Variable length memory chains and
regeneration
Alexsandro Gallo
supervisor: Antonio Galves
MAS 2008, Rennes
The model
Main result
So What??
Probabilistic context trees
Chains with variable lenght memory
Chains with unbounded variable lenght memory
Regeneration scheme
An example of tree
0
←−
1
−→
q
A
A
0 q
q
00
AAq
A
A
1
AAq
10
Finite alphabet A = {0, 1}
The model
Main result
So What??
Probabilistic context trees
Chains with variable lenght memory
Chains with unbounded variable lenght memory
Regeneration scheme
An example of tree
0
←−
1
−→
q
A
A
AAq
0 q
1
A
A
AAq
q
00
10
Finite alphabet A = {0, 1}
Let call this tree τ .
It can be identified with the set
of its leaves: τ = {00, 10, 1}.
These leaves are called contexts.
The model
Main result
So What??
Probabilistic context trees
Chains with variable lenght memory
Chains with unbounded variable lenght memory
Regeneration scheme
Definition of a context tree
0
←−
1
−→
q
A
A
AAq
0 q
1
A
A
AAq
q
00
10
Finite alphabet A = {0, 1}
Let call this tree τ .
It can be identified with the set
of its leaves: τ = {00, 10, 1}.
These leaves are called contexts.
q We call a context tree a set of contexts wich verifies
→ suffix property
→ completeness.
The model
Main result
So What??
Probabilistic context trees
Chains with variable lenght memory
Chains with unbounded variable lenght memory
Regeneration scheme
Definition of a probabilistic context tree
0
←−
1
−→
q
A
A
AAq
0 q
1
A
A p(1|1)
AAq
q
00
10
p(1|00) p(1|01)
Finite alphabet A = {0, 1}
Let call this tree τ .
It can be identified with the set
of its leaves: τ = {00, 10, 1}.
These leaves are called contexts.
q We call a context tree a set of contexts wich verifies
→ suffix property
→ completeness.
q A probabilistic context tree is a pair (τ, p) where p is a set of
transition probabilities: p = {p(1|ω), ω ∈ τ }.
The model
Main result
So What??
Probabilistic context trees
Chains with variable lenght memory
Chains with unbounded variable lenght memory
Regeneration scheme
Let’s construct a stochastic chain with (τ, p)
q
A
A
AAq
0 q
1
A
A p(1|1)
AAq
q
00
A
p(1|00) A
AAq
q
010
110
p(1|010) p(1|011)
We construct the chain using (τ, p) given the past
−1
= . . . 01010
x−∞
The model
Main result
So What??
Probabilistic context trees
Chains with variable lenght memory
Chains with unbounded variable lenght memory
Regeneration scheme
Let’s construct a stochastic chain with (τ, p)
q
A
A
AAq
0 q
1
A
A p(1|1)
AAq
q
00
A
p(1|00) A
AAq
q
010
110
p(1|010) p(1|011)
We construct the chain using (τ, p) given the past
−1
= . . . 01010
x−∞
−1
context cτ (x−∞
) = 010
The model
Main result
So What??
Probabilistic context trees
Chains with variable lenght memory
Chains with unbounded variable lenght memory
Regeneration scheme
Let’s construct a stochastic chain with (τ, p)
q
A
A
AAq
0 q
1
A
A p(1|1)
AAq
q
00
A
p(1|00) A
AAq
q
010
110
p(1|010) p(1|011)
We construct the chain using (τ, p) given the past
−1
= . . . 01010
x−∞
−1
context cτ (x−∞
) = 010
x0 = 1 appears with probability p(1|010)
The model
Main result
So What??
Probabilistic context trees
Chains with variable lenght memory
Chains with unbounded variable lenght memory
Regeneration scheme
Let’s construct a stochastic chain with (τ, p)
q
A
A
AAq
0 q
1
A
A p(1|1)
AAq
q
00
A
p(1|00) A
AAq
q
010
110
p(1|010) p(1|011)
We construct the chain using (τ, p) given the past
0
x−∞
= . . . 010100
The model
Main result
So What??
