1. The document presents a conjecture about the error probability of overestimating the true order k* when learning autoregressive moving average (ARMA) models from samples. 2. The conjecture states that if the estimated order k is greater than the true order k*, the error probability is equal to the probability that a chi-squared distributed random variable with k - k* degrees of freedom is greater than (k - k*)dn, where dn is related to the sample size n. 3. The author provides evidence that a sum of squared estimated ARMA coefficients could be chi-squared distributed, lending credibility to the conjecture.

1. .
.
A Conjecture on Strongly Consistent Learning
Joe Suzuki
Osaka University
July 7, 2009
Joe Suzuki (Osaka University) A Conjecture on Strongly Consistent Learning July 7, 2009 1 / 21

2. Outline
Outline
1 Stochastic Learning with Model Selection
2 Learning CP
3 Learning ARMA
4 Conjecture
5 Summary
Joe Suzuki (Osaka University) A Conjecture on Strongly Consistent Learning July 7, 2009 2 / 21

3. Stochastic Learning with Model Selection
Probability Space (Ω, F, µ)
Ω: the entire set
F: the set of events over Ω
µ : F → [0, 1]: a probability measure (µ(Ω) = 1)
Joe Suzuki (Osaka University) A Conjecture on Strongly Consistent Learning July 7, 2009 3 / 21

4. Stochastic Learning with Model Selection
Stochastic Learning
X : Ω → R: random variable
µX : the measure w.r.t. X
x1, · · · , xn ∈ X(Ω): n samples
Induction/Inference
.
µX −→ x1, · · · , xn (Random Number Generation)
x1, · · · , xn −→ µX (Stochastic Learning)
Joe Suzuki (Osaka University) A Conjecture on Strongly Consistent Learning July 7, 2009 4 / 21

5. Stochastic Learning with Model Selection
Two Problems with Model Selection
Conditional Probability for Y given X
.
.
µY |X
Identify the equivalence relation in X from samples.
x ∼ x′
⇐⇒ µ(Y = y|X = x) = µ(Y = y|X = x′
) for y ∈ Y (Ω)
ARMA for {Xn}∞
n=−∞
.
Xn +
∑k
j=1 λj Xn−j ∼ N(0, σ2) with {λ}k
j=1 (0 ≤ k < ∞)
Identify the true order k from samples.
This paper
compares CP and ARMA to find how close they are.
Joe Suzuki (Osaka University) A Conjecture on Strongly Consistent Learning July 7, 2009 5 / 21

6. Learning CP
CP
A ∈ F
G ⊆ F
CP of A given G (Radon-Nykodim)
.
.
∃G-measurable g : Ω → R s.t µ(A ∩ G) =
∫
G
g(ω)µ(dω) for G ∈ G.
CP µY |X
A := (Y = y), y ∈ Y (Ω)
G: generated by subsets of X.
Assumption
|Y (Ω)| < ∞
Joe Suzuki (Osaka University) A Conjecture on Strongly Consistent Learning July 7, 2009 6 / 21

7. Learning CP
Applications for CP
Stochastic Decision Trees, Stochastic Horn Clauseses Y |X, etc.
.
.
Find G from samples.
G = {G1, G2, G3, G4, G5}
"!
#
"!
#
"!
#
"!
#
"!
#
"!
#
"!
#
"!
#
¡
¡¡
e
ee
¡
¡¡
e
ee
¨¨¨¨¨¨
rrrrrr
Weather
Temp
G3
Wind
G1 G2 G4 G5
Clear
Cloud
Rain
≤ 75 > 75 Yes No
Joe Suzuki (Osaka University) A Conjecture on Strongly Consistent Learning July 7, 2009 7 / 21

8. Learning CP
Applications for CP (cont’d)
Finite Bayesian Networks Xi |Xj , j ∈ π(i)
.
.
Find π(i) ⊆ {1, · · · , i − 1} from samples.
X3
X4
X5
X2
X1
T
E E
d
d
ds
©
d
d
d‚
Joe Suzuki (Osaka University) A Conjecture on Strongly Consistent Learning July 7, 2009 8 / 21

