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Inference via low-dimensional couplings

Alessio Spantini, Daniele Bigoni, Youssef Marzouk

arXiv:1703.06131v4stat.MEstat.COstat.ML

TL;DR

Transport maps can generate samples from target distributions, but their representation and evaluation become challenging in high dimensions. This paper relates Markov structure to sparse and decomposable couplings, then develops transport-based inference methods for non-Gaussian graphical and state-space models.

  • Problem

    Representing and evaluating transport maps becomes challenging in high dimensions, despite their use in coupling tractable reference measures to targets.

    Source: §1 · p. 3

  • Method

    The paper connects Markov properties of the reference-target pair to sparse or decomposable transport maps and applies the structure to inference algorithms.

    Source: §9 · pp. 47–48

  • Results

    The analysis supports low-dimensional couplings and yields variational algorithms for state-space filtering, smoothing, and sequential parameter inference.

    Source: §1 · pp. 3–4

  • Takeaways & Limitations

    For joint state-parameter inference, the proposed algorithms characterize smoothing and joint inference through a decomposable map constructed recursively in one forward pass using local operations.

    Source: §1 · p. 4

    The authors give no rigorous proof that sequential parameter-estimation performance avoids deterioration over time.

    Source: §7.3 · p. 37

Abstract

from arXiv · show

We investigate the low-dimensional structure of deterministic transformations between random variables, i.e., transport maps between probability measures. In the context of statistics and machine learning, these transformations can be used to couple a tractable "reference" measure (e.g., a standard Gaussian) with a target measure of interest. Direct simulation from the desired measure can then be achieved by pushing forward reference samples through the map. Yet characterizing such a map---e.g., representing and evaluating it---grows challenging in high dimensions. The central contribution of this paper is to establish a link between the Markov properties of the target measure and the existence of low-dimensional couplings, induced by transport maps that are sparse and/or decomposable. Our analysis not only facilitates the construction of transformations in high-dimensional settings, but also suggests new inference methodologies for continuous non-Gaussian graphical models. For instance, in the context of nonlinear state-space models, we describe new variational algorithms for filtering, smoothing, and sequential parameter inference. These algorithms can be understood as the natural generalization---to the non-Gaussian case---of the square-root Rauch-Tung-Striebel Gaussian smoother.

1. Introduction

Transport maps couple tractable reference measures to target distributions, but their representation and evaluation become difficult in high dimensions. The paper connects Markov structure to sparse or decomposable maps and develops inference methods for continuous non-Gaussian models.

1. Introduction

  • Transport maps are used in statistics and machine learning for posterior sampling, accelerating MCMC or importance sampling, generative modeling, and density estimation.Reference samples pushed through the map become samples from the target measure.Source: §1 · p. 2
  • High-dimensional transport maps are difficult to construct, represent, and evaluate without additional structure.Maps couple tractable reference measures, such as a standard Gaussian, to target measures and can generate target samples from reference samples.Source: §1 · pp. 2–3
  • Markov properties of the reference-target pair are linked to sparse and/or decomposable transport maps whose low-dimensional structure can be deduced before computation.Sparse components depend on only a few inputs; decomposable maps compose finitely many low-effective-dimensional maps.Source: §1 · p. 3
  • The results support tractable coupling construction and suggest methods for continuous non-Gaussian graphical models, including state-space inference.The paper describes variational algorithms for filtering, smoothing, and sequential parameter inference, related to the square-root Rauch-Tung-Striebel Gaussian smoother.Source: §1 · pp. 3–4
  • The paper develops sparse triangular-map results and decomposable-map theory, then specializes the latter to sequential inference in state-space models.The paper also includes numerical examples and a final discussion.Source: §1 · p. 4

2. Notation

The section establishes notation for random variables, graphs, measures, densities, and maps. It defines pushforward and pullback measures and gives the density formulas when a map is a diffeomorphism.

