NEO: Non Equilibrium Sampling on the Orbits of a Deterministic Transform

Sampling from a complex distribution π and approximating its intractable normalizing constant Z are challenging problems . In this paper , a novel family of importance samplers ( IS ) and Markov chain Monte Carlo ( MCMC ) samplers is derived . Given an invertible map T , these schemes combine ( with weights ) elements from the forward and backward Orbits through points sampled from a proposal distribution ρ . The map T does not leave the target π invariant , hence the name NEO , standing for Non-Equilibrium Orbits . NEO-IS provides unbiased estimators of the normalizing constant and self-normalized IS estimators of expectations under π while NEO-MCMC combines multiple NEO-IS estimates of the normalizing constant and an iterated sampling-importance resampling mechanism to sample from π . For T chosen as a discrete-time integrator of a conformal Hamiltonian system , NEO-IS achieves state-of-the art performance on difficult benchmarks and NEO-MCMC is able to explore highly multimodal targets . Additionally , we provide detailed theoretical results for both methods . In particular , we show that NEO-MCMC is uniformly geometrically ergodic and establish explicit mixing time estimates under mild conditions . 35th Conference on Neural Information Processing Systems ( NeurIPS 2021 ) . 1 Introduction . Consider a target distribution of the form π ( x ) ∝ ρ ( x ) L ( x ) where ρ is a probability density function ( pdf ) on Rd and L is a nonnegative function . Typically , in a Bayesian setting , π is a posterior distribution associated with a prior distribution ρ and a likelihood function L. Another situation of interest is generative modeling where π is the distribution implicitly defined by a Generative Adversarial Networks ( GAN ) discriminator-generator pair where ρ is the distribution of the generator and L is derived from the discriminator [ 31 , 6 ] . In a Variational Auto Encoder ( VAE ) context [ 14 , 5 ] , π could be the true posterior distribution , ρ the approximate posterior distribution output by the encoder and L an importance weight between the true posterior and approximate posterior distributions . We are interested in this paper in sampling from π and approximating its intractable normalizing constant Z = ∫ ρ ( x ) L ( x ) dx . These problems arise in many applications in statistics , molecular dynamics or machine learning , and remain challenging . Many approaches to compute normalizing constants are based on Importance Sampling ( IS ) - see [ 1 , 2 ] and the references therein - and its variations , among others , Annealed Importance Sampling ( AIS ) [ 19 , 34 , 10 ] and Sequential Monte Carlo ( SMC ) [ 9 ] . More recently , Neural IS has also become very popular in machine learning ; see e.g . [ 11 , 18 , 21 , 22 , 32 , 33 ] . Neural IS is an adaptive IS which relies on an importance function obtained by applying a normalizing flow to a reference distribution . The parameters of this normalizing flow are chosen by minimizing a divergence between the proposal and the target ( such as the Kullback–Leibler [ 18 ] or the χ2-divergence [ 1 ] ) . Recent work on the subject proposes to add