Probabilistic context trees
Chains with variable lenght memory
Chains with unbounded variable lenght memory
Regeneration scheme
Let’s construct a stochastic chain with (τ, p)
q
A
A
AAq
0 q
1
A
A p(1|1)
AAq
q
00
A
p(1|00) A
AAq
q
010
110
p(1|010) p(1|011)
We construct the chain using (τ, p) given the past
0
x−∞
= . . . 010100
0
context cτ (x−∞
) = 00
The model
Main result
So What??
Probabilistic context trees
Chains with variable lenght memory
Chains with unbounded variable lenght memory
Regeneration scheme
Let’s construct a stochastic chain with (τ, p)
q
A
A
AAq
0 q
1
A
A p(1|1)
AAq
q
00
A
p(1|00) A
AAq
q
010
110
p(1|010) p(1|011)
We construct the chain using (τ, p) given the past
0
x−∞
= . . . 010100
0
context cτ (x−∞
) = 00
x1 = 1 appears with probability p(1|00)
The model
Main result
So What??
Probabilistic context trees
Chains with variable lenght memory
Chains with unbounded variable lenght memory
Regeneration scheme
Let’s construct a stochastic chain with (τ, p)
q
A
A
AAq
0 q
1
A
A p(1|1)
AAq
q
00
A
p(1|00) A
AAq
q
010
110
p(1|010) p(1|011)
We construct the chain using (τ, p) given the past
1
x−∞
= . . . 0101001
The model
Main result
So What??
Probabilistic context trees
Chains with variable lenght memory
Chains with unbounded variable lenght memory
Regeneration scheme
Let’s construct a stochastic chain with (τ, p)
q
A
A
AAq
0 q
1
A
A p(1|1)
AAq
q
00
A
p(1|00) A
AAq
q
010
110
p(1|010) p(1|011)
We construct the chain using (τ, p) given the past
1
x−∞
= . . . 0101001
1
context cτ (x−∞
)=1
The model
Main result
So What??
Probabilistic context trees
Chains with variable lenght memory
Chains with unbounded variable lenght memory
Regeneration scheme
Let’s construct a stochastic chain with (τ, p)
q
A
A
AAq
0 q
1
A
A p(1|1)
AAq
q
00
A
p(1|00) A
AAq
q
010
110
p(1|010) p(1|011)
We construct the chain using (τ, p) given the past
1
x−∞
= . . . 0101001
1
context cτ (x−∞
)=1
x2 = 1 appears with probability p(1|1)
The model
Main result
So What??
Probabilistic context trees
Chains with variable lenght memory
Chains with unbounded variable lenght memory
Regeneration scheme
Let’s construct a stochastic chain with (τ, p)
q
A
A
AAq
0 q
1
A
A p(1|1)
AAq
q
00
A
p(1|00) A
AAq
q
010
110
p(1|010) p(1|011)
We construct the chain using (τ, p) given the past
1
x−∞
= . . . 0101001
1
context cτ (x−∞
)=1
x2 = 1 appears with probability p(1|1)
..etc... that is the way we construct a chain
with variable lenght memory.
The model
Main result
So What??
Probabilistic context trees
Chains with variable lenght memory
Chains with unbounded variable lenght memory
Regeneration scheme
To conclude on the finite size model
In order to construct the process, it is natural to ask for the suffix
and the completeness properties:
The model
Main result
So What??
Probabilistic context trees
Chains with variable lenght memory
Chains with unbounded variable lenght memory
Regeneration scheme
To conclude on the finite size model
In order to construct the process, it is natural to ask for the suffix
and the completeness properties:
completeness property ⇒ there exists a relevant part of the
past,
The model
Main result
So What??
Probabilistic context trees
Chains with variable lenght memory
Chains with unbounded variable lenght memory
Regeneration scheme
To conclude on the finite size model
In order to construct the process, it is natural to ask for the suffix
and the completeness properties:
completeness property ⇒ there exists a relevant part of the
past,
suffix property ⇒ this relevant past is unique.
The model
Main result
So What??