Notation

  • The notation defines graph neighborhoods and induced subgraphs, as well as cardinality and subset conventions.A ⊂ B includes the possibility A = B.Source: §2 · p. 4
  • The paper mostly considers fully supported probability measures absolutely continuous with respect to Lebesgue measure.For measures ν1 and ν2, ν1 ≪ ν2 denotes absolute continuity.Source: §2 · p. 5
  • For a measurable map T, the pushforward measure is T♯ν = ν ◦ T^-1, while the pullback is T♯ν = ν ◦ T.The pushforward or pullback density is defined when it exists.Source: §2 · p. 5
  • For a diffeomorphism T, the pushforward and pullback densities use the inverse and direct maps, respectively, with the corresponding Jacobian determinants.The section notes that weaker regularity can be handled with modified formulas.Source: §2 · p. 5
  • Boldface capitals denote multivariate random variables; subscripts select components or ranges, and π_X|Y denotes a conditional density.The law of a random variable X is written X♯P, and X ∼ ν means X has law ν.Source: §2 · p. 5

3. Triangular transport maps: a building block

The Knothe–Rosenblatt rearrangement is a monotone triangular coupling that provides a tractable building block for transport maps. Its parameterization supports variational optimization, while its anisotropic structure also helps factorize more general maps.

Triangular transport maps: a building block

  • The Knothe–Rosenblatt rearrangement is the unique monotone increasing lower-triangular map that pushes the reference measure forward to the target, up to reference-null sets.Its kth component depends only on the first k inputs.Source: §3 · p. 6
  • The triangular map’s anisotropic dependence lets it couple arbitrary distributions while serving as a building block for compositions of lower-dimensional maps.The paper describes it as the sparsest possible structure that preserves generality of the coupling.Source: §3 · p. 7
  • Triangular maps can be parameterized by scalar component functions, and the chosen monotone parameterization is invertible for every coefficient choice.The component functions may use polynomial expansions or neural networks; the resulting KL-divergence minimization can be expressed as an optimization problem.Source: §3 · p. 7
  • The KL objective can use an unnormalized target density and be solved with sample-average or stochastic approximation.The reference measure can be chosen to make integration feasible by quadrature, Monte Carlo, or quasi-Monte Carlo.Source: §3 · p. 8
  • With additional regularity, the monotone parameterization yields an unconstrained differentiable optimization problem, with unbiased gradient estimators available from the target log-density gradient or a score method.The score method has higher variance when the log-density gradient is unavailable.Source: §3 · p. 8
  • Transport-map construction and evaluation become increasingly intractable as dimension grows, motivating the search for hidden low-dimensional structure linked to target Markov properties.An alternative construction uses target samples to build an inverse triangular transport, which can be inverted through sequential one-dimensional root-finding problems.Source: §3 · p. 9

4. Markov networks

Markov networks encode conditional independences among random variables using an undirected graph. The section relates these independences to density factorization and motivates their use in identifying low-dimensional transport structure.

Markov networks

  • A Markov network represents conditional independence: if a set S separates A and B in the graph, the corresponding variable groups are conditionally independent given Z_S.Graph nodes represent variables and edges encode probabilistic interactions.Source: §4 · pp. 9–10
  • A graph is an independence map when the distribution satisfies its global Markov property, and minimal I-maps are the sparsest graphs compatible with the conditional independences.I-maps need not be unique.Source: §4 · pp. 10–11
  • Conditional independence corresponds to conditional-density factorization, while factorization by clique potentials implies the global Markov property.The converse requires additional assumptions, such as a continuous and strictly positive density.Source: §4 · p. 11
  • When the target density and its derivatives are available, pairwise conditional independence can be assessed to define a minimal I-map.The procedure adds an edge between each pair unless the corresponding variables are conditionally independent.Source: §4 · pp. 11–12
  • The paper identifies a fundamental connection between distributional Markov properties and the existence of low-dimensional transport maps.For a target specified through potentials, finding a graph according to which it factorizes is straightforward; state-space models are one motivating application.Source: §4 · pp. 11–12

5. Sparsity of triangular transport maps

The section shows how a target measure’s Markov structure can make triangular Knothe–Rosenblatt transports sparser, with bounds derived from an I-map. It connects this structure to Gaussian Markov random field methods and to choosing variable orderings that promote sparsity.

