Galton-Watson trees and probabilities of local neighbourhoods of the root

Transcription

1 probabilities probabilities Courant Institute Mathematical Sciences New York University 14th Annual Northeast Probability Seminar November 20, 2015

2 The First Order (F.O.) World probabilities T rom Watson tree, Poisson(λ) fspring distribution.

3 The First Order (F.O.) World probabilities T rom Watson tree, Poisson(λ) fspring distribution. Constant Symbol: root Equality: x = y, Parent: π(y) = x (x is parent y, binary predicate), Variable Symbols x, y, z..., Boolean,,,,, etc, Quantification x, y over vertices only.

4 The First Order (F.O.) World probabilities T rom Watson tree, Poisson(λ) fspring distribution. Constant Symbol: root Equality: x = y, Parent: π(y) = x (x is parent y, binary predicate), Variable Symbols x, y, z..., Boolean,,,,, etc, Quantification x, y over vertices only. Example a node with exactly one child one grchild.

5 Ehrenfeucht games probabilities Definition 1 Trees T 1, T 2, roots R 1, R 2, # moves = k.

6 Ehrenfeucht games probabilities Definition 1 Trees T 1, T 2, roots R 1, R 2, # moves = k. 2 Spoiler picks any one tree a node from it. Duplicator chooses a node from the other tree.

7 Ehrenfeucht games probabilities Definition 1 Trees T 1, T 2, roots R 1, R 2, # moves = k. 2 Spoiler picks any one tree a node from it. Duplicator chooses a node from the other tree. 3 (x i, y i ) T 1 T 2, 1 i k, pairs nodes selected.

8 Ehrenfeucht games probabilities Definition 1 Trees T 1, T 2, roots R 1, R 2, # moves = k. 2 Spoiler picks any one tree a node from it. Duplicator chooses a node from the other tree. 3 (x i, y i ) T 1 T 2, 1 i k, pairs nodes selected. 4 Duplicator wins if x i = R 1 y i = R 2,

9 Ehrenfeucht games probabilities Definition 1 Trees T 1, T 2, roots R 1, R 2, # moves = k. 2 Spoiler picks any one tree a node from it. Duplicator chooses a node from the other tree. 3 (x i, y i ) T 1 T 2, 1 i k, pairs nodes selected. 4 Duplicator wins if x i = R 1 y i = R 2, π(x j ) = x i π(y j ) = y i,

10 Ehrenfeucht games probabilities Definition 1 Trees T 1, T 2, roots R 1, R 2, # moves = k. 2 Spoiler picks any one tree a node from it. Duplicator chooses a node from the other tree. 3 (x i, y i ) T 1 T 2, 1 i k, pairs nodes selected. 4 Duplicator wins if x i = R 1 y i = R 2, π(x j ) = x i π(y j ) = y i, x i = x j y i = y j.

11 Ehrenfeucht games probabilities Definition 1 Trees T 1, T 2, roots R 1, R 2, # moves = k. 2 Spoiler picks any one tree a node from it. Duplicator chooses a node from the other tree. 3 (x i, y i ) T 1 T 2, 1 i k, pairs nodes selected. 4 Duplicator wins if x i = R 1 y i = R 2, π(x j ) = x i π(y j ) = y i, x i = x j y i = y j. Theorem If Duplicator wins EHR[T 1, T 2, k] then T 1 = A T 2 = A for F.O. A depth k.

12 Ehrenfeucht value probabilities Definition T 1 k T 2 if Duplicator wins EHR[T 1, T 2, k].

13 Ehrenfeucht value probabilities Definition T 1 k T 2 if Duplicator wins EHR[T 1, T 2, k]. Theorem Fix k. Only finitely many equivalence classes.

14 Ehrenfeucht value probabilities Definition T 1 k T 2 if Duplicator wins EHR[T 1, T 2, k]. Theorem Fix k. Only finitely many equivalence classes. Definition Equivalence class T its Ehrenfeucht value.

15 Our results on almost sure theory for F.O. probabilities Theorem Fix k N. Fix a finite tree T 0. A[T 0 ] := { a subtree = T 0 in T }. Conditioned on the tree being infinte, A is almost surely true. Schema A = {A[T 0 ] : T 0 finite tree} gives almost sure theory for infinite trees.

16 Consequence previous result probabilities Corollary Fix k N. Condition on T being infinite. Ehrenfeucht value T depends on the local neighbourhood, radius 3 k+2. For all A = A[T 0 ], P[A] = P[A ] where A only depends on the local neighbourhood.

17 First generation probability conditioned on infiniteness probabilities 1 Only concerned with Γ 1 = {0, 1, 2,... k 1, ω}, ω indicates k.

18 First generation probability conditioned on infiniteness probabilities 1 Only concerned with Γ 1 = {0, 1, 2,... k 1, ω}, ω indicates k. 2 A i = {R has i children}, i = 1, 2,... k 1, ω.

