CHMM Inference in PRISM

Inference with
Constrained Hidden Markov Models
in PRISM

Henning Christiansen Christian Theil Have
Ole Torp Lassen Matthieu Petit

Research group PLIS: Programming, Logic and Intelligent Systems
Department of Communication, Business and Information Technologies
Roskilde University, P.O.Box 260, DK-4000 Roskilde, Denmark

ICLP 2010 in Edinburgh, July 16, 2010

Henning Christiansen, Christian Theil Have, Ole Torp Lassen, Matthieu Petit Inference with CHMMs in PRISM

Outline

1 Motivation and background
Hidden Markov Models
Biological sequence alignment with pair HMMs

2 Constrained HMMs
CHMMs as a constraint program
Decoding algorithm for CHMMs
Implementation in PRISM

3 Summary and future directions


Motivation and background

Motivation

In the LoSt project we explore the use of Probabilistic Logic
programming (with PRISM) for biological sequence analysis
Hidden Markov Models are one of the most widely used models for
sequence analysis
Constrained Hidden Markov Models (CHMMs) allow us to express
relevant prior knowledge and prune the search space


Motivation and background

Contributions and similar work

Contributions
Expressing CHMMs as a constraint problem
Adaptation of Viterbi decoding algorithm for CHMMs
Eﬃcient implementation of CHMMs in PRISM
Similar work
In [SK08] introduces the concept of CHMMs
[CPC08] express relationships in Bayesian Network as a CLP
Constrained Conditional Models [CRR08] augments (discriminative)
models with declarative constraints


Motivation and background Hidden Markov Models


Deﬁnition
A Hidden Markov Model (HMM) is a 4-tuple S, A, T , E , where
S = {s0 , s1 , . . . , sm } is a set of states which includes an initial state
referred to as s0 ;
A = {e1 , e2 , . . . , ek } is a ﬁnite set of emission symbols;
T = {(p(s0 ; s1 ), . . . , p(s0 ; sm )), . . . , (p(sm ; s1 ), . . . , p(sm ; sm ))} is a set
of transition probability distributions representing probabilities to
transit from one state to another;
E = {(p(s1 ; e1 ), . . . , p(s1 ; ek )), . . . , (p(sm ; e1 ), . . . , p(sm ; ek ))} is a set
of emission probability distributions representing probabilities to emit
each symbol from one state.



Hidden Markov Model, graphical example

Postcard
Greetings from wherever, where I am having a
great time. Here is what I have been doing: The
ﬁrst two days, I stayed at the hotel reading a
good book. Then, on the third day I decided to
go shopping. The next three days I did nothing
but lie on the beach. On my last day, I went
shopping for some gifts to bring home and wrote
you this postcard.
Sincerely, Some friend of yours

Observation sequence



Hidden Markov Model run

Definition
A run of an HMM as a pair consisting of a sequence of states
s (0) s (1) . . . s (n) , called a path and a corresponding sequence of emissions
e (1) . . . e (n) , called an observation, such that
s (0) = s0 ;
∀i, 0 ≤ i ≤ n − 1, p(s (i) ; s (i+1) ) > 0
(probability to transit from s (i) to s (i+1) );
∀i, 0 < i ≤ n, p(s (i) ; e (i) ) > 0
(probability to emit e (i) from s (i) ).

Definition
The probability of such a run is defined as

i=1..n p(s (i−1) ; s (i) ) · p(s (i) ; e (i) )



Decoding with Hidden Markov Models
Decoding: Inferring the hidden path given the observation sequence.
argmaxpath P(path|observation)

Computed in linear time using dynamic programming:
The Viterbi algorithm can be seen as keeping track of, for each preﬁx of
an observed emission sequence, the most probable (partial) path leading to
each possible state, and extending those step by step into longer paths,
eventually covering the entire emission sequence.

source: wikipedia

Motivation and background Biological sequence alignment with pair HMMs

A tiny bit of biological motivation

Biologists often want to align sequences (e.g. DNA, proteins, etc.)
which are thought to have descended from a common ancestor
Sequence alignment is fundamental to bioinformatics
We (computer scientists) can just consider biological sequences as a
list of symbols
The number of possible of alignments is exponential
An optimal alignment can be found using a Pair HMM in
O(|sequence|2 ∗ |states|2 )



