Outline Examples and comparisons of rigorous error bounds for MCMC estimates Part I Wojciech Niemiro Nicolaus Copernicus University, Toru´ n, Poland & University of Warsaw, Poland joint work with Bła˙ zej Miasojedow University of Warsaw, Poland & LCTI Telecom ParisTech, France and Agnieszka Perduta-Pilarska Nicolaus Copernicus University, Toru´ n, Poland MCQMC Sydney , February 2012 W. Niemiro, Toru´ n, Poland Examples of error bounds for MCMC
Outline Outline Introduction Computing integrals via MCMC Rigorous accuracy bounds Non-asymptotic bounds Basic bound on MSE Explicit bounds for geometrically ergodic chains Explicit bounds for uniformly ergodic chains Confidence estimation Median of Averages Examples Hierarchical Poisson/Gamma model Hierarchical model of variance components W. Niemiro, Toru´ n, Poland Examples of error bounds for MCMC
Introduction Non-asymptotic bounds Computing integrals via MCMC Confidence estimation Rigorous accuracy bounds Examples Computing integrals via MCMC We are to compute � E π f = π ( f ) = f ( x ) π ( d x ) , X where ◮ X – state space (high dimensional). ◮ π – probability distribution on X (posterior in a Bayesian model). Markov chain X 0 , X 1 , . . . , X n , . . . P ( X n ∈ · ) → π ( · ) ( n → ∞ ) . MCMC estimator n − 1 f n = 1 � ¯ f ( X i ) → E π f ( n → ∞ ) . n i = 0 W. Niemiro, Toru´ n, Poland Examples of error bounds for MCMC
Introduction Non-asymptotic bounds Computing integrals via MCMC Confidence estimation Rigorous accuracy bounds Examples Accuracy bounds Rate of convergence of probability distributions: � P ( X t ∈ · ) − π ( · ) � ≤ ? considered in many papers ... Rate of convergence of sample averages. Mean square error: √ � E (¯ f n − E π f ) 2 ≤ ? MSE = Confidence bounds: P ( | ¯ f n − E π f | > ε ) ≤ ? W. Niemiro, Toru´ n, Poland Examples of error bounds for MCMC
Introduction Non-asymptotic bounds Computing integrals via MCMC Confidence estimation Rigorous accuracy bounds Examples Convergence of probability distributions: Gibbs Sampler in a hierarchical Bayesian model of variance components. Trajectory of one of 1003 coordinates. Time to reach stationarity: ∼ 5-10 steps (visual asessment). W. Niemiro, Toru´ n, Poland Examples of error bounds for MCMC
Introduction Non-asymptotic bounds Computing integrals via MCMC Confidence estimation Rigorous accuracy bounds Examples Convergence of sample averages: Gibbs Sampler in a hierarchical Bayesian model of variance components. The same computation. A graph of cumulative averages. Time to stabilize estimates: at least ∼ 1000 steps (visual asessment). W. Niemiro, Toru´ n, Poland Examples of error bounds for MCMC
Introduction Non-asymptotic bounds Computing integrals via MCMC Confidence estimation Rigorous accuracy bounds Examples Why insist on non-asymptotic bounds? A simulation of an ergodic chain (by courtesy of K. Łatuszy´ nski). W. Niemiro, Toru´ n, Poland Examples of error bounds for MCMC
Introduction Non-asymptotic bounds Computing integrals via MCMC Confidence estimation Rigorous accuracy bounds Examples Why insist on non-asymptotic bounds? A longer version of the same simulation. W. Niemiro, Toru´ n, Poland Examples of error bounds for MCMC
Introduction Basic bound on MSE Non-asymptotic bounds Explicit bounds for geometrically ergodic chains Confidence estimation Explicit bounds for uniformly ergodic chains Examples Assumptions Notation for the transition kernel: P ( x , A ) = P ( X n ∈ A | X n − 1 = x ) . ASSUMPTIONS ◮ Stationary distribution. There exists a probability distribution π on X such that π P = π and P is π -irreducible. ◮ Small set (minorization condition). There exist J ⊆ X with π ( J ) > 0 , a probability measure ν and β > 0 such that P ( x , · ) ≥ β 1 ( x ∈ J ) ν ( · ) . The minorization condition allows us to use an approach based on regeneration due to Nummelin, Athreya & Ney. W. Niemiro, Toru´ n, Poland Examples of error bounds for MCMC
Introduction Basic bound on MSE Non-asymptotic bounds Explicit bounds for geometrically ergodic chains Confidence estimation Explicit bounds for uniformly ergodic chains Examples Regeneration Under the minorization condition we can partition a trajectory of the chain into independent “ tours ” (excursions) of random length: X 0 , . . . , X T 1 − 1 , X T 1 , . . . , X T 2 − 1 , X T 2 , . . . , X T 3 − 1 , . . . � �� � � �� � � �� � T 1 T 2 − T 1 T 3 − T 2 ↑ ↑ R R R ⇐ ⇒ Regeneration ⇐ ⇒ X n ∼ ν ( · ) . The excursions are i.i.d. except for the first one, which can have a different distribution). W. Niemiro, Toru´ n, Poland Examples of error bounds for MCMC