19 First generation probability conditioned on infiniteness probabilities 1 Only concerned with Γ 1 = {0, 1, 2,... k 1, ω}, ω indicates k. 2 A i = {R has i children}, i = 1, 2,... k 1, ω. 3 B = {T is finite}, P[B] = p.

20 First generation probability conditioned on infiniteness probabilities 1 Only concerned with Γ 1 = {0, 1, 2,... k 1, ω}, ω indicates k. 2 A i = {R has i children}, i = 1, 2,... k 1, ω. 3 B = {T is finite}, P[B] = p. 4 For i {0, 1,... k 1} P[A i B c ] =P[A i ] P[A i B] =e λ λi i! (1 pi ).

21 First generation probability conditioned on infiniteness probabilities 1 Only concerned with Γ 1 = {0, 1, 2,... k 1, ω}, ω indicates k. 2 A i = {R has i children}, i = 1, 2,... k 1, ω. 3 B = {T is finite}, P[B] = p. 4 For i {0, 1,... k 1} P[A i B c ] =P[A i ] P[A i B] =e λ λi i! (1 pi ). 5 For ω children: P[A ω B c ] =P[A ω ] P[A ω B] = e λ λj j! [1 pj ]. j=k

22 Some definitions probabilities Definition

23 Some definitions probabilities Definition 1 For 0 i k 1, P i (x) = P[Poi(x) = i] = e x x i i!.

24 Some definitions probabilities Definition 1 For 0 i k 1, P i (x) = P[Poi(x) = i] = e x x i i!. 2 For k, P ω (x) = P[Poi(x) k] = e x j=k x j j!.

25 Some definitions probabilities Definition 1 For 0 i k 1, P i (x) = P[Poi(x) = i] = e x x i i!. 2 For k, P ω (x) = P[Poi(x) k] = e x j=k x j j!. 3 (i + 1)-generation neighbourhood Γ i+1 = {g : Γ i Γ 1 }.

26 Some definitions probabilities Definition 1 For 0 i k 1, P i (x) = P[Poi(x) = i] = e x x i i!. 2 For k, P ω (x) = P[Poi(x) k] = e x j=k x j j!. 3 (i + 1)-generation neighbourhood Γ i+1 = {g : Γ i Γ 1 }. 4 For τ Γ i, P τ (x) = P[i generation neighbourhood τ for Poi(x)].

27 Example + Illustration probabilities

28 Example + Illustration probabilities

29 Example + Illustration probabilities k = 3, g(0) = 0, g(1) = ω, g(2) = 1, g(ω) = 1.

30 Example + Illustration probabilities k = 3, g(0) = 0, g(1) = ω, g(2) = 1, g(ω) = 1. 2 P σ (x) = P g(i) (xp i (x)) P g(ω) (xp ω (x)) i=0 =e x (xp 1 (x)) j j! (xp 2(x)) (xp ω (x)) j=3

31 Recursive computation probabilities probabilities

32 Recursive computation probabilities probabilities Theorem P σ (x) = τ Γ i P g(τ) (xp τ (x)) σ = g Γ i+1.

33 Recursive computation probabilities probabilities Theorem P σ (x) = P g(τ) (xp τ (x)) σ = g Γ i+1. τ Γ i Theorem B = {T is finite}, P[B] = p.

34 Recursive computation probabilities probabilities Theorem P σ (x) = P g(τ) (xp τ (x)) σ = g Γ i+1. τ Γ i Theorem B = {T is finite}, P[B] = p. By duality for, P[{i generation neighbourhood = σ} B] = P σ (pλ).

35 Recursive computation probabilities probabilities Theorem P σ (x) = P g(τ) (xp τ (x)) σ = g Γ i+1. τ Γ i Theorem B = {T is finite}, P[B] = p. By duality for, P[{i generation neighbourhood = σ} B] = P σ (pλ). If P [σ] = P[{i generation neighbourhood = σ} B c ], then P [σ] = P σ(λ) p P σ (pλ). 1 p

36 The probabilities are nice functions probabilities Remark For all i σ Γ i, P σ (x) nice function. Consists polynomials in p, x, e x, base e exponentiation.

37 The probabilities are nice functions probabilities Remark For all i σ Γ i, P σ (x) nice function. Consists polynomials in p, x, e x, base e exponentiation. Example 1 A := {Root has no child with no child}, k 1.

38 The probabilities are nice functions probabilities Remark For all i σ Γ i, P σ (x) nice function. Consists polynomials in p, x, e x, base e exponentiation. Example 1 A := {Root has no child with no child}, k 1. 2 A = {g(0) = 0}.

39 The probabilities are nice functions probabilities Remark For all i σ Γ i, P σ (x) nice function. Consists polynomials in p, x, e x, base e exponentiation. Example 1 A := {Root has no child with no child}, k 1. 2 A = {g(0) = 0}. 3 P[A] = P 0 [λp 0 (λ)] = e λp 0(λ) = e λe λ.

40 probabilities Thank you.