A Factor Graph Approach to Scalable Multi-Output Gaussian Process Regression

Albert Podusenko; Esther G. van Pelt; \.Ismail \c{S}en\"oz; Wouter M. Kouw; Wouter W. L. Nuijten

REVIEW 2 major objections 4 minor 300 references

A Factor Graph Approach to Scalable Multi-Output Gaussian Process Regression

T0 review · 2 major / 4 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read Multi-output GP regression can be computed as exact Gaussian message passing on a nearest-neighbor chain factor graph, with cost linear in the candidate-set size and missing observations handled by omitting local factors.

desk verdict A solid algorithm paper with exact inference on a chain-induced LMC factor graph; the advertised fidelity to the original GP is empirical, not certified by the provided loose bounds. read the letter →

arxiv 2608.11917 v1 pith:OKK3DHKB submitted 2026-08-12 cs.LG

Wouter W. L. Nuijten , Esther G. van Pelt , Albert Podusenko , İsmail Şenöz , Wouter M. Kouw This is my paper · ORCID

classification cs.LG

keywords factorgraphsGaussianprocessesmulti-outputregressionmessagepassinglinearmodelofcoregionalizationstate-spacemodelspartialobservationsnearest-neighborchain

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Multi-output Gaussian process regression normally costs cubic time in the number of observations times outputs and demands special handling when outputs are observed at different inputs. This paper proposes expressing the model as a factor graph in which a nearest-neighbor chain orders a fixed candidate set of $C$ inputs into a one-dimensional sequence; along the chain, latent Matérn processes evolve through linear-Gaussian transitions, and a linear model of coregionalization mixes $L$ latents into $D$ outputs. Inference then becomes exact Gaussian message passing on the chain at $O(C(DL^2 + L^3))$ cost after the chain is built, and missing observations merely omit their local factor rather than forcing covariance-matrix restructuring. The approximation is confined to the chain ordering; inference on the resulting model is exact. The paper argues this is the practical regime for low-dimensional input geometries, where the factor-graph posterior tracks the exact kernel-matrix posterior closely and degrades gracefully as input dimension grows.

What carries the argument

The load-bearing object is a Forney-style factor graph whose backbone is a greedy nearest-neighbor chain over the candidate inputs. Each latent Matérn process is discretized as a linear-Gaussian state-space model, with inter-point distances along the chain serving as the time steps in the transition and process-noise matrices. A deterministic linear-model-of-coregionalization factor mixes the $L$ latent states into the $D$ outputs, and each output at each chain position is an independent scalar Gaussian observation factor. Exact Gaussian message passing, equivalent to Kalman filtering and smoothing, then propagates dense beliefs over the joint $2L$-dimensional latent state, giving the $O(C(DL^2 + L^3))$ per-step cost; the chain construction is what compresses the $M$-dimensional input geometry into the one-dimensional sequence.

What would settle it

Run the paper's synthetic sensor-network benchmark at input dimensions $M=64$ and $M=128$ with the same candidate-set size $C=2000$. If the held-out RMSE gap between the factor-graph posterior and the exact kernel-matrix posterior grows sharply with $M$, beyond the gradual rise from $0.003$ at $M=2$ to $0.114$ at $M=32$, or if the per-sweep wall-clock changes materially with the missingness mask at fixed $C$, $D$, and $L$, then the chain-compression or local-omission claim would be contradicted.

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Extended reading notes

Core claim

The paper's central claim is that, for a fixed candidate set of $C$ inputs, multi-output GP regression with a linear model of coregionalization and Matérn kernels can be rewritten as a factor graph in which inference is exact Gaussian message passing along a nearest-neighbor chain. The cost after the chain is built is $O(C(DL^2 + L^3))$, linear in the number of candidate points for fixed output count and latent rank, and the missingness mask is handled by simply omitting the corresponding scalar observation factors. The only approximation is the chain ordering itself; conditional on that ordering, the posterior is computed exactly. The paper shows that this chain-induced model tracks the exact kernel-matrix posterior closely at low input dimension, with the gap growing gradually as dimension increases, and that on a three-dimensional electricity-forecasting task it matches exact, sparse-variational, and nearest-neighbor baselines in accuracy while scaling linearly and staying invariant to the dropout rate.

Load-bearing premise

The method relies on a greedy nearest-neighbor chain being able to flatten $M$-dimensional input geometry into a one-dimensional sequence without losing the correlations that matter; the paper's own bound on the resulting kernel distortion grows with candidate-set size and becomes uninformative as $M$ grows, so the practical success rests on the empirical observation that the gap stays small.

Editorial extensions

If this is right

At fixed output count $D$ and latent count $L$, per-step inference cost becomes linear in the number of candidate points rather than cubic in $N$ times $D$, making large low-dimensional multi-output datasets feasible with a single forward-backward sweep.
Arbitrary missingness patterns cost nothing extra: unobserved outputs omit their local factor, so the same model and inference call handle any mask without rebuilding covariance matrices.
The chain depends only on the candidate inputs, not on observations, masks, or hyperparameters, so it can be built once and reused across fits, hyperparameter-learning iterations, and streaming updates.
At low input dimension the posterior closely tracks the exact kernel-matrix posterior; at higher dimension the gap grows gradually and remains competitive with inducing-point and nearest-neighbor approximations.
The method applies to Matérn-class kernels that admit state-space representations; outside that class the chain construction does not directly apply.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The chain-induced kernel is effectively a geodesic kernel on the nearest-neighbor path, which suggests a natural testable extension: using multiple chains or random projections to reduce distortion in higher dimensions, something the paper does not explore.
Because chain construction is independent of observations, the approach is well suited to online settings where the candidate set is fixed and data arrive incrementally; the paper only sketches this in its discussion of amortized reuse.
The theoretical bounds are too loose to certify practical error, so the paper's empirical protocol of comparing against an exact kernel-matrix posterior on a dimension sweep is the right arbiter; a stronger bound would be needed before using the method where exact inference is infeasible.
The same local-edit property that makes missingness free could also support active learning or sequential experimental design, where the observation mask is chosen adaptively rather than fixed in advance.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: this is a useful algorithm paper, not a breakthrough. What is new is putting state-space Matérn GPs together with the linear model of coregionalization in a Forney-style factor graph, ordered along a nearest-neighbor chain, so multi-output regression becomes exact linear-Gaussian message passing on that chain. Conditional on the chain, the math is correct: the cost formula O(C(DL^2 + L^3)) follows from block-diagonal dynamics plus a dense observation coupling, and missing outputs are handled by omitting local factors. That modular handling of partial observability is genuinely nice and contrasts with dense kernel slicing. The proofs of Lemma 1, Lemma 2, and Theorem 1 are standard resolvent and spectral-norm arguments, and they check out. The experiments are well controlled: identical hyperparameters across methods, a synthetic dimension sweep up to M=32, ETTh1 forecasting, and code is provided. Credit where due: the authors do not oversell the theory; the limitations section says the bounds are not practical error estimates.

The soft spot is exactly what your stress-test note identifies. The chain compression is the load-bearing approximation, and the theoretical guarantee stops being informative at moderate input dimension. So the claim that the factor-graph posterior tracks the exact GP rests almost entirely on one random-uniform synthetic sweep (C=2000, M up to 32, gap <= 0.12) and one real-data study at M=3. That is a real limitation, especially for structured input geometry like manifolds or clusters. The paper acknowledges it, which is good, but it means the practical value proposition is “works in the tested low-dimensional cases” rather than a certified approximation. The abstract does say “best suited to low input dimension,” so the framing is mostly honest. A secondary gap: Proposition 1 invokes a Rosenkrantz et al. bound for TSP tours and applies it to Hamiltonian paths without derivation. This is a minor flaw in a robustness side-argument, and Appendix D’s empirical sensitivity analysis largely covers it, but the proof should be fixed or the proposition downgraded.

Overall, the central inference result is sound, the implementation is reproducible, and the empirical study is honest. I would point a colleague working on scalable GP inference or sensor-network modeling to it. The paper deserves a serious referee: send it to review and ask the authors to repair Proposition 1, add at least one real-data or structured-geometry case above M=3 if feasible, and make the distinction between exact inference on the chain model and approximate fidelity to the original GP crisper in the abstract.

Referee Report

2 major / 4 minor

Summary. The paper proposes SS-LMC, a Forney-style factor graph formulation of multi-output Gaussian process regression under the linear model of coregionalization. Candidate inputs are ordered by a greedy nearest-neighbor chain; each latent Matérn process is represented in state-space form with linear-Gaussian transitions along the chain; the LMC mixing matrix is a deterministic factor; and per-output scalar observation factors handle arbitrary missingness by simply omitting the local factor. Inference is exact Gaussian message passing (Kalman smoothing) on the chain, at O(C(DL^2 + L^3)) cost after chain construction. The authors prove bounds on the distance between the exact LMC kernel and the chain-induced kernel and on the resulting posterior parameters, and they compare the method against exact kernel-matrix LMC, sparse-variational LMC, and nearest-neighbor LMC baselines on a synthetic input-dimension sweep and on ETTh1 electricity time series forecasting.

Significance. The message-passing construction is clean and the exact-inference part of the claim is mathematically sound: conditional on the chain, posterior computation is standard linear-Gaussian smoothing, and the stated complexity is credible. Missing-data modularity by local factor omission is a genuine and useful advantage over dense covariance restructuring. The experimental comparison is fair in an important respect: all methods share the same fixed LMC hyperparameters and are evaluated with held-out metrics, and the code is provided. The theoretical bounds are honestly described as worst-case and not usable as practical error estimates. If the empirical fidelity at low input dimension holds in broader settings, this is a useful scalable approximation for multi-output GP regression with partial observations. The main weakness is that the chain-fidelity claim is not established by the theory outside small input dimension and is currently supported only by a uniform-random synthetic sweep and one real-data setting at M=3.

major comments (2)

[Section 4.1.1, Theorem 1, Section 5.1, Section 6] The central practical claim that the factor-graph posterior 'tracks the exact kernel-matrix posterior closely' is not supported by the theory for the input dimensions where the method is intended to be used, and the empirical support is narrow. Theorem 1 bounds the posterior gap by eta = O(C^(2-1/M) sum_l ||w_l||^2 alpha_l), which the authors themselves state becomes uninformative as M grows, and Section 6 concedes that the bounds 'do not furnish error estimates one would use in practice.' The supporting evidence is the random-uniform synthetic sweep (C=2000, M<=32, RMSE gap <=0.12) and one real-data setting at M=3. No experiment tests structured candidate geometry such as anisotropic distributions, clusters, or low-dimensional manifolds, which is precisely the regime where a greedy Euclidean nearest-neighbor chain is most likely to distort the input geometry. Please add a benchmark with structured geometry or explicitly scope the 'tracks closely' claim to the uniform low-dimensional regime.
[Appendix H, Proposition 1] The proof of Proposition 1 asserts that Rosenkrantz et al. (1977) prove the greedy nearest-neighbor heuristic produces a Hamiltonian path of length at most (1/2)(ceil(log2 C)+1) times the shortest Hamiltonian path, but the cited result is for nearest-neighbor TSP tours. The Hamiltonian-path version is stated without derivation, and the factor and constant need to be justified or replaced by a direct reference. Since Proposition 1 is the only theoretical control on starting-point variability, this weakens a secondary robustness argument; Appendix D mitigates the concern empirically, but the proposition as stated is not established.

