Math 456: Mathematical Modeling. Tuesday, March 6th, 2018

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1 Math 456: Mathematical Modeling Tuesday, March 6th, 2018

2 Markov Chains: Exit distributions and the Strong Markov Property Tuesday, March 6th, 2018

3 Last time 1. Weighted graphs. 2. Existence of stationary distributions (irreducible chains). 3. Informal definition of transient and recurrent states 4. Definition of stopping times. 5. Statement of the Strong Markov Property.

4 Today Today we will see: 1. Stopping times and Exit distributions 2. The proof of the Strong Markov Property. 3. Probability of return: properties and computations.

5 Notation reminder Probability with respect to initial state Hitting time Recall the notation from last time: Given a Markov Chain X 0, X 1, X 2,... we will write P x [A] := P[A X 0 = x] E x [Y ] := E[Y X 0 = x] Also recall: given a state y, we have the first hitting time T y := min{n 1 X n = y} which is a positive, integer valued random variable.

6 Warm up Consider the Gambler s ruin with a fair coin What is P x [(Winning)]? Using exit times, and writing A = {0, M} What is P x [X TA = M]?

7 Warm up From the total probability formula, it follows that (for x 0, M) P x [X TA = M] = 1 2 P x+1[x TA = M] P x 1[X TA = M] From here, with a little effort, one can show that P x [X TA = M] = x, x = 0, 1,... M

8 Warm up From the total probability formula, it follows that (for x 0, M) P x [X TA = M] = 1 2 P x+1[x TA = M] P x 1[X TA = M] From here, with a little effort, one can show that P x [X TA = M] = x, x = 0, 1,... M What if the coin is not a fair?

9 Exit probabilities We are given: An irreducible Markov chain with state space S A set A S. The state of the chain at the time it first reaches A is given by X TA We may think of A as representing exits to the state space, so X TA represents the location of the chain where we exit.

10 Exit probabilities To study the distribution of X TA, fix y A, and define G y (x) := P[X TA = y X 0 = x] this defines a function G y : S R. This is known as the Green function of the chain (with Dirichlet conditions on A).

11 Exit probabilities Observe: if x A, then T A = 0, so P[X TA = y X 0 = x] = P[X 0 = y X 0 = x] and this is 1 or 0 depending on whether x = y or not, so { 1 if x A, x = y G y (x) = 0 if x A, x y What can we say for x A?

12 Exit probabilities For x A, we divide and conquer. In this case, the chain will move to a new state X 1, and we can condition on this new state. The total probability formula + Markov property then says that P[X TA = y X 0 = x] = x S P[X TA = y X 1 = x ]P[X 1 = x X 0 = x]

13 Exit probabilities Now, the identity P[X TA = y X 0 = x] = x S P[X TA = y X 1 = x ]P[X 1 = x X 0 = x] can be rewritten as G y (x) = x S G y (x )p(x, x ) or, using that p(x, x ) = 1 x (G y (x ) G y (x))p(x, x ) = 0 x S \ A. x S

14 Exit probabilities Example Let us go back to the Gambler s ruin. Assume we have a biased coin (p q) Take y = M. Then for x {1,..., M 1} q(g M (x + 1) G M (x)) + p(g M (x 1) G M (x)) = 0 With some cleverness, we can write this as a recurrence relation: G M (x + 1) G M (x) = p q (G M(x) G M (x 1))

15 Exit probabilities Example Thus, we have for x {1,..., M 1} ( ) p x G M (x + 1) G M (x) = (G M (1) G M (0)) q ( ) p x = G M (1) q this being since G M (0) = 0. Now, we can add up all these identities, from x = 0 up to some arbitrary element of {1,..., M}, obtaining x 1 ( ) p i G M (x) G M (0) = G M (1) q i=0