5.1 Sparsity bounds

  • Theorem 5.1 uses graph operations on a target I-map to bound transport sparsity without computing the maps, assuming a product reference measure.The graph sequence is produced by removing variables and making their neighborhoods cliques; the theorem gives sufficient, not necessary, conditions.Source: §5.1 · pp. 12–13
  • The predicted patterns are bounds rather than exact sparsity patterns: actual transports may be sparser, and the bounds can be sharp.The theorem’s predicted patterns cannot be less sparse than the actual patterns, and matching predicted and actual patterns are attainable under its hypotheses.Source: §5.1 · pp. 13–14
  • The predicted inverse-map pattern contains the direct-map pattern, so parameterizing the inverse can impose more sparsity constraints and still permit pointwise evaluation of the direct map.The paper relates inverse-map sparsity to conditional independence and direct-map sparsity to marginal independence.Source: §5.1 · p. 14
  • For a stochastic-volatility posterior, the predicted inverse map lets each component depend on at most four inputs, regardless of problem dimension N.The stated ordering is (μ, φ, Z0, . . . , ZN).Source: §5.1 · p. 15

5.2 Connection to Gaussian Markov random fields

  • Sparse Cholesky factors for Gaussian Markov random fields provide the linear analogue: the nonlinear triangular-map results generalize their sampling techniques to non-Gaussian distributions.Sampling can use a sparse triangular solve without explicitly storing the dense inverse factor.Source: §5.2 · p. 16

5.3 Ordering of triangular maps

  • Variable ordering changes predicted sparsity; the proposed relaxation maximizes inverse-map sparsity by minimizing graph fill-in, without computing a transport map.The ordering problem is NP-complete, though polynomial-time heuristics such as min-fill are available; finding a good ordering can be negligible compared with transport optimization when an I-map is known.Source: §5.3 · pp. 17–18, Figure 5 · p. 18

6. Decomposability of transport maps

This section connects target-measure Markov structure with decomposable transports, representing general maps as compositions of low-dimensional factors built from triangular maps and graph decompositions. It describes joint or sequential low-dimensional optimization for computing these factors, including sequential computation for Bayesian filtering, smoothing, and joint parameter/state estimation.

6. Decomposability of transport maps

  • A direct transport may be represented as a composition of low-dimensional maps, enabling computation in simpler stages that capture local target-measure features.The analysis covers general, possibly non-triangular transports and characterizes decomposable-map existence and structure from the target’s Markov structure.Source: §6 · pp. 18–19

6.1 Preliminary notions

  • A generalized triangular rearrangement can couple arbitrary measures while its permutation controls the map’s sparsity pattern.A proper graph decomposition requires disjoint nonempty A and B, with a clique separator S; fully connected graphs do not admit such a decomposition.Source: §6.1 · pp. 19–20

6.2 Decomposition and graph sparsification

  • When the target factorizes over a graph with a proper decomposition (A, S, B), Theorem 6.1 identifies transports T = L1 ◦ R, with low-dimensional factors and choices of permutation shaping their sparsity.The theorem constructs a nonempty family of transports from a product reference measure; graph sparsification removes edges incident to A and may add edges within S ∪ B, depending on the permutation.Source: §6.2 · pp. 20–22

6.3 Recursive decompositions

  • If S1 ∪ B1 is not a clique in the updated graph, the decomposition can be refined to add a factor; if it is a clique, no further refinement is available.Recursion terminates after finitely many steps, yielding a composition of low-dimensional generalized triangular factors, although the resulting transport need not be triangular.Source: §6.3 · pp. 23–24
  • For a target distribution factorizing over a tree, a transport exists as a composition of n−1 maps, each with effective dimension at most two, independent of tree size.The decomposition and permutation choices affect factor dimensions and the number of maps; dynamic programming can address this graph-optimization problem, while junction-tree scheduling may require care because graph cliques can grow during recursion.Source: §6.3 · p. 25