Probabilistic context trees
Chains with variable lenght memory
Chains with unbounded variable lenght memory
Regeneration scheme
To conclude on the finite size model
In order to construct the process, it is natural to ask for the suffix
and the completeness properties:
completeness property ⇒ there exists a relevant part of the
past,
suffix property ⇒ this relevant past is unique.
REMARK that the finite size model is nothing else than a
“parcimonieuse” Markov chain, thus we know everything about
existence, uniqueness and perfect simulation of the stationary
measure.
The model
Main result
So What??
Probabilistic context trees
Chains with variable lenght memory
Chains with unbounded variable lenght memory
Regeneration scheme
Some references
On regeneration and prefect simulation for Markov chains
J. G. Propp, D. B. Wilson. Exact sampling with coupled
Markov chains and applications to statistical mechanics.
Random Structures and Algorithms, 9, 223-252, 1996.
The model
Main result
So What??
Probabilistic context trees
Chains with variable lenght memory
Chains with unbounded variable lenght memory
Regeneration scheme
Some references
On regeneration and prefect simulation for Markov chains
J. G. Propp, D. B. Wilson. Exact sampling with coupled
Markov chains and applications to statistical mechanics.
Random Structures and Algorithms, 9, 223-252, 1996.
On Variable length Markov chains
J. Rissanen. A universal data compression system. IEEE
Trans. Inform. Theory, 29(5), 656-664, 1983.
F. G. Leonardi. A generalization of the pst algorithm:
modelling the sparse nature of protein sequences.
Bioinformatics, 22(7), 2006.
A. Galves, E. Löcherbach. Stochastic chains with memory of
variable length. TICSP Series 38, 117-133, 2008.
The model
Main result
So What??
Probabilistic context trees
Chains with variable lenght memory
Chains with unbounded variable lenght memory
Regeneration scheme
Chains with unbounded variable lenght memory
In this case, the size of the contexts is still finite, but
unbounded.
The model
Main result
So What??
Probabilistic context trees
Chains with variable lenght memory
Chains with unbounded variable lenght memory
Regeneration scheme
Chains with unbounded variable lenght memory
In this case, the size of the contexts is still finite, but
unbounded.
It’s a chain of infinite order.
The model
Main result
So What??
Probabilistic context trees
Chains with variable lenght memory
Chains with unbounded variable lenght memory
Regeneration scheme
Chains with unbounded variable lenght memory
In this case, the size of the contexts is still finite, but
unbounded.
It’s a chain of infinite order.
Question of the existence of a stationary process consistent
with (τ, p).
The model
Main result
So What??
Probabilistic context trees
Chains with variable lenght memory
Chains with unbounded variable lenght memory
Regeneration scheme
Chains with unbounded variable lenght memory
In this case, the size of the contexts is still finite, but
unbounded.
It’s a chain of infinite order.
Question of the existence of a stationary process consistent
with (τ, p).
The simplest example is the sparse chain.
The model
Main result
So What??
Probabilistic context trees
Chains with variable lenght memory
Chains with unbounded variable lenght memory
Regeneration scheme
The sparse chain
q
A
A
AAq
q
1
A
p0
A
AAq
q
A
10
A
p1
A
Aq
q
A
100
p2
A
A
Aq
q
A
1000
A
p3
The contexts have the form
k
τ = ∪+∞
k =0 10 .
The transition probabilities
p(1|0k 1) = pk , k ∈ N.
The model
Main result
So What??
Probabilistic context trees
Chains with variable lenght memory
Chains with unbounded variable lenght memory
Regeneration scheme
The sparse chain
q
A
A
The contexts have the form
AAq
k
q
τ = ∪+∞
k =0 10 .
1
A
p0
The transition probabilities
A
A
Aq
q
p(1|0k 1) = pk , k ∈ N.
A
10
A
p1
A
Aq
Remark: when a 1 appears, the process
q
A
100
p2
“forgets” the past.
A
A
Aq
q
A
1000
A
p3
=⇒ the instant where a 1 appears are called
regeneration times.
Probabilistic context trees
Chains with variable lenght memory
Chains with unbounded variable lenght memory
Regeneration scheme
The model
Main result
So What??
The sparse chain is in fact a regeneration process
m
Suppose given x−∞
...