stochastic moves in order to enhance the performance of the normalizing flows [ 33 ] . More recently , the Non-Equilibrium IS ( NEIS ) method has been introduced by [ 23 ] as an alternative to these approaches . Similar to Neural IS , NEIS consists in transporting samples { Xi } Ni=1 from a reference distribution using a family of deterministic mappings . For NEIS , this family is chosen to be an homogeneous differential flow ( φt ) t∈R . In contrast to Neural IS , for any i ∈ [ N ] , the sample Xi is propagated both forward and backward in time along the orbits associated with ( φt ) t∈R until stopping conditions are met . Moreover , the resulting estimator of the normalizing constant is obtained by computing weighted averages of the whole orbit ( φt ( Xi ) ) t∈ [ τ+ , i , τ− , i ] , where τ+ , i , τ− , i are the resulting stopping times , and not only the endpoints φτ+ , i ( X i ) , φτ− , i ( X i ) . In [ 23 ] , the authors provide an application of NEIS with ( φt ) t∈R associated with a conformal Hamiltonian dynamics , and reports impressive numerical results on difficult normalizing constants estimation problems , in particular for high-dimensional multimodal distributions . We propose in this work NEO-IS which alleviates the shortcomings of NEIS . Similar to NEIS , samples are drawn from a reference distribution , typically set to ρ , and are propagated under the forward and backward orbits of a discrete-time dynamical system associated with an invertible transform T. An estimator of the normalizing constant is obtained by reweighting all the points on the whole orbits using the IS rule . Contrary to NEIS , the NEO-IS estimator of Z is unbiased under assumptions that are mild and easy to verify . It is more flexible than NEIS because it does not rely on the accuracy of the discretization of a continuous-time dynamical system . We then show how it is possible to leverage the unbiased estimator of Z defined by NEO-IS to obtain NEO-MCMC , a novel massively parallel MCMC algorithm to sample from π . In a nutshell , NEOMCMC relies on parallel walkers which each estimates the normalizing constant but are allowed to interact through a resampling mechanism . Our contributions can be summarized as follows . ( i ) We present a novel class of IS estimators of the normalizing constant Z referred to as NEO-IS . More broadly , a small modification of this algorithm also allows us to estimate integrals with respect to π . Both finite sample and asymptotic guarantees are provided for these two methodologies . ( ii ) We develop a new massively parallel MCMC method , NEO-MCMC . NEO-MCMC combines NEO-IS unbiased estimator of the normalizing constant with iterated samplingimportance resampling methods . We prove that it is π-reversible and ergodic under very general conditions . We derive also conditions which imply that NEO-MCMC is uniformly geometrically ergodic ( with an explicit expression of the mixing time ) . ( iii ) We illustrate our findings using numerical benchmarks which show that both NEO-IS and NEO-MCMC outperform state-of-the-art ( SOTA ) methods in difficult settings . 