16 Exit probabilities Example Therefore (again, since G M (0) = 0) x 1 ( ) p i G M (x) = G M (1) q i=0 What about G M (1)?. We use the elementary identity Substituting... x 1 ( ) p i = (p/q)x 1 q (p/q) 1 i=0

17 Exit probabilities Example...we obtain for every x {0,..., M} the formula G M (x) = (p/q)x 1 (p/q) 1 G M(1) Since G M (M) = 1, it follows that Therefore, 1 = (p/q)m 1 (p/q) 1 G M(1) G M (1) = (p/q) 1 (p/q) M 1

18 Exit probabilities Example Thus, we have for x {1,..., M 1} ( ) p x G M (x + 1) G M (x) = (G M (1) G M (0)) q ( ) p x = G M (1) q this being since G M (0) = 0. At the same time, we have G M (M) = 1, so 1 G M (M 1) = ( ) p M 1 G M (1) q

19 Exit probabilities Example In conclusion, for every x G M (x) = (p/q)x 1 (p/q) M+1 1

20 The Laplacian Operator To every function f : S R we associate a new one, f(x) := x S(f(x )) f(x))p(x, x ) and known as the Laplacian of f.

21 The Laplacian Operator Then, for a given set A S and y A, we have { Gy (x) = δ xy if x A, G y (x) = 0 if x S \ A. Solving the above amounts to a linear algebra problem involving the transition matrix.

22 The Laplacian Operator This means we can compute an exact formula for the exit distribution for a set A, without even running a single simulation. The flip side is, if one wants to approximate the solution to the above problem, one could run lots of simulations, and tally the exit points.

23 Math 456: Mathematical Modeling Thursday, March 8th, 2018

24 Markov Chains: The Strong Markov Property (...and a bit more about the Laplacian) Thursday, March 8th, 2018

25 Last time 1. A bit on calculus on weighted graphs 2. Definitions: gradient, divergence, and Laplacian. 3. The Laplacian as a measure of a function s smoothness. 4. Exit distributions. Example: Gambler s ruin. 5. Exit distributions and the associated graph Laplacian.

26 Today 1. The Laplacian and stationary distributions. 2. The Strong Markov property. 3. Probability of return: properties and computations.

27 vs. Stationary distributions Last time we defined the Laplacian on a weighted graph. For the graph associated to a Markov chain (G = S, weights given by transition probabilities), for a function f on the state space, its Laplacian is given by f(x) = y G(f(y) f(x))p(x, y)

28 vs. Stationary distributions Let π be a stationary distribution for the chain, and let f be a function solving f = 0. How do the two equations compare? We have π(x) = y G π(y)p(y, x) (f(y) f(x))p(x, y) = 0 y G

29 vs. Stationary distributions Let π be a stationary distribution for the chain, and let f be a function solving f = 0. How do the two equations compare?...rewriting the second one, π(x) = y G π(y)p(y, x) f(x) = y G f(y)p(x, y) where we have used that y p(x, y) = 1 for every x. What is the difference? We have p(x, y) in one, and p(y, x) in the other (and we are summing in y in both cases).

30 vs. Stationary distributions What does this look like in vector notation? Solving f = 0 in all of G means pf = f and as we know, π being stationary means p t π = π The two conditions are very similar, but they are not exactly the same in general! If π t = π, they of course coincide! In general, if p is doubly stochastic then they coincide.

31 Classification of states The Strong Markov Property Recall the Strong Markov Property from last time. Theorem (Strong Markov Property) Let X n denote a Markov Chain. If T is a stopping time and Y n := X T +n, then Y n is also a Markov chain. Moreover, the transition probability for Y n is the same as the one for X n.

32 Classification of states The Strong Markov Property Theorem (Strong Markov Property) Let X n denote a Markov Chain. If T is a stopping time and Y n := X T +n, then Y n is also a Markov chain. Moreover, the transition probability for Y n is the same as the one for X n. What does this mean? It means that given states α 1,..., α n, we have P(Y n+1 = α n+1 Y n = α n,... Y 1 = α 1 ) = p(α n, α n+1 ) where p(x, y) is the transition matrix for the original chain.