Introduction Basic bound on MSE Non-asymptotic bounds Explicit bounds for geometrically ergodic chains Confidence estimation Explicit bounds for uniformly ergodic chains Examples Basic bound for the Mean Square Error THEOREM (Łatuszy´ nski, Miasojedow & Niemiro, 2011) Under Assumptions Stationary distribution and Small set , � � � f n − E π f ) 2 ≤ σ as 1 + C 0 + C 1 n + C 2 E ξ (¯ √ n n , n where σ 2 as = σ 2 as ( P , f ) is the “asymptotic variance” in the CLT: √ n (¯ f n − E π f ) → d N ( 0 , 1 ) ( n → ∞ ) . σ as The leading term in our bound is optimal, because � f n − E π f ) 2 ∼ σ as E ξ (¯ √ n ( n → ∞ ) . Constants C 0 , C 1 , C 2 are expressed in terms of “regenerative tours”. W. Niemiro, Toru´ n, Poland Examples of error bounds for MCMC
Introduction Basic bound on MSE Non-asymptotic bounds Explicit bounds for geometrically ergodic chains Confidence estimation Explicit bounds for uniformly ergodic chains Examples Geometric drift condition Explicit bounds for the MSE for geometrically ergodic chains. Notation for the transition operator: � PV ( x ) = P ( x , d y ) V ( y ) = E ( V ( X n ) | X n − 1 = x ) . X ASSUMPTION ◮ Geometric Drift. There exist a function V : X → [ 1 , ∞ [ , constants λ < 1 and K < ∞ such that � λ V ( x ) for x �∈ J , PV ( x ) ≤ K for x ∈ J . J is a small set which appears in the minorization condition. W. Niemiro, Toru´ n, Poland Examples of error bounds for MCMC
Introduction Basic bound on MSE Non-asymptotic bounds Explicit bounds for geometrically ergodic chains Confidence estimation Explicit bounds for uniformly ergodic chains Examples Explicit bounds under drift condition Bounds on constants σ as , C 0 , C 1 , C 2 (Łatuszy´ nski, Miasojedow & Niemiro, 2011). Under Assumptions Small Set and Geometric Drift , if f is such that � f − π ( f ) � V 1 / 2 = sup x | f ( x ) − π ( f ) | / V ( x ) 1 / 2 ≤ 1 then 1 − λ 1 / 2 π ( V ) + 2 ( K 1 / 2 − λ 1 / 2 − β ) as ≤ 1 + λ 1 / 2 σ 2 π ( V 1 / 2 ) , β ( 1 − λ 1 / 2 ) 1 − λ 1 / 2 π ( V 1 / 2 ) + K 1 / 2 − λ 1 / 2 − β λ 1 / 2 C 0 ≤ , β ( 1 − λ 1 / 2 ) ( 1 − λ 1 / 2 ) 2 ξ ( V ) + 2 ( K 1 / 2 − λ 1 / 2 − β ) 1 C 2 ξ ( V 1 / 2 ) 1 ≤ β ( 1 − λ 1 / 2 ) 2 + β ( K − λ − β ) + 2 ( K 1 / 2 − λ 1 / 2 − β ) 2 , β 2 ( 1 − λ 1 / 2 ) 2 C 2 2 ≤ analogous expression with ξ replaced by ξ P n . W. Niemiro, Toru´ n, Poland Examples of error bounds for MCMC
Introduction Basic bound on MSE Non-asymptotic bounds Explicit bounds for geometrically ergodic chains Confidence estimation Explicit bounds for uniformly ergodic chains Examples Explicit bounds under drift condition Further bounds on π ( V 1 / 2 ) , π ( V ) , ξ P n ( V 1 / 2 ) , ξ P n ( V ) , � f − π ( f ) � V 1 / 2 . Under Assumptions Small Set and Geometric Drift , π ( V 1 / 2 ) ≤ π ( J ) K 1 / 2 − λ 1 / 2 ≤ K 1 / 2 − λ 1 / 2 , 1 − λ 1 / 2 1 − λ 1 / 2 π ( V ) ≤ π ( J ) K − λ 1 − λ ≤ K − λ 1 − λ , K 1 / 2 K 1 / 2 ξ ( V 1 / 2 ) ≤ ξ P n ( V 1 / 2 ) ≤ if then 1 − λ 1 / 2 , 1 − λ 1 / 2 K K ξ P n ( V ) ≤ if ξ ( V ) ≤ then 1 − λ, 1 − λ � π ( J )( K 1 / 2 − λ 1 / 2 ) � � f − π ( f ) � V 1 / 2 ≤ � f � V 1 / 2 1 + ( 1 − λ 1 / 2 ) inf x ∈X V 1 / 2 ( x ) � 1 + K 1 / 2 − λ 1 / 2 � ≤ � f � V 1 / 2 . 1 − λ 1 / 2 W. Niemiro, Toru´ n, Poland Examples of error bounds for MCMC
Introduction Basic bound on MSE Non-asymptotic bounds Explicit bounds for geometrically ergodic chains Confidence estimation Explicit bounds for uniformly ergodic chains Examples Uniformly ergodic chains and perfect sampling If Assumption Small Set is satisfied with J = X then X T 2 − 1 , X T 3 − 1 , . . . are i.i.d. and have exactly π -distribution. For a perfect sampler r f r = 1 � ˇ X T k + 1 − 1 , r k = 1 we obviously have the standard i.i.d. bound � f r − E π f ) 2 ≤ D π f E (ˇ √ r , where D π f = √ Var π f . Remember that to generate r perfect samples we have to run the chain r /β steps (on the average). W. Niemiro, Toru´ n, Poland Examples of error bounds for MCMC
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