2 NEO-IS algorithm . In this section , we derive the NEO-IS algorithm . The two key ingredients for this algorithm are ( 1 ) the reference distribution ρ and ( 2 ) a transformation T assumed to be a C1-diffeomorphism with inverse T−1 . Write , for k ∈ N∗ = N\ { 0 } , Tk = T ◦Tk−1 , T0 = Idd and similarly T−k = T−1 ◦T− ( k−1 ) . For any k ∈ Z , denote by ρk : Rd → R+ the pushforward of ρ by Tk , defined for x ∈ Rd by ρk ( x ) = ρ ( T −k ( x ) ) JT−k ( x ) , where JΦ ( x ) ∈ R+ is the absolute value of the Jacobian determinant of Φ : Rd → Rd evaluated at x . In line with multiple importance sampling à la Owen and Zhou [ 20 ] , we introduce the proposal density ρT ( x ) = Ω −1 ∑ k∈Z $ kρk ( x ) , ( 1 ) where { $ k } k∈Z is a nonnegative sequence and Ω = ∑ k∈Z $ k . Note that we assume in the sequel that the support of the weight sequence defined as { k ∈ Z : $ k 6= 0 } is finite . Thus , the mixture distribution in ( 1 ) is a finite mixture . Given x ∈ Rd , ρT ( x ) is a function of the forward and backward orbit of T through x . For any nonnegative function f , the definition of ρT implies that ∫ f ( y ) ρT ( y ) dy = Ω−1 ∫ ∑ k∈Z $ kf ( T k ( x ) ) ρ ( x ) dx . Assuming that $ 0 > 0 , the ratio ρ ( x ) /ρT ( x ) ≤ $ −10 Ω < ∞ is bounded . We can therefore apply the IS principle which allows to write the identity∫ f ( x ) ρ ( x ) dx = ∫ ( f ( y ) ρ ( y ) ρT ( y ) ) ρT ( y ) dy = ∫ ∑ k∈Z f ( Tk ( x ) ) wk ( x ) ρ ( x ) dx , ( 2 ) where the weights are given by ( see Appendix A.2 for a detailed derivation ) , wk ( x ) = $ kρ ( T k ( x ) ) / { ΩρT ( Tk ( x ) ) } = $ kρ−k ( x ) /∑ i∈Z $ k+iρi ( x ) . ( 3 ) We assume in the sequel that $ 0 > 0 . In particular , note that under this condition , the weights wk are also upper bounded uniformly in x : for any x ∈ Rd , wk ( x ) ≤ $ k/ $ 0 . Equations ( 2 ) and ( 3 ) suggest to estimate the integral ∫ f ( x ) ρ ( x ) dx by INEO $ , N ( f ) = N −1∑N i=1 ∑ k∈Z wk ( X i ) f ( Tk ( Xi ) ) where { Xi } Ni=1 are i.i.d . samples from the proposal ρ , which is denoted by X1 : N iid∼ ρ. Algorithm 1 NEO-IS Sampler 1 . Sample X1 : N iid∼ ρ for i ∈ [ N ] . 2 . For i ∈ [ N ] , compute the path ( Tj ( Xi ) ) j∈Z and weights ( wj ( Xi ) ) j∈Z . 3 . INEO $ , N ( f ) = N −1∑N i=1 ∑ k∈Z wk ( X i ) f ( Tk ( Xi ) ) . This estimator is obtained by a weighted combination of the elements of the independent forward and backward orbits { Tk ( Xi ) } k∈Z with X1 : N iid∼ ρ . This estimator is referred to as NEO-IS . Choosing f ≡ L provides the NEO-IS estimator of the normalizing constant of π : Ẑ $ Xi = ∑ k∈Z L ( T k ( Xi ) ) wk ( X i ) , Ẑ $ X1 : N = N −1∑N i=1 Ẑ $ Xi . ( 4 ) We now study the performance of the NEO-IS estimator . The following two quantities play a fundamental role in the analysis : E $ T = EX∼ρ [ ( ∑ k∈Z wk ( X ) L ( T k ( X ) ) /Z ) 2 ] , M $ T = supx∈Rd ∑ k∈Z wk ( x ) L ( T k ( x ) ) /Z . ( 5 ) Theorem 1 . Ẑ $ X1 : N is an unbiased estimator of Z . If E $ T < ∞ , then , E [ |Ẑ $ X1 : N /Z−1|2 ] = N−1 ( E $ T − 1 ) . If M $ T < ∞ , then , for any δ ∈ ( 0 , 1 ) , with probability 1− δ , √ N ∣∣∣Ẑ $ X1 : N /Z−1∣∣∣ ≤ M $ T √ log ( 2/δ ) /2 . The ( elementary ) proof is postponed to Appendix A.3 . E $ T plays the role of the second-order moment of the importance weights EX∼ρ [ L2 ( X ) ] which is key to the performance of IS algorithms [ 1 , 2 ] . In addition , since the NEO-IS estimator Ẑ $ X1 : N is unbiased , the Cauchy–Schwarz inequality implies that EX∼ρ [ ( ∑ k∈Z wk ( X ) L ( T k ( X ) ) ) 2 ] ≥ Z2 and hence that E $ T ≥ 1 . Note that if ‖L‖∞ = supx∈Rd L ( x ) < ∞ , then since the weights are uniformly bounded by Ω $ −10 , we have M $ T ≤ ‖L‖∞Ω $ −1 0 /Z . Using the NEO-IS estimate Ẑ $ X1 : N of the normalizing constant , we can construct a self-normalized IS estimate of ∫ f ( x ) π ( x ) dx : JNEO $ , N ( f ) = N −1 N∑ i=1 Ẑ $ Xi Ẑ $ X1 : N ∑ k∈Z L ( Tk ( Xi ) ) wk ( X i ) Ẑ $ Xi f ( Tk ( Xi ) ) , ( 6 ) referred to as NEO-SNIS estimator . This expression may seem unnecessarily complicated but highlights the hierarchical structure of the estimator . We combine estimators ( Ẑ $ Xi ) −1∑ k∈Z L ( T k ( Xi ) ) wk ( X i ) f ( Tk ( Xi ) ) evaluated on the forward and backward orbits through the points { Xi } Ni=1 using the normalized weights { Ẑ $ Xi/Ẑ $ X1 : N } Ni=1 . Although the NEO-IS estimator is unbiased , the NEO-SNIS is in general biased . However , for bounded functions , both the bias and the variance of the NEO-SNIS estimator are O ( N−1 ) , with constants proportional to E $ T . For g a π-integrable function , we set π ( g ) = ∫ g ( x ) π ( x ) dx . Theorem 2 . Assume that E $ T < ∞ . Then , for any function g satisfying supx∈Rd |g ( x ) | ≤ 1 on Rd , and N ∈ N E X1 : N iid∼ρ [ |JNEO $ , N ( g ) − π ( g ) |2 ] ≤ 4 ·N−1E $ T , ( 7 ) ∣∣∣E X1 : N iid∼ρ [ JNEO $ , N ( g ) − π ( g ) ] ∣∣∣ ≤ 2 ·N−1E $ T . ( 8 ) If M $ T < ∞ , then for δ ∈ ( 0 , 1 ] , with probability at least 1− δ , √ N |JNEO $ , N ( g ) − π ( g ) | ≤ ‖g‖∞M $ T √ 32 log ( 4/δ ) . ( 9 ) The proof is postponed to Appendix A.4 . These results extend to NEO-SNIS estimators the results known for self-normalized IS estimators ; see e.g. , [ 1 , 2 ] and the references therein . The upper bounds stated in this result suggest it is good practice to keep E $ T /N small in order to obtain sensible approximations . For two pdfs p and q on Rd , denote by Dχ2 ( p , q ) = ∫ { p ( x ) /q ( x ) − 1 } 2q ( x ) dx the χ2-divergence between p and q. Lemma 3 . For any nonnegative sequence ( $ k ) k∈Z , we have E $ T ≤ Dχ2 ( π‖ρT ) + 1 . The proof is postponed to Appendix A.5 . Lemma 3 suggests that accurate sampling requires N to scale linearly with the χ2-divergence between the target π and the extended proposal ρT . Remark 1 . We can extend NEO to non homogeneous flows , replacing the family { Tk : k ∈ Z } with a collection of mappings { Tk : k ∈ Z } . This would allow us to consider further flexible classes of transformations such as normalizing flows ; see e.g . [ 21 ] . The χ2-divergence Dχ2 ( π‖ρT ) provides a natural criterion for learning the transformation . We leave this extension to future work . Conformal Hamiltonian transform The efficiency of NEO relies heavily on the choice of T. Intuitively , a sensible choice of T requires that ( i ) E $ T is small , i.e . ρT should be close to π by Lemma 3 ( see ( 5 ) ) , ( ii ) the inverse T−1 and the Jacobian of T are easy to compute . Following [ 23 ] , we use for T a discretization of a conformal Hamiltonian dynamics . Assume that U ( · ) = − log π ( · ) is continuously differentiable . We consider