minor comments (4)

[Section 4.2 and Table 1] The terminology 'per-step' is used inconsistently: O(C(DL^2 + L^3)) is described in the text as the cost of one smoothing sweep, but Table 1 labels this as 'Per-step inference' and then writes 'Total over N steps' with an extra factor N. Please clarify when N multiplies the sweep cost so readers can verify the scaling claims.
[Figures 3 and 5] The input-dimension axis labels appear as '22 24' in the rendered figures; these should presumably be 2^2 through 2^5 with proper superscripts.
[Section 6] The amortization argument states that the one-off chain construction cost is recovered within a single fit, but no wall-clock time for chain construction is reported. Please report c_pi explicitly so the reader can check the claimed recovery condition R > c_pi / (t_base - t_SS).
[Theorem 1 and Eq. (14)] Theorem 1 assumes scalar observation noise sigma_n^2 I, whereas the model in Eq. (14) uses per-output precisions tau_d^{-1}. Please state whether the experimental 'fixed diagonal noise' is equal across outputs and, if not, how the bounds would be modified for heteroscedastic per-output noise.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the chain-induced model, its exact message-passing inference, and the empirical validation are self-contained; self-citations are background only.

full rationale

The paper's derivation chain starts from standard external results (Matérn-SDE equivalence, FFG message passing, LMC) and builds a chain-induced linear-Gaussian state-space model. The main load-bearing claims are not circular. (i) The O(C(DL^2+L^3)) inference cost is a direct complexity count for exact Kalman smoothing on the linear-Gaussian model in Eqs. (12)-(14): block-diagonal dynamics give the L^3 belief update and the D per-output scalar observation factors give the DL^2 term; it is not obtained by fitting. (ii) The missingness handling is a designed property of the factor graph: observation factors for O_{i,d}=0 are omitted from the joint model, so no covariance restructuring is required; this is a definitional feature, not a fitted result presented as a prediction. (iii) The fidelity claim is supported by Theorem 1, whose proof (Appendices E-G) bounds posterior differences via a resolvent identity and Lipschitz constants, with no fitted parameter; and by Section 5 experiments where all methods share the same fixed hyperparameters and held-out data are genuinely out-of-sample, including an exact kernel-matrix baseline. The chain-construction preprocessing depends only on candidate inputs, not on Y, O, or hyperparameters, so the empirical comparison is not fitting its own output. Self-citations (Şenöz et al. 2021; Bagaev et al. 2023; Kouw 2025; Nguyen et al. 2025) are background or related-work references and none carries the central argument. The admitted looseness of the worst-case bounds and the uninformative bound at larger M are limitations, not circularity; likewise the use of the Rosenkrantz TSP bound for Hamiltonian paths in Appendix H is a possible correctness gap, not circularity.

Assumptions & free parameters 2 free parameters · 3 assumptions · 0 invented entities

The central claim rests mostly on standard results (state-space GP, factor graph message passing) and on the empirical adequacy of the nearest-neighbor chain approximation. The hand-chosen hyperparameters are shared across all methods, so they do not introduce circular fitting.

free parameters (2)

Model hyperparameters (length scales, output scales, mixing matrix, noise variance) = Synthetic: length scales in {1.0, 2.0}, output scales in {2.0, 1.0}, noise fixed; ETT: length scales in {0.5, 1, 2}…
Hand-selected and identical across all compared methods, isolating inference method from hyperparameter choice; not fitted to the held-out targets.
Candidate set size C = 2000 (synthetic), 2N window lengths in ETT (N = 500 to 8000)
A tuning choice for the approximation; the paper reports cost scaling in C but does not learn it.

assumptions (3)

domain assumption Matérn-class kernels with half-integer smoothness admit an exact finite-dimensional linear-Gaussian state-space representation (SDE form).
The chain model requires the latent Matérn processes to be represented as linear-Gaussian transitions via Eq. (2)-(3), cited to Hartikainen and Särkkä (2010). This is a well-established result but it limits the method to Matérn kernels with rational spectral density.
domain assumption The greedy nearest-neighbor chain preserves the low-dimensional input geometry well enough that the chain-induced kernel is close to the Euclidean Matérn kernel.
This is the central approximation; the paper provides only a worst-case bound that is loose (Theorem 1) and relies on empirical evidence. Section 4.1.1 and Section 6 discuss the limitation.
standard math Sum-product message passing on a cycle-free Forney-style factor graph yields exact posterior marginals for linear-Gaussian models.
Standard background for factor graph inference (Kschischang et al., 2001; Loeliger et al., 2007), used throughout Section 4.

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Pith. "Pith review of A Factor Graph Approach to Scalable Multi-Output Gaussian Process Regression." pith.science (2026). https://pith.science/paper/OKK3DHKB

@misc{pith2026260811917,
  author       = {Pith},
  title        = {Pith review of: A Factor Graph Approach to Scalable Multi-Output Gaussian Process Regression},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OKK3DHKB}},
  note         = {Machine review of arXiv:2608.11917}
}

abstract

Multi-output Gaussian process regression scales cubically in the number of observations times outputs, and dense kernel-matrix methods need bespoke handling whenever different outputs are observed at different inputs. We express multi-output Gaussian process regression as a Forney-style factor graph in which a nearest-neighbor chain orders a fixed candidate set of $C$ inputs into a one-dimensional sequence. Along this chain, latent Mat\'ern processes evolve through linear-Gaussian transition factors, while the linear model of coregionalization mixes $L$ latent processes into $D$ outputs through a deterministic mixing factor and per-output scalar observation factors. Posterior computation reduces to exact Gaussian message passing on the chain at cost $\mathcal{O}(C(DL^2 + L^3))$ after chain construction, and missing observations omit their local factor without any covariance-matrix restructuring. The formulation therefore scales in the number of data samples and in the rate of missing observations, while remaining best suited to candidate sets in low input dimension.We compare the factor-graph formulation against an exact kernel-matrix baseline, a sparse-variational inducing-point baseline, and a nearest-neighbor baseline on a synthetic input-dimension sweep and on electricity time series forecasting. At low input dimension the factor-graph posterior tracks the exact kernel-matrix posterior closely, and the gap grows gradually as input dimension increases while staying competitive with both approximate baselines. On the electricity time series our factor-graph formulation matches all three baselines in forecast accuracy while scaling linearly in the number of data points, where the exact kernel-matrix method becomes infeasible and the inducing-point baseline remains substantially slower.

Figures

Figures reproduced from arXiv: 2608.11917 by the authors.

**Figure 2.** Example Forney-style factor graph of SS-LMC, for D = 3. The dashed box indicates a plate repeated over all points in the chain. Messages are carried upwards and joined into a total observation message 4, which updates the chain. If yi,2 and yi,3 are unobserved, the dashed messages (5–11) are absent and do not contribute. 4.1.2. State-Space GP with LMC In state-space form, the full state zi ∈ R 2L is the concatenati… view at source ↗

**Figure 3.** Input-dimension sweep on the synthetic sensor network (C=2000, D=3, L=2, 50/50 train/test, 100 seeds; mean ± std). and forecast the fully held-out next N rows. We compare four methods with identical LMC structure (L=3, ℓl ∈ {0.5, 1, 2}, γ 2 l =1, fixed diagonal noise R=0.2): the proposed SSLMC by reactive message passing, an exact KM-LMC by covariance restructuring, a sparsevariational SVGP-LMC with 64 inducing po… view at source ↗

Figures from the paper (5 more)

**Figure 4.** Figure 4: ETTh1 forecasting at M=3, D=4, L=3, 10 seeds. KM-LMC runs only to N=2000. Timings exclude the one-off chain construction (cf. Section 6) [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]

**Figure 5.** Figure 5: Chain stretch and approximation gap on the sensor network sweep (C=2000, D=3, L=2, 100 seeds). Left: nearest-neighbor chain stretch ∆ (mean, max) vs. input dim M. Middle and right: held-out RMSE and MNLL gap vs. input dim M, as SS-LMC minus exact KM-LMC. Difference is …

**Figure 6.** Figure 6: ETTh1 temporal forecasting (M=1 natural ordering, D=7, L=4, 3 seeds): train on the first half of a length-C window, forecast the held-out second half under per- (timestamp, output) dropout. At M=1 the SS-LMC and KM-LMC posteriors are identical, so the comparison is com…

**Figure 7.** Figure 7: Held-out score against fit+forecast wall clock on ETTh1 (M=3, D=4, L=3, dropout p=0.3, 10 seeds). Lower-left is better. Top row: RMSE; bottom row: MNLL. Columns are C=1000 and C=2000. Dashed line is Pareto front. Bars are ±1 standard deviation over seeds. 17 [PITH_FUL…

**Figure 8.** Figure 8: Spread of SS-LMC’s held-out RMSE across 20 chain starting points (median over 10 seeds per start, then standard deviation across starts). Bars are a bootstrap 95% confidence interval for that standard deviation. Left: vs. input dimension M on the synthetic sensor netwo…

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Works this paper leans on

300 extracted references · 5 canonical work pages

[1]

Adamiat, Sepideh and Kouw, Wouter M. and. Message. Active. doi:10.1007/978-3-031-77138-5_14 , abstract =

doi:10.1007/978-3-031-77138-5_14
[2]

and Shipp, Stewart and Friston, Karl J

Adams, Rick A. and Shipp, Stewart and Friston, Karl J. , year = 2013, journal =. Predictions Not Commands: Active Inference in the Motor System , shorttitle =. doi:10.1007/s00429-012-0475-5 , urldate =

doi:10.1007/s00429-012-0475-5 2013
[3]

and Murphy, Kevin , year = 2018, month = jul, pages =

Alemi, Alexander and Poole, Ben and Fischer, Ian and Dillon, Joshua and Saurous, Rif A. and Murphy, Kevin , year = 2018, month = jul, pages =. Fixing a. Proceedings of the 35th

2018
[4]

Computationally Efficient Convolved Multiple Output

Alvarez, Mauricio A and Lawrence, Neil D , year = 2011, journal =. Computationally Efficient Convolved Multiple Output

2011
[5]

Foundations and Trends in Machine Learning , volume =

Kernels for. Foundations and Trends in Machine Learning , volume =. doi:10.1561/2200000036 , urldate =

doi:10.1561/2200000036
[6]

, year = 2012, month = jun, pages =

Anandkumar, Animashree and Hsu, Daniel and Kakade, Sham M. , year = 2012, month = jun, pages =. A. Proceedings of the 25th

2012
[7]

Anil Meera, Ajith and Lanillos, Pablo , editor =. Towards. Active. doi:10.1007/978-3-031-47958-8_3 , abstract =

doi:10.1007/978-3-031-47958-8_3
[8]

Biosystems , volume =

Natural Hierarchy Emerges from Energy Dispersal , author =. Biosystems , volume =. doi:10.1016/j.biosystems.2008.10.008 , urldate =

doi:10.1016/j.biosystems.2008.10.008 2008

Show all 300 references

[9]