0q
...m − 2
1q
m−1
0q
m
q
q
q
q
...
q
Probabilistic context trees
Chains with variable lenght memory
Chains with unbounded variable lenght memory
Regeneration scheme
The model
Main result
So What??
The sparse chain is in fact a regeneration process
...
0q
...m − 2
1q
m−1
0q
m
q
q
q
q
...
q
Probabilistic context trees
Chains with variable lenght memory
Chains with unbounded variable lenght memory
Regeneration scheme
The model
Main result
So What??
The sparse chain is in fact a regeneration process
...
0q
...m − 2
1q
m−1
0q
0q
m
m+1
q
q
q
...
q
Probabilistic context trees
Chains with variable lenght memory
Chains with unbounded variable lenght memory
Regeneration scheme
The model
Main result
So What??
The sparse chain is in fact a regeneration process
...
0q
...m − 2
1q
m−1
0q
0q
1q
m
m+1
m+2
q
q
...
q
Probabilistic context trees
Chains with variable lenght memory
Chains with unbounded variable lenght memory
Regeneration scheme
The model
Main result
So What??
The sparse chain is in fact a regeneration process
...
0q
...m − 2
1q
m−1
0q
0q
1q
0q
m
m+1
m+2
m+3
q
...
q
Probabilistic context trees
Chains with variable lenght memory
Chains with unbounded variable lenght memory
Regeneration scheme
The model
Main result
So What??
The sparse chain is in fact a regeneration process
...
0q
...m − 2
1q
m−1
0q
0q
1q
0q
0q
m
m+1
m+2
m+3
m+4
...
q
Probabilistic context trees
Chains with variable lenght memory
Chains with unbounded variable lenght memory
Regeneration scheme
The model
Main result
So What??
The sparse chain is in fact a regeneration process
...
0q
...m − 2
1q
m−1
0q
0q
1q
0q
0q
m
m+1
m+2
m+3
m+4
...
1q
m+i
Probabilistic context trees
Chains with variable lenght memory
Chains with unbounded variable lenght memory
Regeneration scheme
The model
Main result
So What??
The sparse chain is in fact a regeneration process
We can decompose the realization in independant blocs,
identically distributed.
...
0q
...m − 2
1q
m−1
0q
0q
1q
0q
0q
m
m+1
m+2
m+3
m+4
...
1q
m+i
If {pk }k ∈N allows infinitely many 1’s, then we have a regeneration
scheme.
The model
Main result
So What??
Presentation of the problem
Main results
The problem of the talk
We want to extand this kind of result to more general trees.
The model
Main result
So What??
Presentation of the problem
Main results
The class of trees we are considering
q
A
A
AAq
q
A
A
AAq
q
A
A
AAq
q
A
A
AAq
q
A
A
We saw that sparse chains are
regeneration processes
τ = ∪k ≥0 10k
The model
Main result
So What??
Presentation of the problem
Main results
The class of trees we are considering
q
A
A
AAq
q
AA h(0)
A
A
AAq
q
A
A
A h(1)
A
AAq
q
A
A
A h(2)
A
AAq
q
A
A h(3)
A
AA
We saw that sparse chains are
regeneration processes
τ = ∪k ≥0 10k
Let’s generalize adding to each context
of the form 10k , a tree τ̂ k of height h(k ):
τ = ∪k ≥0 τ̂ k 10k
where h(k ) is a non decreasing unbounded function.
The model
Main result
So What??
Presentation of the problem
Main results
The class of trees we are considering
q
A
A
We saw that sparse chains are
AAq
q
regeneration processes
AA h(0)
A
τ = ∪k ≥0 10k
A
AAq
q
Let’s generalize adding to each context
A
AA h(1)
A
of the form 10k , a tree τ̂ k of height h(k ):
A
Aq
τ = ∪k ≥0 τ̂ k 10k
q
A
A
A h(2)
where h(k ) is a non decreasing unbounded function.
A
A
Aq
q
A
A h(3)
A
I We are searching for a condition on h(k ) to obtain
AA
a regeneration scheme for this new tree.
The model
Main result
So What??