6.4 Computation of decomposable transports

  • Decomposable transports can be computed by jointly optimizing low-dimensional factors or sequentially solving separate low-dimensional problems, because factor densities may depend on earlier maps.With integrable factors, each map can be optimized only on a low-dimensional portion; the final map R likewise requires optimizing only a low-dimensional portion.Source: §6.4 · pp. 25–26
  • Sequential computation of the maps is available by construction in Bayesian filtering, smoothing, and joint parameter/state estimation, where decomposable transports break inference into smaller steps.For general problems, access to a sequence of suitable factorizations may be difficult.Source: §6.4 · p. 26

7. Sequential inference on state-space models: variational algorithms

For continuous, nonlinear, non-Gaussian state-space models, the section develops a variational algorithm for full-posterior inference. It constructs a decomposable transport in one forward pass from local problems, without a backward pass on the state-space model.

7. Sequential inference on state-space models: variational algorithms

  • The approach addresses Bayesian filtering and smoothing for continuous, nonlinear, non-Gaussian state-space models.Source: §7 · p. 26
  • The algorithm characterizes the full posterior for sequential inference through recursive lag–1 smoothing, building a decomposable transport in one forward pass from local small-dimensional problems without a backward pass.The result extends to joint parameter and state estimation.Source: §7 · p. 26
  • For a rooted graph, maps associated with edges connecting nodes at consecutive depths can be computed in parallel.The passage qualifies that this parallel computation is not always possible.Source: §7 · p. 26

A1 B1 S1

The section develops decomposable transport-map methods for state-space inference, covering posterior representation, sequential joint inference of states and model parameters, and recursively updated fixed-point smoothing. It also connects the approach to the square-root Rauch–Tung–Striebel smoother in the linear Gaussian case.

A1 B1 S1

  • The recursive graph decomposition yields a transport T = L1 ◦ L2 ◦ R with effective dimensions {3, 4, 3}, though the recursion cannot continue once the residual I-map becomes a single clique.The first map separates A1 from S1 ∪ B1; the second decomposition produces a larger clique, and the final residual maps remain low-dimensional with respect to A2.Source: A1 B1 S1 · p. 27

7.1 Smoothing and filtering: the full Bayesian solution

  • Theorem 7.1 represents the full state posterior with nested 2n-dimensional maps, enabling sequential assimilation at constant effort per observation and characterizing filtering and lag–1 smoothing distributions.The maps must have a block upper triangular structure; evaluating the full composition costs more as the time horizon grows.Source: §7.1 · pp. 29–31
  • Numerical map approximations introduce bias, and sequentially computing maps can accumulate error because each map’s target depends on the previous map’s approximation.The paper notes that richer parameterizations can reduce approximation bias at higher computational cost; long-time smoothing stability is investigated empirically in Section 8.Source: §7.1 · pp. 32–33

7.2 The linear Gaussian case: connection with the RTS smoother

  • In linear Gaussian state-space models, the maps have closed forms using square-root Kalman filtering quantities and implement a square-root RTS smoother without revisiting the model.The forward pass stores filtering estimates and additional quantities that can later be used to sample the full posterior or smoothing marginals.Source: §7.2 · p. 34

7.3 Sequential joint parameter and state estimation

  • Theorem 7.2 extends sequential inference to states and hyperparameters using maps of dimension 2n + p, whose composition samples the joint posterior.A recursively maintained parameter map can represent the posterior over static parameters without growing in dimension with time.Source: §7.3 · pp. 35–37
  • One local-computation forward pass gathers the model information needed to sample (πΘ, Z0:k+1|y0:k+1), with accuracy limited by map accuracy and no artificial parameter dynamics or fixed-lag approximation.Source: §7.3 · p. 37
  • The authors provide no rigorous guarantee that sequential parameter-estimation performance avoids deterioration over time, a challenge for exact online inference in general non-Gaussian state-space models.Source: §7.3 · p. 37

7.4 Fixed-point smoothing

  • For πZ0|y0:k+1, recursively updated 3n-dimensional maps yield a single transport map in one forward pass, rather than evaluating k composed 2n-dimensional maps.This characterizes the smoothing marginal for every k ≥ 0.Source: §7.4 · pp. 37–38

8. Numerical illustration

The numerical illustration applies sequential transport maps to joint state–parameter inference in a stochastic-volatility model. The approximations agree closely with reference MCMC results, including for long smoothing windows, while map complexity trades computational cost against accuracy.