Declarative

Antoy, Sergio and Hanus, Michael , editor =. Declarative. Logic. doi:10.1007/11680093_2 , abstract =

doi
[10]

doi:10.1007/978-3-319-20451-2 , urldate =

Readings in. doi:10.1007/978-3-319-20451-2 , urldate =

doi
[11]

and Nouri, Ali and Wingate, David , year = 2009, month = jun, series =

Asmuth, John and Li, Lihong and Littman, Michael L. and Nouri, Ali and Wingate, David , year = 2009, month = jun, series =. A. Proceedings of the

2009
[12]

and Nouri, Ali and Wingate, David , year = 2012, month = may, number =

Asmuth, John and Li, Lihong and Littman, Michael L. and Nouri, Ali and Wingate, David , year = 2012, month = may, number =. A. doi:10.48550/arXiv.1205.2664 , urldate =. arXiv , keywords =:1205.2664 , primaryclass =

2012 doi arXiv
[13]

Planning by Probabilistic Inference , booktitle =

Attias, Hagai , year = 2003, pages =. Planning by Probabilistic Inference , booktitle =

2003
[14]

Reinforcement

Azizzadenesheli, Kamyar and Lazaric, Alessandro and Anandkumar, Animashree , year = 2016, month = jun, pages =. Reinforcement. Conference on

2016
[15]

Reactive

Bagaev, Dmitry and. Reactive. doi:10.48550/arXiv.2112.13251 , urldate =. arXiv , keywords =:2112.13251 , primaryclass =

doi arXiv
[16]

Reactive

Bagaev, Dmitry and De Vries, Bert , editor =. Reactive. Scientific Programming , volume =. doi:10.1155/2023/6601690 , urldate =

2023 doi
[17]

Software Impacts , volume =

Bagaev, Dmitry and. Software Impacts , volume =. doi:10.1016/j.simpa.2022.100299 , urldate =

2022
[18]

doi:10.21105/joss.05161 , urldate =

Bagaev, Dmitry and Podusenko, Albert and de Vries, Bert , year = 2023, month = apr, journal =. doi:10.21105/joss.05161 , urldate =

2023 doi
[19]

ACM Computing Surveys (CSUR) , volume =

A Survey on Reactive Programming , author =. ACM Computing Surveys (CSUR) , volume =. doi:10.1145/2501654.2501666 , urldate =
[20]

and Simpson, Daniel and Rue, H

Bakka, Haakon and Vanhatalo, Jarno and Illian, Janine B. and Simpson, Daniel and Rue, H. Non-Stationary. Spatial Statistics , volume =. doi:10.1016/j.spasta.2019.01.002 , urldate =

2019 doi
[21]

and Daulton, Samuel and Letham, Benjamin and Wilson, Andrew Gordon and Bakshy, Eytan , year = 2020, month = dec, series =

Balandat, Maximilian and Karrer, Brian and Jiang, Daniel R. and Daulton, Samuel and Letham, Benjamin and Wilson, Andrew Gordon and Bakshy, Eytan , year = 2020, month = dec, series =. Proceedings of the 34th

2020
[22]

arXiv , keywords =:2111.10530 , primaryclass =

Kalman Filters as the Steady-State Solution of Gradient Descent on Variational Free Energy , author =. arXiv , keywords =:2111.10530 , primaryclass =

arXiv
[23]

Reactive Probabilistic Programming , booktitle =

Baudart, Guillaume and Mandel, Louis and Atkinson, Eric and Sherman, Benjamin and Pouzet, Marc and Carbin, Michael , year = 2020, month = jun, series =. Reactive Probabilistic Programming , booktitle =. doi:10.1145/3385412.3386009 , urldate =

2020
[24]

doi:10.1098/rstl.1763.0053 , urldate =

Bayes, Thomas , year = 1763, month = dec, journal =. doi:10.1098/rstl.1763.0053 , urldate =
[25]

Beck, Jeff and Ramstead, Maxwell J. D. , year = 2025, month = feb, number =. Dynamic. doi:10.48550/arXiv.2502.21217 , urldate =. arXiv , keywords =:2502.21217 , primaryclass =

2025 doi arXiv
[26]

Unifying

Bellemare, Marc and Srinivasan, Sriram and Ostrovski, Georg and Schaul, Tom and Saxton, David and Munos, Remi , year = 2016, volume =. Unifying. Advances in

2016
[27]

Bellman, Richard , year = 1966, journal =. Dynamic. 1719695 , eprinttype =

1966
[28]

Bulletin of the American Mathematical Society , volume =

The Theory of Dynamic Programming , author =. Bulletin of the American Mathematical Society , volume =. doi:10.1090/S0002-9904-1954-09848-8 , urldate =

1954 doi
[29]

Robustness in Identification and Control , author =

Robust Model Predictive Control:. Robustness in Identification and Control , author =. doi:10.1007/BFb0109870 , abstract =

doi
[30]

Multidimensional Binary Search Trees Used for Associative Searching , author =. Commun. ACM , volume =. doi:10.1145/361002.361007 , urldate =
[31]

Dynamic Programming and Optimal Control:

Bertsekas, Dimitri , year = 2017, edition =. Dynamic Programming and Optimal Control:

2017
[32]

, year = 2015, month = jul, number =

Bezanson, Jeff and Edelman, Alan and Karpinski, Stefan and Shah, Viral B. , year = 2015, month = jul, number =. Julia:. doi:10.48550/arXiv.1411.1607 , urldate =. arXiv , keywords =:1411.1607 , primaryclass =

2015 doi arXiv
[33]

Bingham, Eli and Chen, Jonathan P and Jankowiak, Martin and Obermeyer, Fritz and Pradhan, Neeraj and Karaletsos, Theofanis and Singh, Rohit and Szerlip, Paul and Horsfall, Paul and Goodman, Noah D , year = 2019, journal =. Pyro:

2019
[34]

Pattern Recognition and Machine Learning , author =
[35]

Black, Kevin and Brown, Noah and Driess, Danny and Esmail, Adnan and Equi, Michael and Finn, Chelsea and Fusai, Niccolo and Groom, Lachy and Hausman, Karol and Ichter, Brian and Jakubczak, Szymon and Jones, Tim and Ke, Liyiming and Levine, Sergey and. \. doi:10.48550/arXiv.241...

doi arXiv
[36]

, year = 2011, month = dec, journal =

Blackmore, Lars and Ono, Masahiro and Williams, Brian C. , year = 2011, month = dec, journal =. Chance-. doi:10.1109/TRO.2011.2161160 , urldate =

2011
[37]

and Jordan, Michael I

Blei, David M. and Jordan, Michael I. , year = 2006, month = mar, journal =. Variational Inference for. doi:10.1214/06-BA104 , urldate =

2006 doi
[38]

and Kucukelbir, Alp and McAuliffe, Jon D

Blei, David M. and Kucukelbir, Alp and McAuliffe, Jon D. , year = 2017, month = apr, journal =. Variational. doi:10.1080/01621459.2017.1285773 , urldate =

2017
[39]

Bayesian

Bliznyuk, Nikolay and Ruppert, David and Shoemaker, Christine and Regis, Rommel and Wild, Stefan and Mugunthan, Pradeep , year = 2008, month = jun, journal =. Bayesian. doi:10.1198/106186008X320681 , urldate =

2008 doi
[40]

Blundell, Charles and Cornebise, Julien and Kavukcuoglu, Koray and Wierstra, Daan , year = 2015, month = jun, pages =. Weight. Proceedings of the 32nd

2015
[41]

Biosystems Engineering , volume =

Minimising the Non-Working Distance Travelled by Machines Operating in a Headland Field Pattern , author =. Biosystems Engineering , volume =. doi:10.1016/j.biosystemseng.2008.06.008 , urldate =

2008 doi
[42]

Bolin, David and Lindgren, Finn , year = 2011, journal =. Spatial. 23024839 , eprinttype =

2011
[43]

Artificial Intelligence , volume =

Planning as Heuristic Search , author =. Artificial Intelligence , volume =. doi:10.1016/S0004-3702(01)00108-4 , urldate =

doi
[44]

and Williams, Christopher K

Bonilla, Edwin V and Chai, Kian Ming A. and Williams, Christopher K. I. , year = 2007, volume =. Multi-Task. Advances in

2007
[45]

The International Journal of Robotics Research , volume =

Closing the Learning-Planning Loop with Predictive State Representations , author =. The International Journal of Robotics Research , volume =. doi:10.1177/0278364911404092 , urldate =

doi
[46]

Mat\'ern

Borovitskiy, Viacheslav and Terenin, Alexander and Mostowsky, Peter and Deisenroth, Marc Peter , year = 2020, month = dec, series =. Mat\'ern. Proceedings of the 34th

2020
[47]

Trends in Cognitive Sciences , volume =

Planning as Inference , author =. Trends in Cognitive Sciences , volume =. doi:10.1016/j.tics.2012.08.006 , urldate =

2012 doi
[48]

Generating Sentences from a Continuous Space , booktitle =

Bowman, Samuel and Vilnis, Luke and Vinyals, Oriol and Dai, Andrew and Jozefowicz, Rafal and Bengio, Samy , year = 2016, pages =. Generating Sentences from a Continuous Space , booktitle =

2016
[49]

Bradbury, James and Frostig, Roy and Hawkins, Peter and Johnson, Matthew James and Leary, Chris and Maclaurin, Dougal and Necula, George and Paszke, Adam and VanderPlas, Jake and
[50]

arXiv , keywords =:2606.24039 , primaryclass =

doi:10.48550/arXiv.2606.24039 , urldate =. arXiv , keywords =:2606.24039 , primaryclass =

doi arXiv
[51]

doi:10.48550/arXiv.1606.01540 , urldate =

Brockman, Greg and Cheung, Vicki and Pettersson, Ludwig and Schneider, Jonas and Schulman, John and Tang, Jie and Zaremba, Wojciech , year = 2016, month = jun, number =. doi:10.48550/arXiv.1606.01540 , urldate =. arXiv , keywords =:1606.01540 , primaryclass =

2016 doi arXiv
[52]

van den Broek, Bart and Wiegerinck, Wim and Kappen, Hilbert , year = 2012, month = mar, number =. Risk. doi:10.48550/arXiv.1203.3523 , urldate =. arXiv , keywords =:1203.3523 , primaryclass =

2012 doi arXiv
[53]

and Dhariwal, Prafulla and Neelakantan, Arvind and Shyam, Pranav and Sastry, Girish and Askell, Amanda and Agarwal, Sandhini and

Brown, Tom and Mann, Benjamin and Ryder, Nick and Subbiah, Melanie and Kaplan, Jared D. and Dhariwal, Prafulla and Neelakantan, Arvind and Shyam, Pranav and Sastry, Girish and Askell, Amanda and Agarwal, Sandhini and. Language. Advances in Neural Information Processing Systems...
[54]

Scalable

Bruinsma, Wessel and Perim, Eric and Tebbutt, William and Hosking, Scott and Solin, Arno and Turner, Richard , year = 2020, month = nov, pages =. Scalable. Proceedings of the 37th

