8/7/2019 what is a HMM
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P R I M E R
are a recurring them e in comp utational biology. What are
hiddenand why are they so useful for so many different
problems?
is just aof putting the right label on eachIn gene
identification, we want toasexons, introns, or inter-
In sequence alignment, weto associate residues in a query
ina tar-can always write
ad ho c program for any given problem,t the same frustrating
issues will alwaysOne is that we want to incorporate
of information. Afor instance, ought to combine
and open reading
of information be weighted?issue is to interpret results proba-a
best scoring answer is
but what does the score mean,onfident are we that the best
scor-iscorrect? A third issue is exten-
Themoment we perfect our ad ho cwe wish we had also modeledand a
polyadenylation signal.
a fragileprogram makes it collapse under its
Hidden Markov models (HMMs) are afor making probabilistic
of linear sequence 'labeling' prob-a conceptual toolkit
by draw-
atHoward Hughes Medicaltute & Department of Genetics,
University School of Medicine,Forest Park Blvd., Box 8510, Saint
Louis,
[email protected]
A = 0.25C = 0.25G = 0.25T = 0.25
A = 0C = 0G = 0T = 0
.05
.95ACGJ
= 0.4= 0 1= o!4
Sequence: C T T C A TState path: E E E E E E
E n d
TCAI I I logPI -41.22
P a r s i n g :
P o s t e r i o rd e c o d i n g :46% I
- 4 3 . 9 0- 4 3 . 4 5- 4 3 . 9 4- 4 2 . 5 8- 4 1 . 7 1
28 %
Figure 1 A toy HM M for 5' s p l i c e s i t e r e c o g n i t i
o n . S e e t e x t for e x p l a n a t i o n .
ing an intuitive picture. They are at the heartof a diverse
range of programs, includinggenefinding, profile searches,
multiplesequence alignment and regulatory siteidentification. HMMs
are the Legos of com-putational sequence analysis.A toy H M M : 5'
spl ice s i te recogni t ionAs a simple example, imagine the
followingcaricature of a 5' splice-site recognitionproblem. Assume
we are given a DNAsequence that begins in an exon, containson e 5'
splice site and ends in an intron.The problem is to identify where
the switchfrom exon to intron occurredwhere the5 ' splice site
(5'SS) is.
For us to guess intelligently, the sequencesof exons, splice
sites and introns must have
different statistical pro perties. Let's imaginesome simple
differences: say that exonshave a uniform base composition on
average(25% each base), introns are A/T rich (say,40 % each for
A/T, 10% each for C/G), andth e 5'SS consensus nucleotide is
almostalways a G (say, 9 5 % G and5 % A ).
Starting from this information, we candraw an HM M (Fig. 1) .
The HMM invokesthree states, one for each of the three labelswe
might assign to a nucleotide: E (exon),5 (5'SS) and I (intron).
Each state has itsow n emission probabilities (shown above
thestates), which model the base compositionof exons, introns and
the consensus G at the5'SS. Each state also has transition
probabili-ties (arrows), the probabilities of movingfrom this state
to a new state. The transition
8/7/2019 what is a HMM
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P R I M E Rprobabilities describe the linear order inwhich we
expect the states to occur: one ormore Es, one 5, one or mo re
Is,So, what's hidden?It's useful to imagine an HMM generating
asequence. When we visit a state, we emit aresidue from the state's
emission proba bilitydistribution. Then, we choose which state
tovisit next according to the state's transitionprobability
distribution. The model thusgenerates two strings of information.
Oneis the underlying state path (the labels), aswe transition from
state to state. The otheris the observed sequence (the DNA),
eachresidue being emitted from one state in thestate path.
The state path is a Markov chain, m eaningthat what state we go
to next depends onlyon what state we're in. Since we're only
giventhe observed sequence, this underlying statepath is
hiddenthese are the residue labelsthat we'd like to infer. The
state path is ahidden Markov chain.
The probability P(S,7t|HMM,e) that anHMM with parameters 9
generates a statepath 7t and an observed sequence S is theproduct
of all the emission probabilities andtransition probabilities that
were used.For example, consider the 26-nucleotidesequence and state
path in the middle ofFigure 1, where there are 27 transition s
and26 emissions to tote up. Multiply all 53probabilities together
(and take the log,since these are small numbers) and
you'llcalculate log P(S,7t|HMM,e) = -41,22,
An HMM is a full probabilistic modelthe model parameters and the
overallsequence'scores' are all probabilities. There-fore, we can
use Bayesian prob ability theoryto manipulate these numbers in
standard,powerful ways, including optimizing para-meters and
interpreting the significanceof scores.Finding the best state
pathIn an analysis problem, we're given asequence, and we want to
infer the hiddenstate path. There are potentially many statepaths
that could generate the samesequence. We want to find the one with
thehighest probability.