Model and data

  • The illustration uses a stochastic-volatility model with an autoregressive log-volatility process and daily pound/dollar exchange-rate differences.The data comprise 945 daily differences beginning on 1 October 1981.Source: §8 · p. 38

Sequential inference

  • The sequential algorithm constructs four-dimensional maps for the scalar state and two static parameters, then composes them to sample the full posterior and obtain filtering and smoothing marginals.Each new time step updates the parameter map sequentially, and embedding the maps into identity maps gives the full state–parameter transport.Source: §8 · pp. 38–39

Approximation quality

  • The 947-dimensional transport approximation has KL divergence 0.107, compared with 5.68 for the Laplace approximation.The paper notes that the Laplace approximation cannot be computed sequentially with constant effort per time step.Source: §8 · p. 40

Long-window smoothing

  • At 9009 steps, smoothing remains remarkably stable for low-order marginals, and the joint state approximation is also reported as good.The long-window experiment fixes parameters at posterior medians estimated after 945 days; Table 1 reports the joint-distribution diagnostic.Source: §8 · p. 40, Table 1 · p. 43, Figure 17 · p. 46

Computational cost and accuracy

  • Degree-seven maps require approximately two days for joint state–parameter inference and 40 minutes for 9009-step smoothing; linear maps reduce these times to about one minute and seven minutes, respectively.The paper reports reduced accuracy with linear maps, especially for the full Bayesian solution and parameter marginals, though low-order marginal quantiles can remain indistinguishable from MCMC.Source: §8 · pp. 40–41, Table 1 · p. 43

Approximation quality

  • State smoothing and static-parameter posterior marginals agree remarkably well with reference MCMC, and the state smoothing curves are indistinguishable.The reference chain yields 10^5 effectively independent samples; its acceptance rate is slightly above 75%.Source: §8 · p. 39, Figure 10 · p. 41, Figure 12 · p. 43

9. Discussion

The discussion connects Markov structure to sparse and decomposable transport maps, which can make couplings easier to represent and compute. It also identifies open questions about approximate Markov structure and low-dimensional couplings not captured by the paper’s characterization.

Main contribution

  • Conditional-independence structure links the reference and target measures to sparse or decomposable couplings, supporting simulation from complex distributions in high dimensions.A deterministic coupling pushes reference samples through a transport map to represent the target measure.Source: §9 · p. 47

Sparse transports

  • For a product reference measure, target Markov structure bounds the sparsity of the inverse Knothe–Rosenblatt rearrangement; direct transports are generally dense, with sparsity tied to marginal independence.The inverse rearrangement shares its sparsity pattern with the precision-matrix Cholesky factor for Gaussian Markov random fields.Source: §9 · p. 47

Decomposable transports

  • A target whose Markov network admits a graph decomposition can be coupled to a product reference by an exact composition of finitely many low-dimensional maps.Unlike compositions used merely to increase approximation complexity, this decomposition guarantees low-dimensional component maps.Source: §9 · p. 48

Approximate Markov properties

  • When a sparser graph describes a nearby measure, its structure can guide a more sparse or decomposable approximation, but whether extra Markov properties approximate the target well remains open.The proposed construction minimizes KL divergence over maps patterned by the nearby measure’s graph; the paper also suggests adaptively relaxing sparsity until the approximation is satisfactory.Source: §9 · pp. 48–49

Filtering and smoothing

  • The state-space algorithms characterize the full posterior through recursive lag-1 smoothing in one forward pass, generalizing the square-root Rauch–Tung–Striebel smoother to non-Gaussian models.The discussion describes these as meta-algorithms that can incorporate approximations such as linearizations or restricted map classes, and notes the need for rigorous error analysis.Source: §9 · p. 49