33 Classification of states The Strong Markov Property Theorem (Strong Markov Property) Let X n denote a Markov Chain. If T is a stopping time and Y n := X T +n, then Y n is also a Markov chain. Moreover, the transition probability for Y n is the same as the one for X n. What does this mean? Equivalently, it means that P(Y n+1 = α n+1, Y n = α n,... Y 1 = α 1 ) is equal to p(α n, α n+1 )p(α n 1, α n )... p(α 1, α 2 )P(Y 1 = α 1 ). (we shall skip the proof of this for now and revisit it in a couple of lectures)

34 Probability of return Now, armed with the Strong Markov Property, let us put it to use to analyze various questions about the long time behavior of a chain. Remember: we were trying to study the even that starting from x, we eventually reach some other state y at some (random) time T y, as well as the probability that starting from x, we eventually return to x itself at some later (random) time T x.

35 Probability of return Recall The first hitting time, with the convention that in the event that {X n y n}. T y = min{n 0 X n = y} T y =

36 Probability of return also recall (with the same convention in case the respective event is empty) T k x := min{n > T k 1 x X n = x} T k A := min{n > T k 1 A X n A} This is known as the time of the k-th return to x or the k-th hitting time for A.

37 Probability of return also recall (with the same convention in case the respective event is empty) T k x := min{n > T k 1 x X n = x} T k A := min{n > T k 1 A X n A} This is known as the time of the k-th return to x or the k-th hitting time for A. Evidently, each T k x is a stopping time.

38 Probability of return The probability of returning to x ρ xx := P x [T x < ] The probability of visiting y starting from x. ρ xy := P x [T y < ] Note: ρ xy > 0 if and only if x y.

39 Probability of return Using the Strong Markov Property, we can show the following: Lemma (See equation (1.4) in Durrett) P x [T k x < ] = ρ k xx

40 Probability of return Using the Strong Markov Property, we can show the following: Lemma (See equation (1.4) in Durrett) P x [T k x < ] = ρ k xx Note: In particular, if ρ xx < 1, the probability of returning to the state k times goes to zero exponentially fast as k 0.

41 Probability of return Proof. Observe that, P x [T k x < ] = P x [T k x <, T k 1 x < ] Therefore P x [T k x < ] k 1 = P x [X T k 1 x +n = x for some n, Tx < ] k 1 = P x [X T k 1 x +n = x for some n Tx < ]P[Tx k 1 < ]

42 Probability of return Proof. Let Y n := X T k 1 x +n. By the Strong Markov Property k 1 P x [X T k 1 x +n = x for some n Tx < ] = P[Y n = x for some n Tx k 1 < ] = P[X n = x for some n X 0 = x] = P[T x < X 0 = x] = P x [T x < ] Repeating this argument, we obtain the desired formula.

43 Probability of return Now, reviewing the definition of transient and recurrent from last time, we agree to say that 1. a state x will be called recurrent if ρ xx = a state x will be called transient if ρ xx < 1.

44 Probability of return The previous lemma and these definitions lead us to the following realizations (one trivial, two not so trivial) 1. If x is transient, then the probability of returning to x k times decreases geometrically with k. 2. If x is recurrent and k N, then with probability equal to 1, the chain will be on the state x at least k times. 3. Actually, if x is recurrent, then with probability 1, starting from x, the chain will revisit x infinitely manytimes. This last point is particularly delicate, and the proof uses an important tool in probability known as the Borel-Cantelli lemma.

Math Homework 5 Solutions

Math Homework 5 Solutions Math 45 - Homework 5 Solutions. Exercise.3., textbook. The stochastic matrix for the gambler problem has the following form, where the states are ordered as (,, 4, 6, 8, ): P = The corresponding diagram