the augmented distribution π̃ ( q , p ) ∝ exp { −U ( q ) −K ( p ) } on R2d , where q is the position , p is the momentum , and K ( p ) = pTM−1p/2 is the kinetic energy , with M a positive definite mass matrix . By construction , the marginal distribution of the momentum under π̃ is the target pdf π ( q ) = ∫ π̃ ( q , p ) dp . The conformal Hamiltonian ODE associated with π̃ is defined by dqt/dt = ∇pH ( qt , pt ) = M−1pt , ( 10 ) dpt/dt = −∇qH ( qt , pt ) − γpt = −∇U ( qt ) − γpt , where H ( q , p ) = U ( q ) +K ( p ) , and γ > 0 is a damping constant . Any solution ( qt , pt ) t≥0 of ( 10 ) satisfies setting Ht = H ( qt , pt ) , dHt/dt = −γpTt M−1pt ≤ 0 . Hence , all orbits converge to fixed points that satisfy∇U ( q ) = 0 and p = 0 ; see e.g . [ 12 , 17 ] . In the applications below , we consider the conformal version of the symplectic Euler ( SE ) method of ( 10 ) , see [ 12 ] . This integrator can be constructed as a splitting of the two conformal and conservative parts of the system ( 10 ) . When composing a dissipative with a symplectic operator , we set for all ( q , p ) ∈ R2d , Th ( q , p ) = ( q + hM−1 { e−hγp − h∇U ( q ) } , e−hγp − h∇U ( q ) ) , where h > 0 is a discretization stepsize . This transformation can be connected with classical momentum optimization schemes , see [ 12 , Section 4 ] . By [ 12 , Section 3 ] , for any h > 0 Th is a C1-diffeomorphism on R2d with Jacobian given by JTh ( q , p ) = e−γhd . In addition , its inverse is T −1 h ( q , p ) = ( q − hM−1p , eγh { p+ h∇U ( q − hM−1p ) } ) . Therefore , the weight ( 3 ) of the NEO estimator is given by wk ( q , p ) = $ kρ̃ ( T k h ( q , p ) ) e −γkhd∑ j∈Z $ k+j ρ̃ ( T −j h ( q , p ) ) e γjhd , where ρ̃ ( q , p ) ∝ ρ ( q ) e−K ( p ) . Figure 1 displays for different values of γ on a log-scale the bound E 1 [ 0 : K ] Th − 1 appearing in Theorem 1 as a function of K , here we use the sequence of weights ( $ k ) k∈Z = ( 1 [ 0 : K ] ( k ) ) k∈Z ( i.e . only the K + 1 first elements of the forward orbits are used and are equally weighted ) . For comparison , we also present on the same plot the bounds achieved by averaging K + 1 independent IS estimates , EIS ( K ) − 1 = ( K + 1 ) −1EX∼ρ [ L ( X ) 2 ] . Interestingly , Figure 1 shows that there is a trade-off in the choice of γ which controls the exploration of the state space by the Hamiltonian dynamics since the higher γ , the faster the orbits converge towards the modes . This fast convergence prevents a “ good ” exploration of the space ; e.g . E 1 [ 0 : K ] Th is smaller for γ = 1.0 than for γ = 2.0 when K > 7 . 3 NEO-MCMC algorithm We now derive an MCMC method to sample from π based on the NEO-IS estimator . A natural idea consists in adapting the Sampling Importance Resampling procedure ( SIR ) ( see for example [ 24 , 27 ] ) to the NEO framework . The SIR method to sample JNEO $ , N ( see ( 6 ) ) consists of 4 steps . ( SIR-1 ) Draw independently X1 : N iid∼ ρ and compute the associated forward and backward orbits { Tk ( Xi ) } k∈Z of the point . ( SIR-2 ) Compute the normalizing constants associated with each orbit { Ẑ $ Xi } Ni=1 . ( SIR-3 ) Sample an orbit index IN ∈ [ N ] with probability { Ẑ $ Xi/ ∑N j=1 Ẑ $ Xj } Ni=1 . ( SIR-4 ) Draw the