Probabilistic Performance Analysis of Fault Diagnosis Schemes

where q ∈ (0,1) and
⎧
⎨0.5 if k < 10
p k =
⎩1 otherwise.
Then, the corresponding adjacency matrix is
[ ]
0 1
A = .
1 0
As in Example 4.16, the graph (Θ, A) contains cycles, so Theorem 4.12 does not apply. However, in
this simple case, we can see that the Markov chain θ ∼ ( )
Θ,{Π k },π 0 is tractable. Indeed, consider
a path ϑ 0:l ∈ Θ l+1 , where l ≥ 10. Split the path into two parts, ϑ 0:9 and ϑ 10:l , and let ˆθ be a
Markov chain, such that ˆθ k = θ k−10 , for all k ≥ 0. The first part ϑ 0:9 can take 2 10 different values,
while the second part ϑ 10:l can be considered as a path of the shifted Markov chain ˆθ. Since ˆθ
has the same time-homogeneous distribution as the tractable Markov chain in Example 4.15, the
number of possible paths of the original Markov chain θ must be polynomial.
Before proving Theorems 4.11 and 4.12, we establish a series of lemmas, each of which
is useful in its own right. Then, these lemmas are used to formulate succinct proofs of the
main results.
Supporting Lemmas
The first two lemmas state the notion of tractability in terms of the structure of the transition
probability matrices.
Lemma 4.19. Let θ ∼ ( Θ,{Π k },π 0
)
be a Markov chain, such that Πk is upper-triangular, for
all k. Then, every possible path ϑ 0:l ∈ Θ l+1 satisfies the inequalities
ϑ 0 ≤ ϑ 1 ≤ ··· ≤ ϑ l−1 ≤ ϑ l .
Proof. Let ϑ 0:l ∈ Θ l+1 be a possible path. Then, the inequality
Π l−1 (ϑ l−1 ,ϑ l ) Π l−2 (ϑ l−2 ,ϑ l−1 )··· Π 0 (ϑ 0 ,ϑ 1 ) π 0 (ϑ 0 ) = P(θ 0:l = ϑ 0:l ) > 0.
implies that Π i (ϑ i−1 ,ϑ i ) > 0, for i = 1,2,...,l. Since each Π i is upper triangular, it must be
that ϑ i−1 ≤ ϑ i , for i = 1,2,...,l.
Lemma 4.20. Let θ ∼ ( Θ,{Π k },π 0
)
be a Markov chain, such that Θ = {0,1,...m} and Πk is
upper-triangular, for all k. Then, the number of possible paths ϑ 0:l ∈ Θ l+1 is
l m
m! +O(lm−1 ).
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