Pair HMMs for sequence alignment

A pair HMM is a special kind of HMM that emits two sequences




The match state emit a pair (xi yj ) of symbols




The insert state emits one symbol xi , from sequence x




The delete state emits one symbol yj , from sequence y




The delete state emits one symbol yj , from sequence y
A run of this model produces an alignment of x and y



Alignment with a pair Hidden Markov Model

Observation sequence x Observation sequence y
H G K K G A A Q V K G P K K A Q A

sequence x
sequence y
alignment b




sequence x H
sequence y
alignment b i




sequence x H G
sequence y
alignment b i i




sequence x H G K
sequence y
alignment b i i i




sequence x H G K K
sequence y K
alignment b i i i m




sequence x H G K K G
sequence y K G
alignment b i i i m m




sequence x H G K K G A
sequence y K G P
alignment b i i i m m m




sequence y K G P K
alignment b i i i m m m d




sequence y K G P K K
alignment b i i i m m m d d




sequence x H G K K G A A
sequence y K G P K K A
alignment b i i i m m m d d m




sequence x H G K K G A A Q
sequence y K G P K K A Q
alignment b i i i m m m d d m m




sequence x H G K K G A A Q V
sequence y K G P K K A Q A
alignment b i i i m m m d d m m m

Constrained HMMs


2 Constrained HMMs



Constrained HMMs

Constrained HMMs

Deﬁnition
A constrained HMM (CHMM) is a 5-tuple S, A, T , E , C where,
S, A, T , E is an HMM
C is a set of constraints, each of which is a mapping from HMM runs
into {true, false}.
A run of a CHMM, path, observation is a run of the corresponding HMM
for which C (path, observation) is true.


Constrained HMMs

Constrained HMMs

Why extend an HMM with side-constraints?
To create better, more speciﬁc models with fewer states
Convenient to express prior knowledge in terms of constraints
No need to change underlying HMM
Sometimes it is not possible or feasible to express such constraints as
HMM structure (e.g. all different)
→ infeasibly huge state and parameter space
fewer paths to consider for any given sequence
→ decreased running time


Constrained HMMs CHMMs as a constraint program

Runs of a CHMM as a constraint program

We consider runs of CHMM as a constraint program over ﬁnite domains.
A run of CHMM is a solution of the constraint program
run([s (0) , S1 , . . . , Sn ], [E1 , . . . , En ]),
dom(Si ) = S {s0 } ∧ dom(Ei ) = E

The variable Si represents the HMM state at step i
The variable Ei represents the emission at the step i



Definition of the HMM run program

The run predicate is specified as follows,

run([s (0) , S1 , . . . , Sn ], [E1 , . . . , En ]) is true iff

∃s (1) ∈ dom(S1 ), . . . , ∃s (n) ∈ dom(Sn )∧
∃e (1) ∈ dom(E1 ), . . . , ∃e (n) ∈ dom(En )∧
C (s (0) s (1) . . . s (n) , e (1) . . . e (n) ) is true, s (0) = s0 ∧
p(s (0) ; s (1) ) · p(s (1) ; e (1) ) . . . p(s (n−1) ; s (n) ) · p(s (n) ; e (n) ) > 0.
This states that

s (0) s (1) . . . s (n) and e (1) . . . e (n) is a HMM run which satisfies C .



The HMM structure

The last line in the definition,

p(s (0) ; s (1) ) · p(s (1) ; e (1) ) . . . p(s (n−1) ; s (n) ) · p(s (n) ; e (n) ) > 0
indicates the (local) relationships between Si ,Si+1 and Si ,Ei which
correspond to the structure of the HMM. We define these relationships in
terms of constraints,

trans(Si−1 , Si ) ∧ emit(Si , Ei ), for all i, 1 ≤ i ≤ n

trans(Si , Si+1 ) is true iff emit(Si , Ei ) is true iff
∃s (i) ∈ dom(Si ) ∧ s (i+1) ∈ dom(Si+1 ) ∃s (i) ∈ dom(Si ) ∧ e (i) ∈ dom(Ei )
such that p(s (i) ; s (i+1) ) > 0 such that p(s (i) ; e (i) ) > 0



Revisiting the pair HMM

Consider adding constraints to the pair HMM introduced earlier.

