Learning to Bias: Machine Learning-Enhanced Particle Filters

Apoorv Srivastava; Eric Darve

cs.LGSep 24, 2026

Learning to Bias: Machine Learning-Enhanced Particle Filters

Authors: Apoorv Srivastava, Eric Darve

Organizations: Stanford University Stanford, CA 94305

Abstract

Sequential inference estimates latent states from noisy and incomplete observations. Particle Filters (PFs), a class of Monte Carlo methods based on importance sampling, provide a flexible framework for this task, but often suffer from poor sample efficiency and unfavorable scaling with dimension, partly due to suboptimal proposal distributions. We address these challenges by integrating learned proposals into the PF framework. We introduce Neural Optimal Particle Filters (NOPFs), which learn an amortized approximation to the optimal proposal from offline simulated one-step conditioning tuples. The learned proposal is used as a drop-in replacement in standard PF updates, with samples corrected by standard importance weights so that the method asymptotically targets the same filtering distribution under standard support and density-evaluation assumptions. Across stochastic nonlinear benchmarks of varying inference complexity, NOPFs improve sample efficiency and distributional accuracy over standard PF baselines with modest computational overhead. The approach integrates data-driven proposal learning into classical inference without altering the underlying filtering objective.

Figures & tables

Figure 1

Figure 1: Linear system with biased sensors. Top: true and estimated trajectories with 90% credible intervals for a representative seed. Bottom: aggregate metrics across 10 seeds. Arrows indicate whether larger or smaller values are preferred, and darker shading denotes better performance.

Figure 2: 2D robot localization with range and bearing observations. Top: initial state and later-time posterior summaries for a representative seed with NOPF particles and the EnKF covariance approximation. Bottom: aggregate metrics across 10 seeds.

Figure 3: 2D robot localization with biased range-only observations. Top: posterior snapshots showing whether filters preserve and resolve posterior branch ambiguity over time for a representative seed. Bottom: aggregate metrics across 10 seeds.

Figure 4: 3D robot localization with unordered range-only observations. Top: snapshots showing the evolution of multimodal filtering distribution for a representative seed. Bottom: aggregate metrics across 10 seeds.

Appendix figures & tables12 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 5: Linear system with biased sensors: Metrics as a function of the number of particles

N

Figure 6: 2D robot localization with range and bearing observations: Metrics as a function of the number of particles

N

Figure 7: 2D robot localization with biased range-only observations: Metrics as a function of the number of particles

N

Figure 8: 3D robot localization with unordered range-only observations: Metrics as a function of the number of particles

N

Filter	NMSE $\downarrow$	PNLL $\downarrow$	ES $\downarrow$	ESS $\uparrow$	WT [s] $\downarrow$	MADE $\downarrow$	Joint PICP $\to 0.9$
KF	$0.1179\pm 0.0973$	$-0.6739\pm 0.1572$	$0.2396\pm 0.0601$	–	$\mathbf{0.0124\pm 0.0012}$	–	$0.9069\pm 0.0502$
EnKF	$0.1220\pm 0.1022$	$-0.6696\pm 0.1616$	$0.2424\pm 0.0645$	–	$\mathbf{0.0990\pm 0.0018}$	$\mathbf{0.0245\pm 0.0033}$	$\mathbf{0.9034\pm 0.0558}$
BPF	$0.1298\pm 0.1079$	$-0.6682\pm 0.1608$	$0.2502\pm 0.0572$	$598.5094\pm 47.3294$	$4.5919\pm 0.9028$	$0.0350\pm 0.0054$	$0.8828\pm 0.0634$
UPF	$0.1326\pm 0.1007$	$\mathbf{-0.6764\pm 0.1579}$	$0.2571\pm 0.0736$	$377.0515\pm 19.4465$	$154.5473\pm 3.1899$	$0.0252\pm 0.0029$	$\mathbf{0.8966\pm 0.0685}$
APF	$0.1637\pm 0.1663$	$-0.6559\pm 0.1727$	$0.2616\pm 0.0647$	$377.9505\pm 50.1158$	$4.6464\pm 0.6950$	$0.0299\pm 0.0032$	$0.8879\pm 0.0547$
NuPF	$0.1224\pm 0.1051$	$-0.6561\pm 0.1791$	$0.2468\pm 0.0779$	$601.1692\pm 48.0596$	$4.6895\pm 0.9368$	$0.0357\pm 0.0076$	$0.9069\pm 0.0714$