Further extensions

  • The characterization is not exhaustive: low-rank couplings can be nontrivial only along a rotated low-dimensional subspace and cannot generally be deduced from Markov structure.The discussion points to inverse problems and spatial statistics as settings where data may inform only a few linear combinations of latent parameters.Source: §9 · p. 50

Appendix A. Generalized Knothe-Rosenblatt rearrangement

The appendix defines generalized Knothe–Rosenblatt rearrangements by permuting the triangular order of conditional one-dimensional transports. It relates their computation to an ordinary lower-triangular rearrangement after permuting coordinates.

Classical rearrangement

  • The ordinary Knothe–Rosenblatt rearrangement is lower triangular, with each component depending only on the preceding inputs and its own input.Its coordinatewise increasing rearrangements push reference conditionals to target conditionals.Source: Appendix A · p. 51

Definition

  • The generalized rearrangement builds each output coordinate recursively by increasing rearrangement from the corresponding reference conditional to the target conditional.The resulting map is triangular up to a chosen permutation.Source: Appendix A · p. 52

Computation

  • A generalized rearrangement is obtained by permuting coordinates, applying a lower-triangular rearrangement, and permuting back; thus its computation reduces to the ordinary triangular case.The appendix states that the generalized rearrangement inherits the regularity properties discussed for the classical one.Source: Appendix A · p. 52

Appendix B. Proofs of the main results

Appendix B supplies proofs and supporting lemmas for the paper’s main results, establishing sparsity and low-dimensionality properties of KR rearrangements and pushforward claims for sequential inference maps. Using induction and triangular change-of-variables arguments, it shows the constructed maps are well-defined and push reference measures to the stated filtering and joint marginals, and identifies affine-map parameters with the filtering marginal’s mean and a square root of its covariance.

Proof of Theorem 5.1

  • Theorem 5.1 proves that a version of the KR rearrangement is at least as sparse as predicted by the target measure’s Markov structure.Its proof uses the target’s conditional distributions to show that certain map components are constant along inputs outside the relevant graph neighborhoods.Source: Appendix B · p. 54

Lemma B.1

  • Under Lemma B.1’s independence and matching-distribution assumptions, the generalized KR rearrangement is low-dimensional with respect to A, with ∂jT^k = 0 for j in A and k outside A.The proof reduces the result to a lower-triangular rearrangement and applies Theorem 5.1.Source: Appendix B · p. 54

Proof of Theorem 6.1

  • Theorem 6.1 constructs target pushforward maps from low-dimensional KR maps, and modifies the graph to obtain an I-map for the resulting pullback measure.The construction uses a factorization associated with a proper graph decomposition.Source: Appendix B · pp. 55–56

Corollary B.1

  • Corollary B.1 establishes that each map T in D_i is low-dimensional with respect to A⊥.The proof uses the low-dimensionality of the component maps and their composition.Source: Appendix B · p. 56

Proof of Lemma 6.1

  • Lemma 6.1 relates whether S_i∪B_i is a clique to the existence of a larger proper decomposition, and bounds the effective dimensions of the resulting maps.When S_i∪B_i is not a clique, a larger decomposition can be formed, possibly after adding edges; the proof also gives effective-dimension bounds for its maps.Source: Appendix B · p. 57

Proof of Theorem 7.1

  • Theorem 7.1 proves that the sequential maps are well-defined, their compositions push forward reference measures to the stated posterior targets, and each map yields a lag-one smoothing marginal.The proof establishes the required conditional marginals by induction.Source: Appendix B · pp. 58–59

Proofs of the main results

  • For the affine map k(z_{k+1}) = C_k z_{k+1} + c_k, c_k is the filtering marginal’s mean and C_k is a square root of its covariance.These are the outputs of a square-root Kalman filter at time k+1.Source: Appendix B · p. 60

