path up to k ending in hidden state j.
Again recursions can be found:
The actual sequence of hidden states can be found recursively by
O1 O2 O3 O4 O5 O6 O7 O8 O9 O10 H1 H2 H3

6. What is the probability of specific ”hidden” state?
O1 O2 O3 O4 O5 O6 O7 O8 O9 O10 H1 H2 H3
Let be the probability of the observations from k+1 to n given Hk=j. These will also obey recursions:
The probability of the observations and a specific hidden state can found as:
And of a specific hidden state can found as:

7. Example continued - best path, single hidden state
pb = .5 1
Observations:
Hidden states: or .3
pa = .5 .7 .9
.051
.189 = Max{.7 *.9 *.3, .3 *.1 *.3} b .1
.189
.1191 .3
Single hidden state:
Forward:
Forward - Backward: a b 1
pa = .5
pb = .5
Observations:
Hidden states: a b 1
pa = .5
pb = .5
Observations:
Hidden states:
Backward: a b 1
pa = .5
pb = .5
Observations:
Hidden states: 7

8. Baum-Welch, Parameter Estimation or Training
O1 O2 O3 O4 O5 O6 O7 O8 O9 O10 H1 H2 H3
Objective: Evaluate Transition and Emission Probabilities
Set pij and e( ) arbirarily to non-zero values
Use forward-backward to re-evaluate pij and e( )
Do this until no significant increase in probability of data
To avoid zero probabilities, add pseudo-counts.
Other numerical optimization algorithms can be applied.

9. Fast/Slowly Evolving States Felsenstein & Churchill, 1996
positions
sequences k
slow - rs
fast - rf
HMM:
pr - equilibrium distribution of hidden states (rates) at first position
pi,j - transition probabilities between hidden states
L(j,r) - likelihood for j’th column given rate r.
L(j,r) - likelihood for first j columns given j’th column has rate r.
Make simpler. Show basic probability expressions.
Likelihood Recursions:
Likelihood Initialisations:

10. Recombination HMMs 1 2 3 T
i-1 i L
Data
Trees
Illustrate better

11. Statistical Alignment Steel and Hein,2001 + Holmes and Bruno,2001
Emit functions:
e(##)= p(N1)f(N1,N2)
e(#-)= p(N1), e(-#)= p(N2)
p(N1) - equilibrium prob. of N
f(N1,N2) - prob. that N1 evolves into N2
# # E
# # E *
* lb l/m (1- lb)e-m l/m (1- lb)(1- e-m) (1- l/m) (1- lb) -
# lb l/m (1- lb)e-m l/m (1- lb)(1- e-m) (1- l/m) (1- lb) _
# lb l/m (1- lb)e-m l/m (1- lb)(1- e-m) (1- l/m) (1- lb) # lb
An HMM Generating Alignments

12. Probability of Data given a pedigree.
Elston-Stewart (1971) -Temporal Peeling Algorithm:
Mother
Father
Condition on parental states
Recombination and mutation are Markovian
Lander-Green (1987) - Genotype Scanning Algorithm:
Mother
Father
Condition on paternal/maternal inheritance
Recombination and mutation are Markovian
Comment: Obvious parallel to Wiuf-Hein99 reformulation of Hudson’s 1983 algorithm

13. Further Examples Isochore: Gene Finding: Simple Eukaryotic
Churchill,1989,92
Lp(C)=Lp(G)=0.1, Lp(A)=Lp(T)=0.4, Lr(C)=Lr(G)=0.4, Lr(A)=Lr(T)=0.1
poor
rich
HMM:
Likelihood Recursions:
Likelihood Initialisations:
Gene Finding:
Burge and Karlin, 1996
Simple Eukaryotic
Simple Prokaryotic
Make simpler. Show basic probability expressions.

14. Further Examples Secondary Structure Elements: Profile HMM Alignment:
Goldman, 1996
Further Examples
HMM for SSEs: a  L
.909
.0005
.091
.005
.881
.184
.062
.086
.852
.325
.212
.462 a  L
Adding Evolution:
SSE Prediction:
Make simpler. Show basic probability expressions.
Profile HMM Alignment:
Krogh et al.,1994

15. Summary H1 H2 H3 Definition Three Key Algorithms
O1 O2 O3 O4 O5 O6 O7 O8 O9 O10 H1 H2 H3
Definition
Three Key Algorithms
Summing over Unknown States
Most Probable Unknown States
Marginalizing Unknown States
Key Bioinformatic Applications
Pedigree Analysis
Isochores in Genomes (CG-rich regions)
Profile HMM Alignment
Fast/Slowly Evolving States
Secondary Structure Elements in Proteins
Gene Finding
Statistical Alignment

16. Grammars: Finite Set of Rules for Generating Strings
A starting symbol:
Ordinary letters:
& Variables:
ii. A set of substitution rules applied to variables in the present string:
finished – no variables
Regular
Context Free
Context Sensitive
General (also erasing)

17. Simple String Generators
Terminals (capital) Non-Terminals (small)
i. Start with S S --> aT bS
T --> aS bT 
One sentence – odd # of a’s:
S-> aT -> aaS –> aabS -> aabaT -> aaba
ii. S--> aSa bSb aa bb
One sentence (even length palindromes):
S--> aSa --> abSba --> abaaba

18. Stochastic Grammars i. Start with S. S --> (0.3)aT (0.7)bS
The grammars above classify all string as belonging to the language or not.
All variables has a finite set of substitution rules. Assigning probabilities to the use of each rule will assign probabilities to the strings in the language.
If there is a 1-1 derivation (creation) of a string, the probability of a string can be obtained as the product probability of the applied rules.
i. Start with S. S --> (0.3)aT (0.7)bS
T --> (0.2)aS (0.4)bT (0.2)
*0.2
S -> aT -> aaS –> aabS -> aabaT -> aaba
*0.3
*0.7
*0.3
*0.2
ii. S--> (0.3)aSa (0.5)bSb (0.1)aa (0.1)bb
S -> aSa -> abSba -> abaaba
*0.3
*0.5
*0.1

19. Recommended Literature
Vineet Bafna and Daniel H. Huson (2000) The Conserved Exon Method for Gene Finding ISMB pp. 3-12
S.Batzoglou et al.(2000) Human and Mouse Gene Structure: Comparative Analysis and Application to Exon Prediction. Genome Research
Blayo, Rouze & Sagot (2002) ”Orphan Gene Finding - An exon assembly approach” J.Comp.Biol.
Delcher, AL et al.(1998) Alignment of Whole Genomes Nuc.Ac.Res
Gravely, BR (2001) Alternative Splicing: increasing diversity in the proteomic world. TIGS
Guigo, R.et al.(2000) An Assesment of Gene Prediction Accuracy in Large DNA Sequences. Genome Research
Kan, Z. Et al. (2001) Gene Structure Prediction and Alternative Splicing Using Genomically Aligned ESTs Genome Research
Ian Korf et al.(2001) Integrating genomic homology into gene structure prediction. Bioinformatics vol17.Suppl.1 pages
Tejs Scharling (2001) Gene-identification using sequence comparison. Aarhus University
JS Pedersen (2001) Progress Report: Comparative Gene Finding. Aarhus University
Reese,MG et al.(2000) Genome Annotation Assessment in Drosophila melanogaster Genome Research
Stein,L.(2001) Genome Annotation: From Sequence to Biology. Nature Reviews Genetics

20. Example continued - parameter optimisation20