2020
[55]

and Kim, Chang Sub and McGregor, Simon and Seth, Anil K

Buckley, Christopher L. and Kim, Chang Sub and McGregor, Simon and Seth, Anil K. , year = 2017, month = dec, journal =. The Free Energy Principle for Action and Perception:. doi:10.1016/j.jmp.2017.09.004 , urldate =

2017 doi
[56]

and Nguyen, Cuong V

Bui, Thang D. and Nguyen, Cuong V. and Turner, Richard E. , year = 2017, month = dec, series =. Streaming Sparse. Proceedings of the 31st

2017
[57]

Finding the Outliers in Scanpath Data , booktitle =

Burch, Michael and Kumar, Ayush and Mueller, Klaus and Kervezee, Titus and Nuijten, Wouter and Oostenbach, Rens and Peeters, Lucas and Smit, Gijs , year = 2019, month = jun, series =. Finding the Outliers in Scanpath Data , booktitle =. doi:10.1145/3317958.3318225 , urldate =

2019
[58]

Exploration by

Burda, Yuri and Edwards, Harrison and Storkey, Amos and Klimov, Oleg , year = 2018, month = oct, number =. Exploration by. doi:10.48550/arXiv.1810.12894 , urldate =. arXiv , keywords =:1810.12894 , primaryclass =

2018 doi arXiv
[59]

and Lee, Daniel and Goodrich, Ben and Betancourt, Michael and Brubaker, Marcus and Guo, Jiqiang and Li, Peter and Riddell, Allen , year = 2017, journal =

Carpenter, Bob and Gelman, Andrew and Hoffman, Matthew D. and Lee, Daniel and Goodrich, Ben and Betancourt, Michael and Brubaker, Marcus and Guo, Jiqiang and Li, Peter and Riddell, Allen , year = 2017, journal =. doi:10.18637/jss.v076.i01 , urldate =

2017 doi
[60]

Lectures on

Caticha, Ariel , year = 2008, month = jul, number =. Lectures on. doi:10.48550/arXiv.0808.0012 , urldate =. arXiv , keywords =:0808.0012 , primaryclass =

2008 doi arXiv
[61]

Stochastic Versions of the Em Algorithm: An Experimental Study in the Mixture Case , shorttitle =

Celeux,. Stochastic Versions of the Em Algorithm: An Experimental Study in the Mixture Case , shorttitle =. Journal of Statistical Computation and Simulation , volume =. doi:10.1080/00949659608811772 , urldate =

doi
[62]

Branching

Champion, Th. Branching. Neural Networks , volume =. doi:10.1016/j.neunet.2022.03.036 , urldate =

2022 doi
[63]

Chang, Huiwen and Zhang, Han and Barber, Jarred and Maschinot, A. J. and Lezama, Jose and Jiang, Lu and Yang, Ming-Hsuan and Murphy, Kevin and Freeman, William T. and Rubinstein, Michael and Li, Yuanzhen and Krishnan, Dilip , year = 2023, month = jan, number =. Muse:. doi:10.4...

2023 doi arXiv
[64]

Chen, Fan and Wang, Huan and Xiong, Caiming and Mei, Song and Bai, Yu , year = 2023, month = jul, pages =. Lower. Proceedings of the 40th

2023
[65]

Physical Review E , volume =

Loop Calculus in Statistical Physics and Information Science , author =. Physical Review E , volume =. doi:10.1103/PhysRevE.73.065102 , urldate =

doi
[66]

Advances in Neural Information Processing Systems , volume =

Minigrid & Miniworld:. Advances in Neural Information Processing Systems , volume =
[67]

Diffusion Policy:

Chi, Cheng and Xu, Zhenjia and Feng, Siyuan and Cousineau, Eric and Du, Yilun and Burchfiel, Benjamin and Tedrake, Russ and Song, Shuran , year = 2025, month = sep, journal =. Diffusion Policy:. doi:10.1177/02783649241273668 , urldate =

2025 doi
[68]

International Journal of Systems Science , volume =

Every Good Regulator of a System Must Be a Model of That System , author =. International Journal of Systems Science , volume =. doi:10.1080/00207727008920220 , urldate =

doi
[69]

Temporal

Corenflos, Adrien and Zhao, Zheng and S. Temporal. 2022 25th. doi:10.23919/FUSION49751.2022.9841306 , urldate =

2022
[70]

Introduction to Algorithms , author =
[71]

doi:10.48550/arXiv.2605.08110 , urldate =

Coscia, Dario and L. doi:10.48550/arXiv.2605.08110 , urldate =. arXiv , keywords =:2605.08110 , primaryclass =

doi arXiv
[72]

International

Cox, Marco and. International
[73]

American journal of physics , volume =

Probability, Frequency and Reasonable Expectation , author =. American journal of physics , volume =
[74]

and Heskes, T

Cseke, B. and Heskes, T. , year = 2011, month = may, journal =. Properties of. doi:10.1613/jair.3195 , urldate =

2011 doi
[75]

and Ramaker, B

Cutler, Richard R. and Ramaker, B. L. , year = 1979, journal =. Dynamic

1979
[76]

Active Inference on Discrete State-Spaces:

Da Costa, Lancelot and Parr, Thomas and Sajid, Noor and Veselic, Sebastijan and Neacsu, Victorita and Friston, Karl , year = 2020, month = dec, journal =. Active Inference on Discrete State-Spaces:. doi:10.1016/j.jmp.2020.102447 , urldate =

2020
[77]

Da Costa, Lancelot and Tenka, Samuel and Zhao, Dominic and Sajid, Noor , year = 2024, month = jan, number =. Active. doi:10.48550/arXiv.2401.12917 , urldate =. arXiv , keywords =:2401.12917 , primaryclass =

2024 doi arXiv
[78]

doi:10.5194/essd-15-317-2023 , urldate =

Earth System Science Data , volume =. doi:10.5194/essd-15-317-2023 , urldate =

2023 doi
[79]

IEEE Transactions on Geoscience and Remote Sensing , volume =

Bayesian. IEEE Transactions on Geoscience and Remote Sensing , volume =. doi:10.1109/TGRS.2024.3434443 , urldate =

2024
[80]

and Gelfand, Alan E

Datta, Abhirup and Banerjee, Sudipto and Finley, Andrew O. and Gelfand, Alan E. , year = 2016, month = apr, journal =. Hierarchical. doi:10.1080/01621459.2015.1044091 , urldate =

2016
[81]

Differentiable Expected Hypervolume Improvement for Parallel Multi-Objective

Daulton, Samuel and Balandat, Maximilian and Bakshy, Eytan , year = 2020, month = dec, series =. Differentiable Expected Hypervolume Improvement for Parallel Multi-Objective. Proceedings of the 34th

2020
[82]

Parallel

Daulton, Samuel and Balandat, Maximilian and Bakshy, Eytan , year = 2021, month = dec, series =. Parallel. Proceedings of the 35th

2021
[83]

, year = 2007, month = jun, pages =

Dauwels, J. , year = 2007, month = jun, pages =. On. doi:10.1109/ISIT.2007.4557602 , abstract =

2007
[84]

doi:10.1109/MMAR.2012.6347921 , urldate =

Consistent Control Hierarchies with Top Layers Represented by Timed Event Graphs , booktitle =. doi:10.1109/MMAR.2012.6347921 , urldate =

2012
[85]

Theory of Probability:

De Finetti, Bruno , year = 1974, publisher =. Theory of Probability:

1974
[86]

Entropy , volume =

Spatial and. Entropy , volume =. doi:10.3390/e26010083 , urldate =

doi
[87]

De Vries, Bert , year = 2026, month = mar, number =. Active. doi:10.48550/arXiv.2603.20927 , urldate =. arXiv , keywords =:2603.20927 , primaryclass =

2026 doi
[88]

Expected

De Vries, Bert and Nuijten, Wouter and. Expected. doi:10.48550/arXiv.2504.14898 , urldate =. arXiv , keywords =:2504.14898 , primaryclass =

doi
[89]

Journal of neural engineering , volume =

The Neural Optimal Control Hierarchy for Motor Control , author =. Journal of neural engineering , volume =
[90]

and Langmore, Ian and Tran, Dustin and Brevdo, Eugene and Vasudevan, Srinivas and Moore, Dave and Patton, Brian and Alemi, Alex and Hoffman, Matt and Saurous, Rif A

Dillon, Joshua V. and Langmore, Ian and Tran, Dustin and Brevdo, Eugene and Vasudevan, Srinivas and Moore, Dave and Patton, Brian and Alemi, Alex and Hoffman, Matt and Saurous, Rif A. , year = 2017, month = nov, number =. doi:10.48550/arXiv.1711.10604 , urldate =. arXiv , keyw...

2017 doi arXiv
[91]

IJCAI : proceedings of the conference , volume =

Hidden. IJCAI : proceedings of the conference , volume =
[92]

Stochastic

Dritsas, Ioannis , year = 2011, month = feb, publisher =. Stochastic

2011
[93]

and Del Bello, U

Drusch, M. and Del Bello, U. and Carlier, S. and Colin, O. and Fernandez, V. and Gascon, F. and Hoersch, B. and Isola, C. and Laberinti, P. and Martimort, P. and Meygret, A. and Spoto, F. and Sy, O. and Marchese, F. and Bargellini, P. , year = 2012, month = may, journal =. Sen...

2012 doi
[94]

and Sutskever, Ilya and Abbeel, Pieter , year = 2016, month = nov, number =

Duan, Yan and Schulman, John and Chen, Xi and Bartlett, Peter L. and Sutskever, Ilya and Abbeel, Pieter , year = 2016, month = nov, number =. doi:10.48550/arXiv.1611.02779 , urldate =. arXiv , keywords =:1611.02779 , primaryclass =

2016 doi arXiv
[95]

Duane, Simon and Kennedy, A. D. and Pendleton, Brian J. and Roweth, Duncan , year = 1987, month = sep, journal =. Hybrid. doi:10.1016/0370-2693(87)91197-X , urldate =

1987 doi
[96]

Optimal Learning:

Duff, Michael O'Gordon , year = 2002, school =. Optimal Learning:

2002
[97]

Durrande, Nicolas and Adam, Vincent and Bordeaux, Lucas and Eleftheriadis, Stefanos and Hensman, James , year = 2019, month = apr, pages =. Banded. Proceedings of the

2019
[98]

and Choo, Xuan and Bekolay, Trevor and DeWolf, Travis and Tang, Yichuan and Rasmussen, Daniel , year = 2012, month = nov, journal =

Eliasmith, Chris and Stewart, Terrence C. and Choo, Xuan and Bekolay, Trevor and DeWolf, Travis and Tang, Yichuan and Rasmussen, Daniel , year = 2012, month = nov, journal =. A. doi:10.1126/science.1225266 , urldate =

2012 doi
[99]

doi:10.48550/arXiv.2112.10751 , urldate =

Emmons, Scott and Eysenbach, Benjamin and Kostrikov, Ilya and Levine, Sergey , year = 2022, month = may, number =. doi:10.48550/arXiv.2112.10751 , urldate =. arXiv , keywords =:2112.10751 , primaryclass =

2022 doi arXiv
[100]

Resolving Uncertainty on the Fly:

Engstr. Resolving Uncertainty on the Fly:. doi:10.48550/arXiv.2311.06417 , urldate =. arXiv , keywords =:2311.06417 , primaryclass =

doi arXiv
[101]

Scalable Global Optimization via Local

Eriksson, David and Pearce, Michael and Gardner, Jacob R and Turner, Ryan and Poloczek, Matthias , year = 2019, month = dec, number =. Scalable Global Optimization via Local. Proceedings of the 33rd

2019
[102]

Dynamical

Esaki, Kanako and Matsumura, Tadayuki and Minusa, Shunsuke and Shao, Yang and Yoshimura, Chihiro and Mizuno, Hiroyuki , editor =. Dynamical. Active. doi:10.1007/978-3-031-47958-8_2 , abstract =

doi
[103]

Hierarchical

Findeisen, W. Hierarchical
[104]

and Datta, Abhirup and Cook, Bruce D

Finley, Andrew O. and Datta, Abhirup and Cook, Bruce D. and Morton, Douglas C. and Andersen, Hans E. and Banerjee, Sudipto , year = 2019, month = apr, journal =. Efficient. doi:10.1080/10618600.2018.1537924 , urldate =

2019
[105]

Annealing in Variational Inference Mitigates Mode Collapse:

Fogliani, Luigi and Loureiro, Bruno and Gabri. Annealing in Variational Inference Mitigates Mode Collapse:. doi:10.48550/arXiv.2602.12923 , urldate =. arXiv , keywords =:2602.12923 , primaryclass =

doi
[106]

David , year = 2001, journal =

Forney, G. David , year = 2001, journal =. Codes on Graphs:

2001
[107]

Deep Active Inference Agents Using

Fountas, Zafeirios and Sajid, Noor and Mediano, Pedro and Friston, Karl , year = 2020, volume =. Deep Active Inference Agents Using. Advances in

2020
[108]

and Sudderth, Erik B

Fox, Emily B. and Sudderth, Erik B. and Jordan, Michael I. and Willsky, Alan S. , year = 2011, journal =. A. 23024915 , eprinttype =

2011
[109]

, year = 2018, month = jul, number =

Frazier, Peter I. , year = 2018, month = jul, number =. A. doi:10.48550/arXiv.1807.02811 , urldate =. arXiv , keywords =:1807.02811 , primaryclass =

2018 doi arXiv
[110]

and Daunizeau, Jean and Kilner, James and Kiebel, Stefan J

Friston, Karl J. and Daunizeau, Jean and Kilner, James and Kiebel, Stefan J. , year = 2010, month = mar, journal =. Action and Behavior: A Free-Energy Formulation , shorttitle =. doi:10.1007/s00422-010-0364-z , urldate =

2010 doi
[111]

Cognitive Neuroscience , volume =

Active Inference and Epistemic Value , author =. Cognitive Neuroscience , volume =. doi:10.1080/17588928.2015.1020053 , urldate =

2015
[112]

Neuroscience & Biobehavioral Reviews , volume =

Active Inference and Learning , author =. Neuroscience & Biobehavioral Reviews , volume =. doi:10.1016/j.neubiorev.2016.06.022 , urldate =

2016 doi
[113]

Friston, Karl and FitzGerald, Thomas and Rigoli, Francesco and Schwartenbeck, Philipp and Pezzulo, Giovanni , year = 2017, month = jan, journal =. Active. doi:10.1162/NECO_a_00912 , urldate =

2017 doi
[114]

and Lin, Marco and Frith, Christopher D

Friston, Karl J. and Lin, Marco and Frith, Christopher D. and Pezzulo, Giovanni and Hobson, J. Allan and Ondobaka, Sasha , year = 2017, month = oct, journal =. Active. doi:10.1162/neco_a_00999 , urldate =

2017 doi
[115]

Friston, Karl J. and Salvatori, Tommaso and Isomura, Takuya and Tschantz, Alexander and Kiefer, Alex and Verbelen, Tim and Koudahl, Magnus and Paul, Aswin and Parr, Thomas and Razi, Adeel and Kagan, Brett J. and Buckley, Christopher L. and Ramstead, Maxwell J. D. , year = 2025...

2025 doi
[116]

arXiv:1805.07092 [stat] , eprint =

Bayesian Model Reduction , author =. arXiv:1805.07092 [stat] , eprint =

arXiv
[117]

and Ramstead, Maxwell J

Friston, Karl J. and Ramstead, Maxwell J. D. and Kiefer, Alex B. and Tschantz, Alexander and Buckley, Christopher L. and Albarracin, Mahault and Pitliya, Riddhi J. and Heins, Conor and Klein, Brennan and Millidge, Beren and Sakthivadivel, Dalton A. R. and Smithe, Toby St Clere...

doi arXiv
[118]

Collective Intelligence , volume =

Designing Ecosystems of Intelligence from First Principles , author =. Collective Intelligence , volume =
[119]

Neuroscience & Biobehavioral Reviews , volume =

Federated Inference and Belief Sharing , author =. Neuroscience & Biobehavioral Reviews , volume =. doi:10.1016/j.neubiorev.2023.105500 , urldate =

2023
[120]

doi:10.48550/arXiv.1906.10184 , urldate =

A Free Energy Principle for a Particular Physics , author =. doi:10.48550/arXiv.1906.10184 , urldate =. arXiv , keywords =:1906.10184 , primaryclass =

1906 doi arXiv
[121]

The Free-Energy Principle: A Unified Brain Theory? , shorttitle =

Friston, Karl , year = 2010, month = feb, journal =. The Free-Energy Principle: A Unified Brain Theory? , shorttitle =. doi:10.1038/nrn2787 , urldate =

2010 doi
[122]

Physics of Life Reviews , volume =

Path Integrals, Particular Kinds, and Strange Things , author =. Physics of Life Reviews , volume =. doi:10.1016/j.plrev.2023.08.016 , urldate =

2023 doi
[123]

and Daunizeau, Jean and Kiebel, Stefan J

Friston, Karl J. and Daunizeau, Jean and Kiebel, Stefan J. , year = 2009, month = jul, journal =. Reinforcement. doi:10.1371/journal.pone.0006421 , urldate =

2009 doi
[124]

Sophisticated

Friston, Karl and Da Costa, Lancelot and Hafner, Danijar and Hesp, Casper and Parr, Thomas , year = 2021, month = mar, journal =. Sophisticated. doi:10.1162/neco_a_01351 , urldate =

2021 doi
[125]

Friston, K. J. and. NeuroImage , volume =. doi:10.1016/j.neuroimage.2008.02.054 , abstract =

2008 doi
[126]

Friston, Karl , year = 2011, month = nov, journal =. What. doi:10.1016/j.neuron.2011.10.018 , urldate =

2011 doi
[127]

Bayesian

Garnett, Roman , year = 2023, publisher =. Bayesian. doi:10.1017/9781108348973 , urldate =

2023 doi
[128]

Ge, Hong and Xu, Kai and Ghahramani, Zoubin , year = 2018, month = mar, pages =. Turing:. Proceedings of the

2018
[129]

A Concise Introduction to Models and Methods for Automated Planning , author =
[130]

Model-Free, Model-Based, and General Intelligence , booktitle =

Geffner, Hector , year = 2018, pages =. Model-Free, Model-Based, and General Intelligence , booktitle =

2018
[131]

Gelman, Andrew and Lee, Daniel and Guo, Jiqiang , year = 2015, month = oct, journal =. Stan:. doi:10.3102/1076998615606113 , urldate =

2015 doi
[132]

Bayesian

Ghavamzadeh, Mohammad and Mannor, Shie and Pineau, Joelle and Tamar, Aviv , year = 2015, month = nov, journal =. Bayesian. doi:10.1561/2200000049 , urldate =

2015 doi
[133]

Nonseparable,

Gneiting, Tilmann , year = 2002, month = jun, journal =. Nonseparable,. doi:10.1198/016214502760047113 , urldate =

2002 doi
[134]

Strictly

Gneiting, Tilmann and Raftery, Adrian E , year = 2007, month = mar, journal =. Strictly. doi:10.1198/016214506000001437 , urldate =

2007 doi
[135]

Regression with

Goldberg, Paul and Williams, Christopher and Bishop, Christopher , year = 1997, volume =. Regression with. Advances in

1997
[136]

and Bonawitz, Keith and Tenenbaum, Joshua B

Goodman, Noah and Mansinghka, Vikash and Roy, Daniel M. and Bonawitz, Keith and Tenenbaum, Joshua B. , year = 2014, month = jul, number =. Church: A Language for Generative Models , shorttitle =. doi:10.48550/arXiv.1206.3255 , urldate =. arXiv , keywords =:1206.3255 , primaryclass =

2014 doi arXiv
[137]

Human movement science , volume =

Evidence for a Distributed Hierarchy of Action Representation in the Brain , author =. Human movement science , volume =. doi:10.1016/j.humov.2007.05.009 , urldate =

2007 doi
[138]

Practical

Graves, Alex , year = 2011, volume =. Practical. Advances in

2011
[139]

Achieving

Griffiths, Ryan-Rhys and Aldrick, Alexander and. Achieving. Machine Learning: Science and Technology , volume =. doi:10.1088/2632-2153/ac298c , abstract =

doi
[140]

and Chater, Nick and Kemp, Charles and Perfors, Amy and Tenenbaum, Joshua B

Griffiths, Thomas L. and Chater, Nick and Kemp, Charles and Perfors, Amy and Tenenbaum, Joshua B. , year = 2010, month = aug, journal =. Probabilistic Models of Cognition: Exploring Representations and Inductive Biases , shorttitle =. doi:10.1016/j.tics.2010.05.004 , urldate =

2010 doi
[141]

Efficient

Guez, Arthur and Silver, David and Dayan, Peter , year = 2012, volume =. Efficient. Advances in

2012
[142]

Nonlinear

Gultekin, San and Kitts, Brendan and Flores, Aaron and Paisley, John , year = 2023, month = mar, number =. Nonlinear. doi:10.48550/arXiv.2303.04450 , urldate =. arXiv , keywords =:2303.04450 , primaryclass =

2023 doi arXiv
[143]

arXiv preprint arXiv:1803.10122 , eprint =

World Models , author =. arXiv preprint arXiv:1803.10122 , eprint =

arXiv
[144]

Reinforcement

Haarnoja, Tuomas and Tang, Haoran and Abbeel, Pieter and Levine, Sergey , year = 2017, month = jul, pages =. Reinforcement. Proceedings of the 34th

2017
[145]

Haarnoja, Tuomas and Zhou, Aurick and Abbeel, Pieter and Levine, Sergey , year = 2018, month = jul, pages =. Soft. Proceedings of the 35th

2018
[146]

Dream to

Hafner, Danijar and Lillicrap, Timothy and Ba, Jimmy and Norouzi, Mohammad , year = 2020, month = mar, number =. Dream to. doi:10.48550/arXiv.1912.01603 , urldate =. arXiv , keywords =:1912.01603 , primaryclass =

2020 doi arXiv
[147]

Learning

Hafner, Danijar and Lillicrap, Timothy and Fischer, Ian and Villegas, Ruben and Ha, David and Lee, Honglak and Davidson, James , year = 2019, month = may, pages =. Learning. Proceedings of the 36th