Proof of Theorem 7.2

  • The proof establishes by induction that the maps M_i and T_i are well-defined and that fM_i pushes η_{XΘ,X_{i+1}} forward to π_{Θ,Z_{i+1}|y_{0:i+1}} for every i ≥ 0.The change of variables is justified by factoring fM_i into generalized triangular maps and applying Lemma A.1 iteratively.Source: Appendix B · pp. 61–62
  • By induction, (T_{k+1})♯η_{XΘ,X_{0:k+2}} = π_{Θ,Z_{0:k+2}|y_{0:k+2}}; every map in C_{k+1} satisfies Lemma A.1’s hypothesis, completing the induction and proof of Part 2.Source: Appendix B · p. 62

Appendix C. Algorithms for inference on state-space models

The appendix presents pseudocode for recursive transport-map inference in state-space models, along with procedures for sampling from the resulting joint and filtering distributions.

Appendix C. Algorithms for inference on state-space models

  • Algorithms 1 and 2 provide building blocks for approximating a transport map from an unnormalized target density and projecting a map onto monotone transformations.The appendix describes these algorithms alongside Algorithms 3–5.Source: Appendix C · pp. 62–63
  • Algorithm 3 recursively approximates the full Bayesian joint smoothing distribution for states and static parameters with a decomposable transport map.For models without static parameters, the algorithm drops the formal dependence on Θ; a standard normal reference is used, though more general choices are possible.Source: Appendix C · p. 63
  • Algorithms 4 and 5 sample from the approximate joint distribution and its filtering marginal, respectively.The filtering marginal is πΘ,Zk+1|y0:k+1.Source: Appendix C · p. 63

Algorithm 1 (Computation of a monotone map)

The listed procedures fit a monotone triangular map to a target density or to an existing map, and the appendix also introduces a recursive joint state-parameter inference algorithm.

Algorithm 1 (Computation of a monotone map)

  • Algorithm 1 finds coefficients for a parametric triangular monotone map that optimize the stated criterion for an unnormalized target density.The supplied passage refers to the objective as (3.7).Source: Algorithm 1 (Computation of a monotone… · p. 63

Algorithm 2 (Regression of a monotone map)

Algorithm 4 (Sample the smoothing distribution)

The procedure samples from the smoothing distribution by applying the previously computed maps to a standard Gaussian reference draw.

Algorithm 4 (Sample the smoothing distribution)

  • Sampling uses the maps computed by Algorithm 3 to generate a draw from the smoothing distribution.The procedure starts with a standard normal sample in dθ + k·dz dimensions and applies the maps in reverse index order.Source: Algorithm 4 (Sample the smoothing… · p. 65

Algorithm 5 (Sample the filtering distribution)

Algorithm 5 samples the filtering distribution by drawing a standard Gaussian input and applying the maps computed in Algorithm 3.

Algorithm 5 (Sample the filtering distribution)

Appendix D. Additional results for the stochastic volatility model

Linear maps make these stochastic-volatility inference runs substantially faster, but accuracy depends on the target: state smoothing and filtering remain close to high-order references, while parameter marginals can be inadequate.

Appendix D. Additional results for the stochastic volatility model

  • Linear-map reruns are substantially faster but less accurate than nonlinear-map reruns across the joint state/parameter and long-time smoothing problems.Source: Appendix D · p. 65
  • For state smoothing, linear maps incur only a minimal accuracy loss relative to high-order maps, more pronounced at earlier times.Source: Figure 18 · p. 66
  • Linear-map filtering marginals look remarkably similar to the seventh-order reference despite the large computational-cost difference.Source: Figure 19 · p. 66
  • Linear maps inadequately characterize static parameter marginals, especially around the transition at time 74.Source: Figure 21 · p. 67, Figure 20 · p. 67
  • For sequential parameter inference, linear maps give plausible but somewhat inaccurate results, making nonlinear transformations better suited to this problem.Source: Figure 22 · p. 68
  • Linear maps accurately characterize long-time smoothing marginals across all times, at a fraction of the cost of a high-order nonlinear transformation.Source: Figure 24 · p. 69
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