Palm Theory, Random Measures and Stein Couplings Based on joint - PowerPoint PPT Presentation

Palm Theory, Random Measures and Stein Couplings Based on joint work with Adrian R¨ ollin and Aihua Xia Louis H. Y. Chen National University of Singapore Workshop on Stein’s Method in Machine Learning and Statistics International Conference on Machine Learning 2019 15 June 2019 Long Beach, California L. H. Y. Chen (NUS) Random Mesaures and Stein Couplings Stein’s Method in ML 1 / 33

Outline Stein’s Method Palm Theory A General Normal Apprximation Theorem Application to Random Measures and Stochastic Geometry Application to Stein Couplings L. H. Y. Chen (NUS) Random Mesaures and Stein Couplings Stein’s Method in ML 2 / 33

Stein’s Lemma Lemma 1 (Stein, 1960’s) Let W be a random variable and let σ 2 > 0 . Then W ∼ N (0 , σ 2 ) if and only if E { σ 2 f ′ ( W ) − Wf ( W ) } = 0 for all bounded absolutely continuous functions f with bounded f ′ . Proof. (i) Only if : By integration by parts. (ii) If : It suffices to consider the case σ 2 = 1 . Let h ∈ C B and let f h be the unique C 1 B solution of f ′ ( w ) − wf ( w ) = h ( w ) − E h ( Z ) where Z ∼ N (0 , 1) . Then E h ( W ) − E h ( Z ) = E { f ′ h ( W ) − Wf h ( W ) } = 0 . This implies W ∼ N (0 , 1) . L. H. Y. Chen (NUS) Random Mesaures and Stein Couplings Stein’s Method in ML 3 / 33

Stein’s Method for Normal Approximation Stein (1972), Proc. Sixth Berkeley Symposium Let W be such that E W = 0 and Var( W ) = 1 . If W is not distributed N (0 , 1) , then E { f ′ ( W ) − Wf ( W ) } � = 0 . How to quantify the discrepancy between L ( W ) and N (0 , 1) ? Choose f to be a bounded solution, f h , of f ′ ( w ) − wf ( w ) = h ( w ) − E h ( Z ) (Stein equation) , where Z ∼ N (01 , ) and h ∈ G , a suitable separating class of functions. Then E h ( W ) − E h ( Z ) = E { f ′ h ( W ) − Wf h ( W ) } . The distance induced by G is defined as d G ( W, Z ) := sup | E h ( W ) − E h ( Z ) | h ∈G | E { f ′ h ( W ) − Wf h ( W ) }| . = sup h ∈G L. H. Y. Chen (NUS) Random Mesaures and Stein Couplings Stein’s Method in ML 4 / 33

Separating Classes of Functions These separating classes of functions defined are of interest. G W { h ; | h ( u ) − h ( v ) | ≤ | u − v | , u, v ∈ R } , := G K := { h ; h ( w ) = 1 for w ≤ x and = 0 for w > x, x ∈ R } , G TV := { h ; h ( w ) = I ( w ∈ A ) , A is a Borel subset of R } . The distances induced by these three separating classes are respectively called the Wasserstein distance, the Kolmogorov distance, and the total variation distance. It is customary to denote d G W , d G K and d G T V respectively by d W , d K and d TV . L. H. Y. Chen (NUS) Random Mesaures and Stein Couplings Stein’s Method in ML 5 / 33

Approximation by Other Distributions Stein’s ideas are very general and applicable to approximations by other distributions. Examples are Poisson (Chen (1975)), binomial (Stein (1986)), compound Poisson (Barbour, Chen and Loh (1992)), multivariate normal (Barbour (1990), G¨ otze (1991)), Poisson process (Barbour and Brown (1992)), multinomial (Loh (1992)), exponential (Chatterjee, Fulman and R¨ ollin (2011)). Stein expanded his method into a definitive theory in the monograph, Approximate Computation of Expectations , IMS Lecture Notes Monogr. Ser. 7. Inst. Math. Statist. (1986). L. H. Y. Chen (NUS) Random Mesaures and Stein Couplings Stein’s Method in ML 6 / 33

Palm Measures Let Γ be a locally compact separable metric space. Let Ξ be a random measure on Γ with finite intensity measure Λ , that is, for every Borel subset A ⊂ Γ , Λ( A ) = E Ξ( A ) and Λ(Γ) < ∞ . For α ∈ Γ , there exists a random measure Ξ α such that �� � �� � E f ( α, Ξ)Ξ( dα ) = E f ( α, Ξ α )Λ( dα ) Γ Γ for f ( · , · ) ≥ 0 (or for real-valued f ( · , · ) for which the expectations exist). (Campbell equation) Ξ α is called the Palm measure associated with Ξ at α . L. H. Y. Chen (NUS) Random Mesaures and Stein Couplings Stein’s Method in ML 7 / 33

Palm Measures If Ξ is a simple point process, the distribution of Ξ α can be interpreted as the conditional distribution of Ξ given that a point of Ξ has occurred at α . If Ξ is a Poisson point process, then L (Ξ α ) = L (Ξ + δ α ) a.e. Λ . If Λ( { α } ) > 0 , then Ξ( { α } ) is a non-negative random variable with positive mean and Ξ α ( { α } ) is a Ξ( { α } ) -size-biased random variable. Let Y = Ξ( { α } ) , Y s = Ξ α ( { α } ) and let µ = E Ξ( { α } ) . The Campbell equation gives E Y f ( Y ) = µ E f ( Y s ) for all f for which the expectations exist. This implies that ν s ( dy ) = y µν ( dy ) . In general, we may interpret the Palm measure as a “size-biased random measure”. L. H. Y. Chen (NUS) Random Mesaures and Stein Couplings Stein’s Method in ML 8 / 33