iteration index KN on the IN -th orbit with probability { L ( Tk ( XIN ) ) wk ( XI N ) /Ẑ $ XIN } k∈Z . The resulting draw is denoted by UN = TK N ( XI N ) . By construction , for any bounded function f , we get that E [ f ( UN ) ∣∣X1 : N , IN ] = { Ẑ $ XIN } −1∑k∈Z wk ( XIN ) L ( Tk ( XIN ) ) which implies E [ f ( UN ) ∣∣X1 : N ] = JNEO $ , N ( f ) ( see ( 6 ) ) . Using Theorem 2 , we therefore obtain |E [ f ( UN ) ] −∫ f ( z ) π ( z ) dz| ≤ 101/2‖f‖∞E $ T N−1 , showing that the law of the random variable µN = Law ( UN ) converges in total variation to π as N →∞ , ‖µN − π‖TV = sup ‖f‖∞≤1 |µN ( f ) − π ( f ) | ≤ 101/2E $ T N−1 . ( 11 ) Algorithm 2 NEO-MCMC Sampler At step n ∈ N∗ , given the conditioning orbit point Yn−1 . Step 1 : Update the conditioning point 1 . Set X1n = Yn−1 and for any i ∈ { 2 , . . . , N } , sample Xin iid∼ ρ . 2 . Sample the orbit index In with probability proportional to ( Ẑ $ Xin ) i∈ [ N ] , ( 4 ) . 3 . Set Yn = XInn . Step 2 : Output a sample 4 . Sample index Kn with probability proportional to { wk ( Yn ) L ( Tk ( Yn ) ) /Ẑ $ Yn } k∈Z 5 . Output Un = TKn ( Yn ) . Based on [ 3 ] , we now derive the NEOMCMC procedure , which in a nutshell consists in iterating the SIR procedure while keeping a conditioning point ( or equivalently , orbit ) ; see Appendix C. The convergence of NEO-MCMC does not rely on letting N →∞ : the NEO-MCMC works as soon as N ≥ 2 , although as we will see below the mixing time decreases as N increases . This procedure is summarized in Algorithm 2 . The NEO-MCMC procedure is an iterated algorithm which produces a sequence { ( Yn , Un ) } n∈N of points in Rd . The n-th iteration of the NEO-MCMC algorithm consists in two main steps : 1 ) updating the conditioning point Yn−1 → Yn 2 ) sampling Un by selecting a point in the orbit { Tk ( Yn ) } k∈Z of the conditioning point . Compared to SIR , only the generation of the points ( step ( SIR-1 ) ) is modified : we set X1n = Yn−1 ( the conditioning point ) , and then draw X2 : Nn iid∼ ρ . The sequence { Yn } n∈N defined by Algorithm 2 is a Markov chain : P ( Yn ∈ A |Y0 : n−1 ) = P ( Yn ∈ A |Yn−1 ) = P ( Yn , A ) where P ( y , A ) = ∫ δy ( dx 1 ) N∏ j=2 ρ ( xj ) dxj N∑ i=1 Ẑ $ xi∑N j=1 Ẑ $ xj 1A ( x i ) , y ∈ Rd , A ∈ B ( Rd ) . ( 12 ) Note that this Markov kernel describes the way , at stage n + 1 , the conditioning point Yn+1 is selected given Yn , which depends only on the estimator of the normalizing constants associated with each orbit , but not on the sample Un selected on the conditioning orbit . In addition , given the conditioning point Yn at the n-th iteration , the conditional distribution of the output sample Un is P ( Un ∈ B | In , X1 : Nn ) = P ( Un ∈ B |Yn ) = Q ( Yn , B ) where Q ( y , B ) = ∑ k∈Z wk ( y ) L ( T k ( y ) ) Ẑ $ y 1B ( T k ( y ) ) , y ∈ Rd , B ∈ B ( Rd ) . ( 13 ) With these notations , if the Markov chain is started at Y0 = y , then for any n ∈ N , the law of the n-th conditioning point is P ( Yn ∈ A |Y0 = y ) = Pn ( y , A ) and the law of the n-th sample is P ( Un ∈ B |Y0 ) = PnQ ( y , B ) . Define π̃ the pdf given for y ∈ Rd by π̃ ( y ) = ρ ( y ) Z ∑ k∈Z wk ( y ) L ( T k ( y ) ) = ρ ( y ) Ẑ $ y Z . ( 14 ) The following theorem shows that , for any initial