9. Learning CP
Learning CP
zi := (xi , yi ) ∈ Z(Ω) := X(Ω) × Y (Ω)
From zn := (z1, · · · , zn) ∈ Zn(Ω), find a minimal G.
G∗: true
ˆGn: estimated
Assumption
.
.
|G∗| ∞
Two Types of Errors
G∗ ⊂ ˆGn (Over Estimation)
G∗ ̸⊆ ˆGn (Under Estimation)
Joe Suzuki (Osaka University) A Conjecture on Strongly Consistent Learning July 7, 2009 9 / 21

10. Learning CP
Example: Quinlan’s Q4.5
(a) G∗
¡¡ ee ¡¡ ee
¨¨¨¨
rrrr
Weather
Temp
3
Wind
1 2 4 5
Clear
Cloud
Rain
≤ 75 75 Yes No
(b) Overestimation
¡¡ ee ¡¡ ee
¨¨¨¨
rrrr
Weather
Temp
Wind
Wind
1 2 5 6
3 4
Clear
Cloud
Rain
≤ 75 75 Yes No
Yes No
¡¡ ee
(c) Underestimation
¡¡ ee
¨¨¨¨
rrrr
Weather
Temp
3
4
1 2
Clear
Cloud
Rain
≤ 75 75
(d) Underestimation
¨¨¨¨
rrrr
Weather
1
Wind
Wind
4 5
¡¡ ee
¡¡ ee
2 3
Clear
Cloud
Rain
Yes No
¡¡ ee
Joe Suzuki (Osaka University) A Conjecture on Strongly Consistent Learning July 7, 2009 10 / 21

11. Learning CP
Information Criteria for CP
Given zn ∈ Z(Ω), find G minimizing
I(G, zn
) := H(G, zn
) +
k(G)
2
dn
H(G, zn): Empirical Entropy (Fitness of zn to G)
k(G): the # of Parameters (Simplicity of G)
dn ≥ 0:
dn
n
→ 0
dn = log n BIC/MDL
dn = 2 AIC
Joe Suzuki (Osaka University) A Conjecture on Strongly Consistent Learning July 7, 2009 11 / 21

12. Learning CP
Consistency
Consistency ( ˆGn −→ G∗ (n → ∞))
.
.
Weak Consistency Probability Convergence (O(1) dn o(n))
Strong Consistency Almost Surely Convergence (MDL/BIC etc.)
AIC (dn = 2) is not consistent because {dn} is too small !
Problem
What is the minimum {dn} for Strong Consistency ?
Answer (Suzuki, 2006)
dn = 2 log log n
(the Law of Iterated Logarithms)
Joe Suzuki (Osaka University) A Conjecture on Strongly Consistent Learning July 7, 2009 12 / 21

13. Learning CP
Error Probability for CP
G∗ ⊂ G (Over Estimation)
.
.
µ{ω ∈ Ω|I(G, Zn
(ω)) I(G∗
, Zn
(ω))}
= µ{ω ∈ Ω|χ2
K(G)−K(G∗)(ω) (K(G) − K(G∗
))dn}
χ2
K(G)−K(G∗) ∼ χ2 of freedom K(G) − K(G∗)
G∗ ̸⊆ G (Under Estimation)
µ{ω ∈ Ω|I(G, Zn
(ω)) I(G∗
, Zn
(ω))
diminishes exponentially with n
Joe Suzuki (Osaka University) A Conjecture on Strongly Consistent Learning July 7, 2009 13 / 21

14. Learning CP
Applications for ARMA
Time Series Xi |Xi−k, · · · , Xi−1
.
.
Xi +
k∑
j=1
λj Xi−j ∼ N(0, σ2
)
Find k from samples.
Gaussian Bayesian Networks Xi |Xj , j ∈ π(i)
Xi +
∑
j∈π(i)
λj,i Xj ∼ N(0, σ2
i )
Find π(i) ⊆ {1, · · · , i − 1} from samples.
Joe Suzuki (Osaka University) A Conjecture on Strongly Consistent Learning July 7, 2009 14 / 21