Normal approximation for random measures It has applications to stochastic geometry as many problems therein can be formulated in terms of random measures. By taking f to be absolutely continuous from R to R such that f ′ is bounded, the Campbell equation implies � E | Ξ | f ( | Ξ | ) = E f ( | Ξ α | )Λ( dα ) , where | Ξ | = Ξ(Γ) . Γ Assume that Ξ and Ξ α , α ∈ Γ , are defined on the same probability space. L. H. Y. Chen (NUS) Random Mesaures and Stein Couplings Stein’s Method in ML 9 / 33

Normal approximation for random measures Let B 2 = Var( | Ξ | ) , and define W = | Ξ | − λ W α = | Ξ α | − λ , , B B and let ∆ α = W α − W. Then 1 � [ f ( W α ) − f ( W )]Λ( dα ) E Wf ( W ) = B E Γ � ∞ f ′ ( W + t ) ˆ = K ( t ) dt E −∞ where K ( t ) = 1 � ˆ [ I (∆ α > t > 0) − I (∆ α < t ≤ 0)]Λ( dα ) . B Γ L. H. Y. Chen (NUS) Random Mesaures and Stein Couplings Stein’s Method in ML 10 / 33

Normal approximation for random measures Let f x be the bounded unique solution of the Stein equation f ′ ( w ) − f ( w ) = I ( w ≤ x ) − P ( Z ≤ x ) , Z ∼ N (0 , 1) . Then | P ( W ≤ x ) − P ( Z ≤ x ) | d K ( W, Z ) = sup x ∈ R | E { f ′ x ( W ) − Wf x ( W ) }| = sup x ∈ R � ∞ x ( W + t ) ˆ � � E { f ′ f ′ � x ( W ) − K ( t ) dt } = sup � . x ∈ R −∞ L. H. Y. Chen (NUS) Random Mesaures and Stein Couplings Stein’s Method in ML 11 / 33

A General Theorem Theorem 2 Let W be such that E W = 0 and Var( W ) = 1 . Suppose there exists a random function ˆ K ( t ) such that � ∞ f ′ ( W + t ) ˆ E Wf ( W ) = E K ( t ) dt −∞ for all absolutely continuous functions f with bounded f ′ . Let ˆ K ( t ) = ˆ K in ( t ) + ˆ K out ( t ) where ˆ K in ( t ) = 0 for | t | > 1 . Define K ( t ) = E ˆ K ( t ) , K in ( t ) = E ˆ K in ( t ) , and K out ( t ) = E ˆ K out ( t ) . Then d K ( W, Z ) ≤ 2 r 1 + 11 r 2 + 5 r 3 + 10 r 4 + 7 r 5 , where Z ∼ N (0 , 1) . L. H. Y. Chen (NUS) Random Mesaures and Stein Couplings Stein’s Method in ML 12 / 33

A General Theorem In Theorem 2, � 2 � 1 � 2 �� ( ˆ K in ( t ) − K in ( t )) dt r 1 = , E | t |≤ 1 � | tK in ( t ) | dt, r 2 = | t |≤ 1 � ∞ | ˆ K out ( t ) | dt, r 3 = E −∞ � ( ˆ K in ( t ) − K in ( t )) 2 dt, r 4 = E | t |≤ 1 � 1 � � 2 | t | ( ˆ K in ( t ) − K in ( t )) 2 dt r 5 = E . | t |≤ 1 L. H. Y. Chen (NUS) Random Mesaures and Stein Couplings Stein’s Method in ML 13 / 33

Applications Completely random measures. A random measure Ξ on Γ is completely random if Ξ( A 1 ) , · · · , Ξ( A k ) are independent whenever A 1 , · · · , A k ∈ B (Γ) are pairwise disjoint (Kingman, Pacific J. Math. 1967). Excursion random measures Ξ( dt ) = I (( t, X t ) ∈ E ) dt, E ∈ B ([0 , T ] × S ) . Number of maximal points of a Poisson point process in a region. Ginibre-Voronoi tessellation. Stein couplings: ( G, W ′ , W ) such that E [ Gf ( W ′ ) − Gf ( W )] = E Wf ( W ) for absolutely continuous f with f ( x ) = O (1 + | x | ) . Stein couplings include local dependence, exchangeable pairs, size-bias couplings and others. L. H. Y. Chen (NUS) Random Mesaures and Stein Couplings Stein’s Method in ML 14 / 33

Excursion Random Measures S a metric space. { X t : 0 ≤ t ≤ T } an l -dependent S -valued random process, that is, { X t : 0 ≤ t ≤ a } and { X t : b ≤ t ≤ T } are independent if b − a > l . Define the excursion random measure Ξ( dt ) = I (( t, X t ) ∈ E ) dt, E ∈ B ([0 , T ] × S ) Theorem 3 Let µ = E Ξ([0 , T ]) , B 2 = Var( | Ξ | ) and W = | Ξ | − µ . We have B � l 3 / 2 µ 1 / 2 + l 2 µ � d K ( W, Z ) = O , B 2 B 3 where Z ∼ N (0 , 1) . L. H. Y. Chen (NUS) Random Mesaures and Stein Couplings Stein’s Method in ML 15 / 33

Ginibre-Voronoi Tessellation Points are randomly scattered on a plane. Cells are formed by drawing lines symmetrically between every two adjacent points. Voronoi diagram (20 points and their Voronoi cells ) A lot of interest in the literature in the total edge length of the cells (or of the tessellation). L. H. Y. Chen (NUS) Random Mesaures and Stein Couplings Stein’s Method in ML 16 / 33