2019
[148]

Hamelijnck, Oliver and Wilkinson, William and Loppi, Niki and Solin, Arno and Damoulas, Theodoros , year = 2021, volume =. Spatio-. Advances in

2021
[149]

Unsupervised

Han, Zhao-Yu and Wang, Jun and Fan, Heng and Wang, Lei and Zhang, Pan , year = 2018, month = jul, journal =. Unsupervised. doi:10.1103/PhysRevX.8.031012 , urldate =

2018 doi
[150]

Kalman Filtering and Smoothing Solutions to Temporal

Hartikainen, Jouni and S. Kalman Filtering and Smoothing Solutions to Temporal. 2010. doi:10.1109/MLSP.2010.5589113 , urldate =

2010
[151]

Learning to

Heess, Nicolas and Tarlow, Daniel and Winn, John , year = 2013, volume =. Learning to. Advances in

2013
[152]

and Tschantz, Alexander , year = 2022, month = may, journal =

Heins, Conor and Millidge, Beren and Demekas, Daphne and Klein, Brennan and Friston, Karl and Couzin, Iain D. and Tschantz, Alexander , year = 2022, month = may, journal =. Pymdp:. doi:10.21105/joss.04098 , urldate =

2022 doi
[153]

Spin Glass Systems as Collective Active Inference , booktitle =

Heins, Conor and Klein, Brennan and Demekas, Daphne and Aguilera, Miguel and Buckley, Christopher L , year = 2022, pages =. Spin Glass Systems as Collective Active Inference , booktitle =

2022
[154]

Computers & Chemical Engineering , series =

Stochastic Model Predictive Control --- How Does It Work? , author =. Computers & Chemical Engineering , series =. doi:10.1016/j.compchemeng.2017.10.026 , urldate =

2017 doi
[155]

, author =

Entropy Search for Information-Efficient Global Optimization. , author =. Journal of Machine Learning Research , volume =
[156]

Proceedings of

Predictive. Proceedings of
[157]

, year = 2006, month = jun, journal =

Heskes, T. , year = 2006, month = jun, journal =. Convexity. doi:10.1613/jair.1933 , urldate =

2006 doi
[158]

Hesp, Casper and Ramstead, Maxwell and Constant, Axel and Badcock, Paul and Kirchhoff, Michael and Friston, Karl , editor =. A. Evolution,. doi:10.1007/978-3-030-00075-2_7 , abstract =

doi
[159]

Session 8 Formalisms for Artificial Intelligence a Universal Modular Actor Formalism for Artificial Intelligence , booktitle =

Hewitt, Carl and Bishop, Peter and Steiger, Richard , year = 1973, volume =. Session 8 Formalisms for Artificial Intelligence a Universal Modular Actor Formalism for Artificial Intelligence , booktitle =

1973
[160]

IEEE Open Journal of Signal Processing , pages =

A. IEEE Open Journal of Signal Processing , pages =. doi:10.1109/OJSP.2026.3695788 , urldate =

2026
[161]

and Gelman, Andrew , year = 2011, month = nov, number =

Hoffman, Matthew D. and Gelman, Andrew , year = 2011, month = nov, number =. The. doi:10.48550/arXiv.1111.4246 , urldate =. arXiv , keywords =:1111.4246 , primaryclass =

2011 doi arXiv
[162]

Hoffman, Matthew D and Gelman, Andrew , year = 2014, journal =. The

2014
[163]

Matrix Analysis , author =
[164]

Advances in

Houthooft, Rein and Chen, Xi and Chen, Xi and Duan, Yan and Schulman, John and De Turck, Filip and Abbeel, Pieter , year = 2016, volume =. Advances in

2016
[165]

and Zhang, Tong , year = 2012, month = sep, journal =

Hsu, Daniel and Kakade, Sham M. and Zhang, Tong , year = 2012, month = sep, journal =. A Spectral Algorithm for Learning. doi:10.1016/j.jcss.2011.12.025 , urldate =

2012 doi
[166]

P and Gao, X and Ferreira, L

Huete, A and Didan, K and Miura, T and Rodriguez, E. P and Gao, X and Ferreira, L. G , year = 2002, month = nov, journal =. Overview of the Radiometric and Biophysical Performance of the. doi:10.1016/S0034-4257(02)00096-2 , urldate =

2002 doi
[167]

Memoized

Hughes, Michael C and Sudderth, Erik , year = 2013, volume =. Memoized. Advances in

2013
[168]

Scalable

Hughes, Michael C and Stephenson, William T and Sudderth, Erik , year = 2015, volume =. Scalable. Advances in

2015
[169]

Intelligence, Physical and Amin, Ali and Aniceto, Raichelle and Balakrishna, Ashwin and Black, Kevin and Conley, Ken and Connors, Grace and Darpinian, James and Dhabalia, Karan and DiCarlo, Jared and Driess, Danny and Equi, Michael and Esmail, Adnan and Fang, Yunhao and Finn, ...

doi arXiv
[170]

Nature Communications , volume =

Experimental Validation of the Free-Energy Principle with in Vitro Neural Networks , author =. Nature Communications , volume =. doi:10.1038/s41467-023-40141-z , urldate =

doi
[171]

Kullback--

Ito, Kaito and Kashima, Kenji , year = 2022, month = jun, journal =. Kullback--. doi:10.1080/18824889.2022.2095827 , urldate =

2022
[172]

Risk-Sensitive Control as Inference with

Ito, Kaito and Kashima, Kenji , year = 2024, month = nov, number =. Risk-Sensitive Control as Inference with. doi:10.48550/arXiv.2411.01827 , urldate =. arXiv , keywords =:2411.01827 , primaryclass =

2024 doi arXiv
[173]

Vision Research , series =

Bayesian Surprise Attracts Human Attention , author =. Vision Research , series =. doi:10.1016/j.visres.2008.09.007 , urldate =

2008 doi
[174]

Neural computation , volume =

Observable Operator Models for Discrete Stochastic Time Series , author =. Neural computation , volume =
[175]

Categorical Reparameterization with Gumbel-Softmax , booktitle =
[176]

doi:10.48550/arXiv.2006.01959 , urldate =

Jaques, Miguel and Burke, Michael and Hospedales, Timothy , year = 2021, month = apr, number =. doi:10.48550/arXiv.2006.01959 , urldate =. arXiv , keywords =:2006.01959 , primaryclass =

2021 doi arXiv
[177]

Jaynes, E. T. , year = 2003, month = apr, edition =. Probability. doi:10.1017/CBO9780511790423 , urldate =

2003 doi
[178]

Jin, Xuanjin and Zeng, Chendong and Zhu, Shengfa and Liu, Chunxiao and Cai, Panpan , year = 2025, month = nov, journal =. Hi-. doi:10.1109/LRA.2025.3619747 , urldate =

2025
[179]

Optimal Coverage Path Planning for Arable Farming on

Jin, Jian and Tang, Lie , year = 2010, journal =. Optimal Coverage Path Planning for Arable Farming on

2010
[180]

Jin, Chi and Kakade, Sham and Krishnamurthy, Akshay and Liu, Qinghua , year = 2020, volume =. Sample-. Advances in

2020
[181]

Backpropagation , author =

Forward Models:. Backpropagation , author =
[182]

and Ghahramani, Zoubin and Jaakkola, Tommi S

Jordan, Michael I. and Ghahramani, Zoubin and Jaakkola, Tommi S. and Saul, Lawrence K. , editor =. An. Learning in. doi:10.1007/978-94-011-5014-9_5 , urldate =

doi
[183]

Machine learning , volume =

An Introduction to Variational Methods for Graphical Models , author =. Machine learning , volume =
[184]

Artificial Intelligence , volume =

Planning and Acting in Partially Observable Stochastic Domains , author =. Artificial Intelligence , volume =. doi:10.1016/S0004-3702(98)00023-X , urldate =

doi
[185]

A New Approach to Linear Filtering and Prediction Problems , author =
[186]

Biological Cybernetics , volume =

Planning and Navigation as Active Inference , author =. Biological Cybernetics , volume =. doi:10.1007/s00422-018-0753-2 , urldate =

doi
[187]

Machine Learning , volume =

Optimal Control as a Graphical Model Inference Problem , author =. Machine Learning , volume =. doi:10.1007/s10994-012-5278-7 , urldate =

doi
[188]

Journal of Statistical Mechanics: Theory and Experiment , volume =

Path Integrals and Symmetry Breaking for Optimal Control Theory , author =. Journal of Statistical Mechanics: Theory and Experiment , volume =. doi:10.1088/1742-5468/2005/11/P11011 , urldate =

2005 doi
[189]

and Watanabe, K

Katahira, K. and Watanabe, K. and Okada, M. , year = 2008, month = jan, journal =. Deterministic Annealing Variant of Variational. doi:10.1088/1742-6596/95/1/012015 , urldate =

2008 doi
[190]

Proceedings of the 21st

Katt, Sammie and Nguyen, Hai and Oliehoek, Frans and Amato, Christopher , year = 2022, pages =. Proceedings of the 21st

2022
[191]

Bayesian

Katt, Sammie and Oliehoek, Frans and Amato, Christopher , year = 2018, month = nov, number =. Bayesian. doi:10.48550/arXiv.1811.05612 , urldate =. arXiv , keywords =:1811.05612 , primaryclass =

2018 doi arXiv
[192]

and Amato, Christopher , year = 2017, month = jul, pages =

Katt, Sammie and Oliehoek, Frans A. and Amato, Christopher , year = 2017, month = jul, pages =. Learning in. Proceedings of the 34th

2017
[193]

Katzfuss, Matthias and Guinness, Joseph , year = 2021, journal =. A. 26997951 , eprinttype =

2021
[194]

Entropy , volume =

An Active Inference Model of Collective Intelligence , author =. Entropy , volume =. doi:10.3390/e23070830 , urldate =. arXiv , keywords =:2104.01066 , primaryclass =

arXiv
[195]

International Journal of Applied Earth Observation and Geoinformation , volume =

Average Variograms to Guide Soil Sampling , author =. International Journal of Applied Earth Observation and Geoinformation , volume =. doi:10.1016/j.jag.2004.07.005 , urldate =

2004 doi
[196]

Kingma, Diederik and Ba, Jimmy , year = 2015, address =. Adam:. International

2015
[197]

and Welling, Max , year = 2022, month = dec, number =

Kingma, Diederik P. and Welling, Max , year = 2022, month = dec, number =. Auto-. doi:10.48550/arXiv.1312.6114 , urldate =. arXiv , keywords =:1312.6114 , primaryclass =

2022 doi arXiv
[198]

Improved

Kingma, Durk P and Salimans, Tim and Jozefowicz, Rafal and Chen, Xi and Sutskever, Ilya and Welling, Max , year = 2016, volume =. Improved. Advances in

2016
[199]

Kirchhoff, Michael and Parr, Thomas and Palacios, Ensor and Friston, Karl and Kiverstein, Julian , year = 2018, month = jan, journal =. The. doi:10.1098/rsif.2017.0792 , urldate =

2018
[200]

and Pouget, Alexandre , year = 2004, month = dec, journal =

Knill, David C. and Pouget, Alexandre , year = 2004, month = dec, journal =. The. doi:10.1016/j.tins.2004.10.007 , urldate =