condition y ∈ Rd , the distribution of the variable Yn converges in total variation to π̃ and that the distribution of Un converges to π. Theorem 4 . The Markov kernel P is reversible with respect to the distribution π̃ , ergodic and Harris positive , i.e. , for all y ∈ Rd , limn→∞ ‖Pn ( y , · ) − π̃‖TV = 0 . In addition , π = π̃Q and limn→∞ ‖PnQ ( y , · ) − π‖TV = 0 . Moreover , for any bounded function g and any y ∈ Rd , limn→∞ n −1∑n−1 i=0 g ( Ui ) = π ( g ) , P-almost surely , where { Ui } i∈N is defined in Algorithm 2 with Y0 = y . The proof is postponed to Appendix A.6 . Remark 2 . We may provide another sampling procedure of { Yn } n∈N . Define the pdf on the extended space [ N ] × RdN by π̌ ( i , x1 : N ) = N−1π̃ ( xi ) ∏N j=1 , j 6=i ρ ( x j ) . Consider a Gibbs sampler targeting π̌ consisting in ( a ) sampling X1 : N\ { In−1 } n | ( In−1 , Xn−1 ) ∼ ∏ j 6=In−1 ρ ( x j ) , ( b ) sampling In|X1 : Nn ∼ Cat ( { Ẑ $ Xin / ∑N j=1 Ẑ $ Xjn } Ni=1 and ( c ) set Yn = XInn . This algorithm is a Gibbs sampler on π̌ and we easily verify that the distribution of { Yn } n∈N is the same as Algorithm 2 . The next theorem provides non asymptotic quantitative bounds on the convergence in total variation . The main interest of NEO-MCMC algorithm is motivated empirically from observed behaviour : the mixing time of the corresponding Markov chain improves as N increases . This behaviour is quantified theoretically in the next theorem . Moreover , this improvement is obtained with little extra computational overhead , since sampling N points from the proposal distribution ρ , computing the forward and backward orbits of the points and evaluating the normalizing constants { Ẑ $ Xin } Ni=1 can be performed in parallel . Theorem 5 . Assume thatM $ T < ∞ , see ( 5 ) . Set N = ( N − 1 ) / ( 2M $ T +N − 2 ) and κN = 1− N . Then , for any y ∈ Rd and k ∈ N , ‖P k ( y , · ) − π̃‖TV ≤ κkN and ‖P kQ ( y , · ) − π‖TV ≤ κkN . Instead of sampling the new points X2 : Nn independently from ρ ( Step 1 in Algorithm 2 ) , it is possible to draw the proposals X1 : Nn conditional to the current point Yn−1 ; see [ 29 , 8 , 26 , 25 ] for related works . Following [ 25 ] , we use a reversible Markov kernel with respect to the proposal ρ , i.e. , such that ρ ( x ) m ( x , x′ ) = ρ ( x′ ) m ( x′ , x ) , assuming for simplicity that this kernel has density m ( x , x′ ) . If ρ = N ( 0 , σ2 Idd ) , an appropriate choice is an autoregressive kernel m ( x , x′ ) = N ( x′ ; αx , σ2 ( 1− α2 ) Idd ) . More generally , we can use a Metropolis–Hastings kernel with invariant distribution ρ . In this case , r1 ( x1 , x1 : N\ { 1 } ) = ∏N j=2m ( x j−1 , xj ) and for each i ∈ [ 2 : N ] , ri ( x i , x1 : N\ { i } ) = i−1∏ j=1 m ( xj+1 , xj ) N∏ j=i+1 m ( xj−1 , xj ) . ( 15 ) Since m is reversible with respect to ρ , for all i , j ∈ [ N ] , ρ ( xi ) ri ( xi , x1 : N\ { i } ) = ρ ( xj ) rj ( x j , x1 : N\ { j } ) where ri ( xi ; x1 : N\ { i } ) defines the the conditional distribution of X1 : N\ { i } given Xi = xi . The only modification in Algorithm 2 is Step 1 , which is replaced by : Draw Un ∈ [ N ] uniformly , set XUnn = Yn−1 and sample X 1 : N\ { Un } n ∼ rUn ( XUnn , · ) . The validity of this procedure is established in Appendix A.6 . 