For instance..
In a biological context, we may want to only
consider alignments with a limited number of
insertions and deletions given the assumption
that the two sequences are closely related.

C= {cardinality atmost(Nd , [S1 , . . . , Sn ], delete),
cardinality atmost(Ni , [S1 , . . . , Sn ], insert)} .

The constraint cardinality atmost(N, L, X ) is satisﬁed whenever L is a list
of elements, out of which at most N are equal to X .



Computation of best alignment with constraints


Constrained HMMs Decoding algorithm for CHMMs

Decoding with a CHMM

Decoding is the process of ﬁnding the most probable state sequence given
a known sequence of observations.

Consider a given observation e (1) . . . e (n) , a CHMM S, A, T , E , C , and its
constraint program

run([s (0) , S1 , . . . , Sn ], [e (1) , . . . , e (n) ]).

The most probable path is computed by ﬁnding the valuation s (1) , . . . , s (n)
that maximizes the objective function: the run probability.
In standard HMMs this is usually accomplished in polynomial time using
the Viterbi dynamic programming algorithm.
But we need to consider additional constraints, C .



An algorithm for decoding with CHMMs

The computation starts from an initial set of 5-tuples

{ s (0) , 0, 1, , C ∧ trans(s (0) , S1 ) ∧ 1≤i≤n−1 trans(Si , Si+1 ) ∧ 1≤i≤n emit(Si , ei ) }.






Then, the following two rules are applied as long as possible






trans ctr
Σ := Σ ∪ { s , i +1, p · p(s; s ) · p(s ; e (i+1) ), π s , σ ∧ Si+1 = s }
whenever s, i, p, π, σ ∈ Σ, p(s; s ), p(s ; e (i+1) ) > 0 and
check constraints(σ ∧ Si+1 = s ) and prune ctr does not apply.






trans ctr
Σ := Σ ∪ { s , i +1, p · p(s; s ) · p(s ; e (i+1) ), π s , σ ∧ Si+1 = s }
whenever s, i, p, π, σ ∈ Σ, p(s; s ), p(s ; e (i+1) ) > 0 and
check constraints(σ ∧ Si+1 = s ) and prune ctr does not apply.

prune ctr
Σ := Σ { s, i +1, p , π , σ }
whenever there is another s, i +1, p, π, σ ∈ Σ with
p ≥ p and sol(σ ) ⊆ sol(σ)



Illustration of the trans ctr transition rule
Constraints:
cardinality atmost(1, [S1 , . . . , Sn ], delete)
cardinality atmost(1, [S1 , . . . , Sn ], insert)

check constraints


Constraints:

check constraints → OK


Constraints:

Path extended


Constraints:

check constraints → fail


Constraints:

Path not extended


Illustration of the prune ctr rule

Two (partial) paths, p and p , reach the
same state at time i




sol(σ ) ⊆ sol(σ): Are all states
reachable from p at any time j > i also
reachable from p?




reachable from p?
no →




reachable from p?
no →
rule not does apply (keep both paths)




reachable from p?
no →
yes →




reachable from p?
no →
yes → Is P(p) ≥ P(p )?




reachable from p?
no →
yes




reachable from p?
no →
yes →
discard lowest probability path p




reachable from p?
no →
yes →
no →




reachable from p?
no →
yes →
no →
rule does not apply (keep both paths)


Constrained HMMs Implementation in PRISM

PRISM

PRogramming In Statistical Modelling developed by Sato, Kameya
and Zhou
An extension of Prolog with random variables, called MSWs
Provides eﬃcient generalized inference algorithms (Viterbi, EM, etc)
using tabling



Example HMM in PRISM

values/2 Example HMM in PRISM
declares the outcomes values(trans(_), [sunny,rainy]).
of random variables values(emit(_), [shop,beach,read]).

msw/2 hmm(L):- run_length(T),hmm(T,start,L).
simulates a random
variable, stochastically hmm(0,_,[]).
selecting one of the hmm(T,State,[Emit|EmitRest]) :-
outcomes T > 0,
msw(trans(State),NextState),
Model in Prolog msw(emit(NextState),Emit),
Speciﬁes relation T1 is T-1,
between variables hmm(T1,NextState,EmitRest).

run_length(7).