2004 doi
[201]

, year = 2006, month = feb, journal =

Knowles, J. , year = 2006, month = feb, journal =. doi:10.1109/TEVC.2005.851274 , urldate =

2006
[202]

Biological Cybernetics , volume =

The Structure of Images , author =. Biological Cybernetics , volume =. doi:10.1007/BF00336961 , urldate =

doi
[203]

Probabilistic

Koller, Daphne and Friedman, Nir , year = 2009, month = jul, publisher =. Probabilistic

2009
[204]

A Factor Graph Approach to Signal Modelling, System Identification and Filtering , author =
[205]

Kostrikov, Ilya and Nair, Ashvin and Levine, Sergey , year = 2021, month = oct, number =. Offline. doi:10.48550/arXiv.2110.06169 , urldate =. arXiv , keywords =:2110.06169 , primaryclass =

2021 doi arXiv
[206]

Realising

Koudahl, Magnus and. Realising. doi:10.48550/arXiv.2306.08014 , urldate =. arXiv , keywords =:2306.08014 , primaryclass =

doi arXiv
[207]

, year = 2025, month = aug, pages =

Kouw, Wouter M. , year = 2025, month = aug, pages =. Bayesian Autoregression to Optimize Temporal. Proceedings of the

2025
[208]

and Nisslbeck, Tim N

Kouw, Wouter M. and Nisslbeck, Tim N. and Nuijten, Wouter W. L. , editor =. Message. Active. doi:10.1007/978-3-032-16955-6_16 , abstract =

doi
[209]

Sensors , volume =

Kragh, Mikkel Fly and Christiansen, Peter and Laursen, Morten Stigaard and Larsen, Morten and Steen, Kim Arild and Green, Ole and Karstoft, Henrik and J. Sensors , volume =. doi:10.3390/s17112579 , urldate =

doi
[210]

Probabilistic Programming:

Krapu, Christopher and Borsuk, Mark , year = 2019, month = apr, journal =. Probabilistic Programming:. doi:10.1016/j.envsoft.2019.01.014 , urldate =

2019 doi
[211]

IEEE Transactions on information theory , volume =

Factor Graphs and the Sum-Product Algorithm , author =. IEEE Transactions on information theory , volume =. doi:10.1109/18.910572 , urldate =

doi
[212]

and Leibler, R

Kullback, S. and Leibler, R. A. , year = 1951, month = mar, journal =. On. doi:10.1214/aoms/1177729694 , urldate =

1951
[213]

Kurach, Karol and Raichuk, Anton and Sta. Google. doi:10.48550/arXiv.1907.11180 , urldate =. arXiv , keywords =:1907.11180 , primaryclass =

1907 doi arXiv
[214]

Kurniawati, Hanna and Hsu, David and Lee, Wee Sun and others , year = 2008, volume =. Sarsop:. Robotics:

2008
[215]

Advances in

Kwon, Jeongyeol and Efroni, Yonathan and Caramanis, Constantine and Mannor, Shie , year = 2021, volume =. Advances in

2021
[216]

Behavioral and Brain Sciences , volume =

Building Machines That Learn and Think like People , author =. Behavioral and Brain Sciences , volume =. doi:10.1017/S0140525X16001837 , urldate =

doi
[217]

Simple and

Lakshminarayanan, Balaji and Pritzel, Alexander and Blundell, Charles , year = 2017, volume =. Simple and. Advances in

2017
[218]

Lanczos, Cornelius , year = 1970, publisher =. The

1970
[219]

Planning Algorithms , author =
[220]

doi:10.52202/079017-3705 , file =

What Type of Inference Is Planning? , booktitle =. doi:10.52202/079017-3705 , file =

doi
[221]

Automatica , volume =

State-Space Interpretation of Model Predictive Control , author =. Automatica , volume =. doi:10.1016/0005-1098(94)90159-7 , urldate =

doi
[222]

, year = 1966, month = jun, journal =

Lefkowitz, I. , year = 1966, month = jun, journal =. Multilevel. doi:10.1115/1.3645868 , urldate =

1966 doi
[223]

Reinforcement

Levine, Sergey , year = 2018, month = may, number =. Reinforcement. doi:10.48550/arXiv.1805.00909 , urldate =. arXiv , keywords =:1805.00909 , primaryclass =

2018 doi arXiv
[224]

Computers & Chemical Engineering , series =

Chance Constrained Programming Approach to Process Optimization under Uncertainty , author =. Computers & Chemical Engineering , series =. doi:10.1016/j.compchemeng.2007.05.009 , urldate =

2007 doi
[225]

2407.00397 , primaryclass =

Learning Time-Varying Multi-Region Brain Communications via Scalable Markovian Gaussian Processes , author =. 2407.00397 , primaryclass =

arXiv
[226]

Lin, Wu and Khan, Mohammad Emtiyaz and Schmidt, Mark , year = 2019, month = oct, number =. Stein's. doi:10.48550/arXiv.1910.13398 , urldate =. arXiv , keywords =:1910.13398 , primaryclass =

2019 doi arXiv
[227]

An Explicit Link between

Lindgren, Finn and Rue, H. An Explicit Link between. Journal of the Royal Statistical Society: Series B (Statistical Methodology) , volume =. doi:10.1111/j.1467-9868.2011.00777.x , urldate =

2011
[228]

Lindley, D. V. , year = 1956, month = dec, journal =. On a. doi:10.1214/aoms/1177728069 , urldate =

1956
[229]

Predictive

Littman, Michael and Sutton, Richard S , year = 2001, volume =. Predictive. Advances in

2001
[230]

Emergent

Liu, Siqi and Lever, Guy and Merel, Josh and Tunyasuvunakool, Saran and Heess, Nicolas and Graepel, Thore , year = 2019, month = feb, number =. Emergent. doi:10.48550/arXiv.1902.07151 , urldate =. arXiv , keywords =:1902.07151 , primaryclass =

2019 doi arXiv
[231]

Liu, Qinyuan and Wang, Zidong and He, Xiao and Zhou, D. H. , year = 2015, month = sep, journal =. Event-. doi:10.1109/TAC.2015.2390554 , urldate =

2015
[232]

Hierarchical Optimal Control of a 7-

Liu, Dan and Todorov, Emanuel , year = 2009, pages =. Hierarchical Optimal Control of a 7-. 2009

2009
[233]

Liu, Qinghua and Chung, Alan and Szepesvari, Csaba and Jin, Chi , year = 2022, month = jun, pages =. When. Proceedings of

2022
[234]

, year = 2007, month = jun, journal =

Loeliger, Hans-Andrea and Dauwels, Justin and Hu, Junli and Korl, Sascha and Ping, Li and Kschischang, Frank R. , year = 2007, month = jun, journal =. The. doi:10.1109/JPROC.2007.896497 , urldate =

2007
[235]

IEEE Signal Processing Magazine , volume =

An Introduction to Factor Graphs , author =. IEEE Signal Processing Magazine , volume =. doi:10.1109/MSP.2004.1267047 , abstract =

2004 arXiv
[236]

Reinforcement

Lu, Xiuyuan and Van Roy, Benjamin and Dwaracherla, Vikranth and Ibrahimi, Morteza and Osband, Ian and Wen, Zheng , year = 2023, month = jul, journal =. Reinforcement. doi:10.1561/2200000097 , urldate =

2023 doi
[237]

Advances in Neural Information Processing Systems , author =

The. Advances in Neural Information Processing Systems , author =
[238]

Lukashchuk, Mykola and Nuijten, Wouter W. L. and Bagaev, Dmitry and. Riemannian Black Box Variational Inference , booktitle =
[239]

and Thomas, Andrew and Best, Nicky and Spiegelhalter, David , year = 2000, month = oct, journal =

Lunn, David J. and Thomas, Andrew and Best, Nicky and Spiegelhalter, David , year = 2000, month = oct, journal =. doi:10.1023/A:1008929526011 , urldate =

2000 doi
[240]

Luttinen, Jaakko , year = 2016, month = jan, journal =

2016
[241]

MacKay, David J. C. , year = 2003, month = sep, publisher =. Information

2003
[242]

MacKay, David J. C. , year = 1992, month = jul, journal =. Information-. doi:10.1162/neco.1992.4.4.590 , urldate =

1992 doi
[243]

The Concrete Distribution:

Maddison, C and Mnih, A and Teh, Y , year = 2017, publisher =. The Concrete Distribution:. Proceedings of the International Conference on Learning

2017
[244]

Bayesian

Maddox, Wesley J and Balandat, Maximilian and Wilson, Andrew G and Bakshy, Eytan , year = 2021, volume =. Bayesian. Advances in

2021
[245]

and Steppe, Kathy , year = 2019, month = feb, journal =

Maes, Wouter H. and Steppe, Kathy , year = 2019, month = feb, journal =. Perspectives for. doi:10.1016/j.tplants.2018.11.007 , urldate =

2019 doi
[246]

, year = 2021, journal =

Maestrini, Luca and Wand, Matt P. , year = 2021, journal =. The. doi:10.1111/anzs.12339 , urldate =

2021 doi
[247]

Risk-Averse Heteroscedastic

Makarova, Anastasiia and Usmanova, Ilnura and Bogunovic, Ilija and Krause, Andreas , year = 2021, month = dec, series =. Risk-Averse Heteroscedastic. Proceedings of the 35th

2021
[248]

Variational

Mandt, Stephan and McInerney, James and Abrol, Farhan and Ranganath, Rajesh and Blei, David , year = 2016, month = may, pages =. Variational. Proceedings of the 19th

2016
[249]

Venture: A Higher-Order Probabilistic Programming Platform with Programmable Inference , shorttitle =

Mansinghka, Vikash and Selsam, Daniel and Perov, Yura , year = 2014, month = mar, number =. Venture: A Higher-Order Probabilistic Programming Platform with Programmable Inference , shorttitle =. doi:10.48550/arXiv.1404.0099 , urldate =. arXiv , keywords =:1404.0099 , primaryclass =

2014 doi arXiv
[250]

, year = 2003, publisher =

Martin, Robert C. , year = 2003, publisher =. Agile

2003
[251]

, year = 2012, eprint =

Mathys, Christoph D. , year = 2012, eprint =. Hierarchical. doi:10.3929/ETHZ-A-007595146 , urldate =

2012 doi
[252]

and Lomakina, Ekaterina I

Mathys, Christoph D. and Lomakina, Ekaterina I. and Daunizeau, Jean and Iglesias, Sandra and Brodersen, Kay H. and Friston, Karl J. and Stephan, Klaas E. , year = 2014, month = nov, journal =. Uncertainty in Perception and the. doi:10.3389/fnhum.2014.00825 , urldate =

2014
[253]

Geoderma , volume =

On Digital Soil Mapping , author =. Geoderma , volume =. doi:10.1016/S0016-7061(03)00223-4 , urldate =

doi
[254]

, year = 1998, pages =

McGovern, Amy and Precup, Doina and Ravindran, Balaraman and Singh, Satinder and Sutton, Richard S. , year = 1998, pages =. Hierarchical Optimal Control of. Proceedings of the