For example, if we were given the HMMand the 26-nucleotide
sequence in Figure 1,there are 14 possible paths that have non-zero
probability, since the 5'SS must fall onone of 14 internal As or
Gs, Figure 1 enu-merates the six highest-scoring paths (those
with G at the 5'SS), The best one has a logprobability of-41,22,
which infers that themost likely 5'SS position is at the fifth
G,
For most problems, there are so manypossible state sequences
that we could notafford to enumerate them. The efficientViterbi
algorithm is guaranteed to find themost probable state path given a
sequenceand an H MM , The Viterbi algorithm is adynamic programming
algorithm quitesimilar to those used for standard
sequencealignment.Beyond best scoring alignmentsFigure 1 shows that
on e alternative statepath differs only slightly in score from
put-ting the 5'SS at the fifth G (log probabilitiesof-4 1,71 versus
-41,22) , How confident arewe that the fifth G is the right
choice?
This is an example of an advantage ofprobabilistic modeling: we
can calculate ourconfidence directly. The probability thatresidue i
was emitted by state k is the sumof the probabilities of all the
state pathsthat use state k to generate residue i (that is,TC; = k
in the state path n) , normalized by thesum over all possible state
paths. In our toymode l, this is just on e state path in
thenumerator and a sum over 14 state paths inthe denominator. We
get a probability of46 % that the best-scoring fifth G is
correctand 28% that the sixth G position is correct(Fig, 1,
bottom). This is called posteriordecoding. For larger problems,
posteriordecoding uses two dynamic programmingalgorithms called
Forward and Backward,which are essentially like Viterbi, but
theysum over possible paths instead of choosingthe best.Making more
realistic modelsMaking an HMM means specifying fourthings: (i) the
symbol alphabet, K differentsymbols (e,g,, ACGT, K= 4); (ii) the
num berof states in the model, M; (iii) emissionprobabilities e,(x)
for each state ;, that sumto one over K symbols x, Xe,( x) = 1;
and(iv) transition probabilities tj(j) for each state1 going to any
other state j (including itself)that sum to one over the M state
s;, X f,( j) = 1,Any model that has these properties is an HMM,This
means that one can make a newHMM just by drawing a picture
corres-ponding to the problem at hand, likeFigure 1, This graphical
simplicity lets onefocus clearly on the biological definition ofa
problem.
For example, in our toy splice-site mmaybe we're not happy with
our discrimtion power; maybe we want to add a realistic
six-nucleotide consensus GTRat the 5' splice site. We can put a
rosix HMM states in place of '5' statmodel a six-base ungapped
consensus mparameterizing the emission probabion known 5' splice
sites. And maybewant to model a complete intron, inclua 3' splice
site; we just add a row of statethe 3'SS consensus, and add a 3'
exon to let the observed sequence end in an instead of an intron.
Then maybe we wabuild a complete gene model,,,whawe add, it's just
a matter of drawing we want.The catchHMMs don't deal well with
correlabetween residues, because they assumeeach residue depends
only on one ulying state. An example where HMMusually inappropriate
is RNA seconstructure analysis, Gonserved RNA pairs induce
long-range pairwise cortions; one position might be any resbut the
base-paired partner must be plementary. An HMM state path haway of
'remembering' what a distant generated.
Sometimes, one can bend the ruleHMMs without breaking the
algoritFor instance, in genefinding, one wanemit a correlated
triplet codon insteathree independent residues; HMM rithms can
readily be extended to tremitting states. However, the basic
Htoolkit can only be stretched so far. BeHMMs, there are more
powerful (thless efficient) classes of probabilistic mfor sequence
analysis,1 . R abiner, L ,R, A tu toria l on h idden M arkov
mod
seiected appl icat ions in speech recogni t ion,; E 7 7 , 2 5 7
- 2 8 6 ( 1 9 8 9 ),
2 , Durb in , R,, Eddy, S,R,, K rogh, A, & M i tch
isoBiological Seq uence Analysis: Probabilistic MoProteins and
Nucieic Acids {Cambridge UniPress, Cambridge UK, 1 998) ,
naturebiotechnologyWondering how some othermathematical
technique really worksSend suggestions for future primers
taskthegeek@natureny,com.