Adding constraint checking to the HMM

HMM with constraint checking
hmm(T,State,[Emit|EmitRest],StoreIn) :-
T > 0,
msw(trans(State),NxtState),
msw(emit(NxtState),Emit),
check_constraints([NxtState,Emit],StoreIn,StoreOut),
T1 is T-1,
hmm(T1,NxtState,EmitRest,StoreOut).

Call to check constraints/3 after each distinct sequence of msw
applications
Side-constaints: The constraints are assumed to be declared
elsewhere and not interleaved with model speciﬁcation
Extra Store argument in the probabilistic predicate



Checking the constraints

The goal check constraints/3 calls constraint checkers for all
constraints declared on the model.
For instance, with our example pair HMM constraint,

C= {cardinality atmost(Nd , [S1 , . . . , Sn ], delete),
cardinality atmost(Ni , [S1 , . . . , Sn ], insert)} .

We have the following incremental constraint checker implementation
A cardinality atmost constraint checker
init_constraint_store(cardinality_atmost(_,_), 0).
check_sat(cardinality_atmost(U,Max), U, In, Out) :-
Out is In + 1,Out =< Max.
check_sat(cardinality_atmost(X,_),U,S,S) :- X = U.



A library of global constraints for Hidden Markov Models

Our implementation contains a few well-known global constraints adapted
to Hidden Markov Models.
Global constraints
cardinality lock to sequence
all different lock to set

In addition, the implementation provides operators which may be used to
apply constraints to a limited set of variables.
Constraint operators
state specific
emission specific
forall subseq (sliding window operator)
for range (time step range operator)



Improving tabling with PRISM
Problem: The extra Store argument makes PRISM table multiple goals
(for diﬀerent constraint stores) when it should only store one.

hmm(T,State,[Emit|EmitRest],Store)
To get rid of the extra argument, check constraints dynamically
maintains it as a stack using assert/retract:

check_constraints(Update) :-
get_store(StoreBefore),
check_constraints(Update,StoreBefore,StoreAfter),
forward_store(StoreAfter).

get_store(S) :- store(S), !.
forward_store(S) :-
asserta(store(S)) ;
retract(store(S)),fail.


Impact of using a separate constraint store stack


Summary and future directions


2 Constrained HMMs




Summary

We have demonstrated
That CHMMs can be a useful tool to
Incorporate prior knowledge in probabilistic model
Prune the search space and allow faster decoding
That CHMMs may be useful in biological sequence analysis
How CHMMs can be formulated as a constraint problem
An adaptation of the Viterbi algorithm for CHMMs
How CHMMs can be eﬃciently implemented in PRISM



Future directions

Improve method to employ existing state-of-the-art CSP techniques
Soft constraints
Parameter learning for CHMMs
Grammar constraints
More biological applications



Questions

Questions?



Bibliography I

V.S. Costa, D. Page, and J. Cussens.
CLP(BN): Constraint logic programming for probabilistic knowledge.
Probabilistic Inductive Logic Programming, LNAI 4911:156–188, 2008.
M-W Chang, L-A Ratinov, and D. Rizzolo, N. Roth.
Learning and inference with constraints.
In Proc. of AAAI Conference on Artiﬁcial Intelligence, pages
1513–1518, Chicago, USA, July 2008.
T. Sato and Y. Kameya.
New advances in logic-based probabilistic by PRISM.
In Probabilistic Inductive Logic Programming, LNCS, pages 118–155.
Springer, 2008.


CHMM Inference in PRISM

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