1998
[255]

Stochastic

Mesbah, Ali , year = 2016, month = dec, journal =. Stochastic. doi:10.1109/MCS.2016.2602087 , urldate =

2016
[256]

arXiv preprint arXiv:1702.04267 , eprint =

On Detecting Adversarial Perturbations , author =. arXiv preprint arXiv:1702.04267 , eprint =

arXiv
[257]

and Buckley, Christopher L

Millidge, Beren and Tschantz, Alexander and Seth, Anil K. and Buckley, Christopher L. , editor =. On the. Active. doi:10.1007/978-3-030-64919-7_1 , abstract =

doi
[258]

, year = 2021, month = feb, journal =

Millidge, Beren and Tschantz, Alexander and Buckley, Christopher L. , year = 2021, month = feb, journal =. Whence the. doi:10.1162/neco_a_01354 , urldate =

2021 doi
[259]

, year = 2013, month = jan, number =

Minka, Thomas P. , year = 2013, month = jan, number =. Expectation. doi:10.48550/arXiv.1301.2294 , urldate =. arXiv , keywords =:1301.2294 , primaryclass =

2013 doi arXiv
[260]

Nature , volume =

Human-Level Control through Deep Reinforcement Learning , author =. Nature , volume =. doi:10.1038/nature14236 , urldate =

doi
[261]

Mnih, Volodymyr and Kavukcuoglu, Koray and Silver, David and Graves, Alex and Antonoglou, Ioannis and Wierstra, Daan and Riedmiller, Martin , year = 2013, month = dec, number =. Playing. doi:10.48550/arXiv.1312.5602 , urldate =. arXiv , keywords =:1312.5602 , primaryclass =

2013 doi arXiv
[262]

Admissibility

Moor, Thomas and Raisch, J. Admissibility. IFAC Proceedings Volumes , series =. doi:10.1016/S1474-6670(17)36456-X , urldate =

doi
[263]

Proceedings of the IEEE , volume =

Cramming More Components onto Integrated Circuits , author =. Proceedings of the IEEE , volume =
[264]

The Eleventh International Conference on Learning Representations , author =
[265]

Dynamic Bayesian Networks: Representation, Inference and Learning , author =
[266]

and Weiss, Yair and Jordan, Michael I

Murphy, Kevin P. and Weiss, Yair and Jordan, Michael I. , year = 1999, month = jul, series =. Loopy Belief Propagation for Approximate Inference: An Empirical Study , shorttitle =. Proceedings of the

1999
[267]

, year = 2022, month = mar, publisher =

Murphy, Kevin P. , year = 2022, month = mar, publisher =. Probabilistic

2022
[268]

, year = 2023, month = aug, publisher =

Murphy, Kevin P. , year = 2023, month = aug, publisher =. Probabilistic

2023
[269]

, year = 2013, month = jun, number =

Murray, Lawrence M. , year = 2013, month = jun, number =. Bayesian. doi:10.48550/arXiv.1306.3277 , urldate =. arXiv , keywords =:1306.3277 , primaryclass =

2013 doi arXiv
[270]

doi:10.48550/arXiv.2006.09359 , urldate =

Nair, Ashvin and Gupta, Abhishek and Dalal, Murtaza and Levine, Sergey , year = 2021, month = apr, number =. doi:10.48550/arXiv.2006.09359 , urldate =. arXiv , keywords =:2006.09359 , primaryclass =

2021 doi arXiv
[271]

Variational

Nazaret, Achille and Blei, David , year = 2022, month = sep, number =. Variational. doi:10.48550/arXiv.2209.10091 , urldate =. arXiv , keywords =:2209.10091 , primaryclass =

2022 doi arXiv
[272]

arXiv: Learning , urldate =

Neiswanger, Willie and Kandasamy, Kirthevasan and P. arXiv: Learning , urldate =
[273]

Nguyen, Hoang Minh Huu and. A. IEEE Open Journal of Signal Processing , volume =. doi:10.1109/OJSP.2025.3585440 , urldate =

2025
[274]

Nickisch, Hannes and Solin, Arno and Grigorevskiy, Alexander , year = 2018, month = jul, pages =. State. Proceedings of the 35th

2018
[275]

arXiv preprint arXiv:2511.18955 , eprint =

Active Inference Is a Subtype of Variational Inference , author =. arXiv preprint arXiv:2511.18955 , eprint =
[276]

Nuijten, Wouter W. L. and. Expected. Transactions on Machine Learning Research , issn =
[277]

Nuijten, Wouter W. L. and Bagaev, Dmitry and. Entropy. An International and Interdisciplinary Journal of Entropy and Information Studies , volume =
[278]

Nuijten, Wouter W. L. and Lukashchuk, Mykola and. A. Active. doi:10.1007/978-3-032-16955-6_5 , abstract =

doi
[279]

Nuijten, Wouter W. L. and Menkovski, Vlado , year = 2024, pages =. Node Classification in Random Trees , booktitle =

2024
[280]

International Workshop on Active Inference , author =

Reactive Environments for Active Inference Agents with. International Workshop on Active Inference , author =
[281]

Nuijten, Wouter W. L. and. Sophisticated. 2026. doi:10.48550/arXiv.2607.19518 , urldate =. arXiv , keywords =:2607.19518 , primaryclass =

2026 doi arXiv
[282]

Nuijten, Wouter W. L. and Lukashchuk, Mykola and van de Laar, Thijs and de Vries, Bert , year = 2026, month = jun, eprint =. What. Conference on. doi:10.48550/arXiv.2606.04935 , urldate =

2026 doi arXiv
[283]

Making Sense of Reinforcement Learning and Probabilistic Inference , booktitle =
[284]

Variational

O' Donoghue, Brendan , year = 2021, volume =. Variational. Advances in

2021
[285]

Journal of Field Robotics , volume =

Coverage Path Planning Algorithms for Agricultural Field Machines , author =. Journal of Field Robotics , volume =. doi:10.1002/rob.20300 , urldate =

doi
[286]

Computers and Electronics in Agriculture , volume =

Shape-Describing Indices for Agricultural Field Plots and Their Relationship to Operational Efficiency , author =. Computers and Electronics in Agriculture , volume =. doi:10.1016/j.compag.2013.08.014 , urldate =

2013 doi
[287]

doi:10.48550/arXiv.2303.08774 , urldate =

OpenAI and Achiam, Josh and Adler, Steven and Agarwal, Sandhini and Ahmad, Lama and Akkaya, Ilge and Aleman, Florencia Leoni and Almeida, Diogo and Altenschmidt, Janko and Altman, Sam and Anadkat, Shyamal and Avila, Red and Babuschkin, Igor and Balaji, Suchir and Balcom, Valer...

doi arXiv
[288]

and Braun, Daniel A

Ortega, Pedro A. and Braun, Daniel A. , year = 2011, month = jul, number =. Information,. doi:10.48550/arXiv.1107.5766 , urldate =. arXiv , keywords =:1107.5766 , primaryclass =

2011 doi arXiv
[289]

Information,

Ortega, Daniel Alexander and Braun, Pedro Alejandro , editor =. Information,. Artificial. doi:10.1007/978-3-642-22887-2_28 , abstract =

doi
[290]

A Practical Introduction to Tensor Networks:

Or. A Practical Introduction to Tensor Networks:. Annals of Physics , volume =. doi:10.1016/j.aop.2014.06.013 , urldate =

2014 doi
[291]

and Wen, Zheng , year = 2019, journal =

Osband, Ian and Roy, Benjamin Van and Russo, Daniel J. and Wen, Zheng , year = 2019, journal =. Deep

2019
[292]

Osband, Ian and Russo, Daniel and Van Roy, Benjamin , year = 2013, volume =. (. Advances in

2013
[293]

Randomized

Osband, Ian and Aslanides, John and Cassirer, Albin , year = 2018, volume =. Randomized. Advances in

2018
[294]

Oseledets, I. V. , year = 2011, month = jan, journal =. Tensor-. doi:10.1137/090752286 , urldate =

2011 doi
[295]

A Philosophy of Software Design , author =
[296]

Palacios, Ensor Rafael and Razi, Adeel and Parr, Thomas and Kirchhoff, Michael and Friston, Karl , year = 2020, month = feb, journal =. On. doi:10.1016/j.jtbi.2019.110089 , urldate =

2020
[297]

Palmieri, Francesco A. N. , year = 2013, month = aug, journal =. A. arXiv , keywords =:1308.5576 , primaryclass =

2013 arXiv
[298]

Palmieri, Francesco A. N. and Pattipati, Krishna R. and Gennaro, Giovanni Di and Fioretti, Giovanni and Verolla, Francesco and Buonanno, Amedeo , year = 2022, journal =. A. doi:10.1109/ACCESS.2022.3148127 , abstract =

2022
[299]

, year = 2022, month = mar, publisher =

Parr, Thomas and Pezzulo, Giovanni and Friston, Karl J. , year = 2022, month = mar, publisher =. Active. doi:10.7551/mitpress/12441.001.0001 , urldate =

2022 doi
[300]

Parr, Thomas and Friston, Karl and Zeidman, Peter , year = 2024, month = mar, journal =. Active. doi:10.3390/a17030118 , urldate =

2024 doi

Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

[1] [1]

Adamiat, Sepideh and Kouw, Wouter M. and. Message. Active. doi:10.1007/978-3-031-77138-5_14 , abstract =

doi:10.1007/978-3-031-77138-5_14

[2] [2]

and Shipp, Stewart and Friston, Karl J

Adams, Rick A. and Shipp, Stewart and Friston, Karl J. , year = 2013, journal =. Predictions Not Commands: Active Inference in the Motor System , shorttitle =. doi:10.1007/s00429-012-0475-5 , urldate =

doi:10.1007/s00429-012-0475-5 2013

[3] [3]

and Murphy, Kevin , year = 2018, month = jul, pages =

Alemi, Alexander and Poole, Ben and Fischer, Ian and Dillon, Joshua and Saurous, Rif A. and Murphy, Kevin , year = 2018, month = jul, pages =. Fixing a. Proceedings of the 35th

2018

[4] [4]

Computationally Efficient Convolved Multiple Output

Alvarez, Mauricio A and Lawrence, Neil D , year = 2011, journal =. Computationally Efficient Convolved Multiple Output

2011

[5] [5]

Foundations and Trends in Machine Learning , volume =

Kernels for. Foundations and Trends in Machine Learning , volume =. doi:10.1561/2200000036 , urldate =

doi:10.1561/2200000036

[6] [6]

, year = 2012, month = jun, pages =

Anandkumar, Animashree and Hsu, Daniel and Kakade, Sham M. , year = 2012, month = jun, pages =. A. Proceedings of the 25th

2012

[7] [7]

Anil Meera, Ajith and Lanillos, Pablo , editor =. Towards. Active. doi:10.1007/978-3-031-47958-8_3 , abstract =

doi:10.1007/978-3-031-47958-8_3

[8] [8]

Biosystems , volume =

Natural Hierarchy Emerges from Energy Dispersal , author =. Biosystems , volume =. doi:10.1016/j.biosystems.2008.10.008 , urldate =

doi:10.1016/j.biosystems.2008.10.008 2008