Regeneration times
Theorem:
If (τ, p) verifies:
Presentation of the problem
Main results
The model
Main result
So What??
Presentation of the problem
Main results
Regeneration times
Theorem:
If (τ, p) verifies:
regularity: ∀ω ∈ τ, < p(1|ω) < 1 − The model
Main result
So What??
Presentation of the problem
Main results
Regeneration times
Theorem:
If (τ, p) verifies:
regularity: ∀ω ∈ τ, < p(1|ω) < 1 − k
1+δ
1
h(k ) ≤ 1−
, for some positive δ and k suficiently large,
The model
Main result
So What??
Presentation of the problem
Main results
Regeneration times
Theorem:
If (τ, p) verifies:
regularity: ∀ω ∈ τ, < p(1|ω) < 1 − k
1+δ
1
h(k ) ≤ 1−
, for some positive δ and k suficiently large,
then for all ω ∈ τ , there exists infinitely many instants
. . . < σ0ω ≤ 0 < σ1ω < σ2ω . . .
wich are regeneration times for the realisation of (τ, p) in relation
to ω.
The model
Main result
So What??
Presentation of the problem
Main results
Extension to more general trees
This result still hold for more more general form of trees, and I
explain this now on the blackboard.
The model
Main result
So What??
Presentation of the problem
Main results
References on regeneration scheme
F. Comets, R. Fernández, and P. A. Ferrari. Processes with
long memory: Regenerative construction and perfect
simulation. Ann. Appl. Probab. vol. 12 3:921-943, 2002.
I The regeneration scheme is not visible
G., Galves (in preparation)
I The regeneration scheme is visible
The model
Main result
So What??
Why is this an important result?
Perspectives
What does the result say?
The processes constructed with the class of trees we consider
have a regeneration scheme, this implies:
The model
Main result
So What??
Why is this an important result?
Perspectives
What does the result say?
The processes constructed with the class of trees we consider
have a regeneration scheme, this implies:
existence of a stationary measure for our process
The model
Main result
So What??
Why is this an important result?
Perspectives
What does the result say?
The processes constructed with the class of trees we consider
have a regeneration scheme, this implies:
existence of a stationary measure for our process
unicity of this measure
The model
Main result
So What??
Why is this an important result?
Perspectives
What does the result say?
The processes constructed with the class of trees we consider
have a regeneration scheme, this implies:
existence of a stationary measure for our process
unicity of this measure
perfect simulation
The model
Main result
So What??
Why is this an important result?
Perspectives
What does the result say?
The processes constructed with the class of trees we consider
have a regeneration scheme, this implies:
existence of a stationary measure for our process
unicity of this measure
perfect simulation
I without any condition on transition probabilities!!!
The model
Main result
So What??
Why is this an important result?
Perspectives
What does the result say?
The processes constructed with the class of trees we consider
have a regeneration scheme, this implies:
existence of a stationary measure for our process
unicity of this measure
perfect simulation
I without any condition on transition probabilities!!!
.....except the regularity condition, wich is rather restrictive, but
very current in the litterature...
The model
Main result
So What??
Why is this an important result?
Perspectives
What we conclude?
This result
I minimizes the importance of the notion of continuity rate in the
study of infinite order stochastic process,
The model
Main result
So What??
Why is this an important result?
Perspectives
What we conclude?
This result
I minimizes the importance of the notion of continuity rate in the
study of infinite order stochastic process,
I deals with a quite big class of trees since h(k ) can grow
exponencially fast,
The model
Main result
So What??
Why is this an important result?
Perspectives
What we conclude?
This result
I minimizes the importance of the notion of continuity rate in the
study of infinite order stochastic process,
I deals with a quite big class of trees since h(k ) can grow
exponencially fast,
I should give a new approach to the bootstrap for infinite
memory processes on finite alphabet.
The model
Main result
So What??
Why is this an important result?
Perspectives
What shall we do know
I 1. Estimation of this kind of trees,
The model
Main result
So What??
Why is this an important result?
Perspectives
What shall we do know
I 1. Estimation of this kind of trees,
I 2. bootstrap?.
The model
Main result
So What??
Why is this an important result?
Perspectives
END...