4 Continuous-time version of NEO and NEIS The NEO framework can be thought of as an extension of NEIS introduced in [ 23 ] . NEIS focuses on normalizing constant estimation and should be therefore compared with NEO-IS . In [ 23 ] , the authors do not consider possible extensions of these ideas to sampling problems . We consider here how NEO could be adapted to continuous-time dynamical system . Proofs of the statements and detailed technical conditions are postponed to Appendix B . Consider the Ordinary Differential Equation ( ODE ) ẋt = b ( xt ) , where b : Rd → Rd is a smooth vector field . Denote by ( φt ) t∈R the flow of this ODE ( assumed to be well-behaved ) . Under appropriate regularity condition Jφt ( x ) = exp ( ∫ t 0 ∇ · b ( φs ( x ) ) ds ) ; see Lemma S5 . Let $ : R → R+ be a nonnegative smooth function with finite support , with Ωc = ∫∞ −∞ $ ( t ) dt . The continuous-time counterpart of the proposal distribution ( 1 ) is ρcT ( x ) = ( Ω c ) −1 ∫∞ −∞ $ ( t ) ρ ( φ−t ( x ) ) Jφ−t ( x ) dt , which is a continuous mixture of the pushforward of the proposal ρ by the flow of ( φs ) s∈R . Assuming for simplicity that ρ ( x ) > 0 for all x ∈ Rd , then ρcT ( x ) > 0 for all x ∈ Rd , and using again the IS formula , for any nonnegative function f , ∫ f ( x ) ρ ( x ) dx = ∫ f ( x ) ρ ( x ) ρcT ( x ) ρcT ( x ) dx = ∫ [ ∫ ∞ −∞ wct ( x ) f ( φt ( x ) ) dt ] ρ ( x ) dx , ( 16 ) wct ( x ) = $ ( t ) ρ ( φt ( x ) ) Jφt ( x ) /∫ ∞ −∞ $ ( s+ t ) ρ ( φs ( x ) ) Jφs ( x ) ds . ( 17 ) These relations are the continuous-time counterparts of ( 2 ) . Eqs . ( 16 ) - ( 17 ) define a version of NEIS [ 23 ] , with a finite support weight function $ ; see Appendices B.2 and B.3 for weight functions with infinite support . This identity is of theoretical interest but must be discretized to obtain a computationally tractable estimator . For h > 0 , denote by Th an integrator with stepsize h > 0 of the ODE ẋ = b ( x ) . We may construct NEO-IS and NEO-SNIS estimators based on the transform T ← Th and weights $ k ← $ ( kh ) . We might show that for any bounded function f and for any x ∈ Rd , limh↓0 ∑ k∈Z wk ( x ) f ( T k h ( x ) ) = ∫∞ −∞ w c t ( x ) f ( φt ( x ) ) dt , where we omitted here the dependency in h of wk . Therefore , taking h ↓ 0+ , the NEO-IS converges to the continuous-version ( 16 ) - ( 17 ) . There is however an important difference between NEO and the NEIS method in [ 23 ] which stems from the way ( 16 ) - ( 17 ) are discretized . Compared to NEIS , NEO-IS using T← Th and weights $ k ← $ ( kh ) is unbiased for any stepsize h > 0 . NEIS uses an approach inspired by the nested-sampling approach , which amounts to discretizing the integral in ( 16 ) also in the state-variable x ; see [ 28 , 7 ] . This discretization is biased which prevents the use of this approach to develop MCMC sampling algorithm ; see Appendix B .

Summary: The paper puts forward a new family of algorithms, NEO, for computing normalizing constants and sampling from complex distributions. NEO-IS is an importance sampling estimator of the normalizing constant. NEO-MCMC combines NEO-IS unbiased estimator of the normalizing constant with iterated sampling- importance resampling methods.