15. Learning ARMA
Learning ARMA
k ≥ 0
{λj }k
j=1: λi ∈ R
σ2 ∈ R0
{Xi }∞
i=−∞: Xi +
k∑
j=1
λj Xi−j = ϵi ∼ N(0, σ2
)
Given xn := (x1, · · · , xn) ∈ X1(Ω) × · · · × Xn(Ω), estimate
k: known {λj }k
j=1, σ2
k: Unknown k as well as {λj }k
j=1, σ2
Joe Suzuki (Osaka University) A Conjecture on Strongly Consistent Learning July 7, 2009 15 / 21

16. Learning ARMA
Yule-Walker
If k is known, solve {ˆλj,k}k
j=1 and ˆσ2
k w.r.t.
¯x :=
1
n
n∑
i=1
xi
cj :=
1
n
n−j∑
i=1
(xi − ¯x)(xi+j − ¯x) , j = 0, · · · , k







−1 c1 c2 · · · ck
0 c0 c1 · · · ck−1
0 c1 c0 · · · ck−2
...
...
...
...
...
0 ck−1 ck−2 · · · c0














ˆσ2
k
ˆλ1,k
ˆλ2,k
...
ˆλk,k







=







−c0
−c1
−c2
...
−ck







Joe Suzuki (Osaka University) A Conjecture on Strongly Consistent Learning July 7, 2009 16 / 21

17. Learning ARMA
Information Criteria for ARMA
If k is unknown, given xn ∈ Xn(Ω), find k minimiziing
I(k, xn
) :=
1
2
log ˆσ2
k +
k
2
dn
ˆσ2
k: obtain using Yule-Walker
dn ≥ 0:
dn
n
→ 0
dn = log n BIC/MDL
dn = 2 AIC
dn = 2 log log n Hannan-Quinn (1979)
=⇒the minimum {dn} satisfying Strong Consistency
Suzuki (2006) was inspired by Hannan-Quinn (1979) !
Joe Suzuki (Osaka University) A Conjecture on Strongly Consistent Learning July 7, 2009 17 / 21

18. Learning ARMA
Error Probability for ARMA
k∗: true Order
k∗ k (Under Estimation)
.
µ{ω ∈ Ω|I(k, Xn
(ω)) I(k∗, Xn
(ω))}
diminishes exponentially with n
Joe Suzuki (Osaka University) A Conjecture on Strongly Consistent Learning July 7, 2009 18 / 21

19. Conjecture
Error Probability for ARMA (cont’d)
Conjecture: k∗ k (Over Estimation)
.
µ{ω ∈ Ω|I(k, Xn
(ω)) I(k∗, Xn
(ω))}
= µ{ω ∈ Ω|χ2
k−k∗
(ω) (k − k∗)dn}
χ2
k−k∗
∼ χ2 of freedom k − k∗
Joe Suzuki (Osaka University) A Conjecture on Strongly Consistent Learning July 7, 2009 19 / 21

20. Conjecture
In fact, ...
For k k∗ and large n (use ˆσ2
k = (1 − ˆλ2
k,k)σ2
k−1),
1
2
log ˆσ2
k +
k
2
dn
k∗
2
log σ2
k∗
+
k∗
2
dn
⇐⇒
k∑
j=k∗+1
log
ˆσ2
j
ˆσ2
j−1
(k − k∗)dn
⇐⇒
k∑
j=k∗+1
log(1 − ˆλ2
k,k) dn
⇐⇒
k∑
j=k∗+1
ˆλ2
k,k (k − k∗)dn
It is very likely that
∑k
j=k∗+1
ˆλ2
k,k ∼ χ2
k−k∗
although ˆλk,k ̸∼ N(0, 1).
Joe Suzuki (Osaka University) A Conjecture on Strongly Consistent Learning July 7, 2009 20 / 21

21. Summary
Summary
CP and ARMA are similar
.
.
1 Results in ARMA can be applied to CP
2 Results in Gausssian BN can be applied to finite BN.
The conjecture is likely enough ?
.
.
Answer: Perhaps. Several evidences.
{Zi }n
i=1: independent
Zi ∼ N(0, 1)
rankA = n − k, A2
= A =⇒ t
[Z1, · · · , Zn]A[Z1, · · · , Zn] ∼ χ2
n−k
Joe Suzuki (Osaka University) A Conjecture on Strongly Consistent Learning July 7, 2009 21 / 21