Appendix

Table 1: Mean

\pm

standard deviation across 10 seeds for the linear system with biased sensors benchmark (Section 4.1 ). Bold marks the top two per column (closest to

0.9

for Joint PICP).

Filter	NMSE $\downarrow$	PNLL $\downarrow$	ES $\downarrow$	ESS $\uparrow$	WT [s] $\downarrow$	MADE $\downarrow$	Joint PICP $\to 0.9$
EnKF	$\mathbf{0.0540\pm 0.0193}$	$\mathbf{-9.6921\pm 0.4545}$	$\mathbf{0.4309\pm 0.0838}$	–	$\mathbf{0.9094\pm 0.0094}$	$\mathbf{0.0397\pm 0.0039}$	$\mathbf{0.8796\pm 0.0863}$
BPF	$0.1163\pm 0.0541$	$1.1781\pm 2.3617$	$0.7519\pm 0.2399$	$8.3675\pm 0.9149$	$\mathbf{18.0811\pm 0.0524}$	$0.2018\pm 0.1127$	$0.0755\pm 0.0962$
UPF	$0.0717\pm 0.0463$	$31.3362\pm 122.3115$	$0.4694\pm 0.0915$	$198.3859\pm 13.3273$	$1064.7424\pm 2.5994$	$0.0698\pm 0.0786$	$0.7204\pm 0.2411$
APF	$0.0723\pm 0.0212$	$-1.1114\pm 1.5949$	$0.5849\pm 0.0773$	$8.5483\pm 0.9656$	$19.8896\pm 3.9272$	$0.1593\pm 0.0165$	$0.0184\pm 0.0326$
NuPF	$0.0869\pm 0.0493$	$0.8330\pm 1.7710$	$0.6247\pm 0.1931$	$8.6296\pm 1.0483$	$19.0552\pm 0.0677$	$0.1702\pm 0.0448$	$0.1367\pm 0.0545$
ISPF	$0.0559\pm 0.0198$	$-8.0896\pm 0.9456$	$0.4434\pm 0.0930$	$\mathbf{574.4442\pm 52.8567}$	$81.0166\pm 4.3176$	$0.0603\pm 0.0071$	$0.7408\pm 0.1902$

Appendix

Table 2: Mean

\pm

standard deviation across 10 seeds for the 2D robot localization benchmark with range and bearing observations (Section 4.2 ). Bold marks the top two per column (closest to

0.9

for Joint PICP).

Filter	NMSE $\downarrow$	PNLL $\downarrow$	ES $\downarrow$	ESS $\uparrow$	WT [s] $\downarrow$	MADE $\downarrow$	Joint PICP $\to 0.9$
EnKF	$0.6767\pm 0.0642$	$-3.0860\pm 0.2682$	$1.5510\pm 0.1880$	–	$\mathbf{0.6236\pm 0.0395}$	$0.1856\pm 0.0344$	$\mathbf{0.8793\pm 0.1044}$
BPF	$0.0242\pm 0.0201$	$-4.9084\pm 0.1347$	$0.3296\pm 0.1510$	$1525.8024\pm 72.6469$	$\mathbf{22.3428\pm 2.6978}$	$0.1467\pm 0.0748$	$0.7086\pm 0.3181$
UPF	$0.7444\pm 0.7660$	$18.7829\pm 41.4584$	$2.4642\pm 1.9179$	$1.2591\pm 0.0547$	$1125.3255\pm 84.5301$	$1.4938\pm 1.4605$	$0.0000\pm 0.0000$
APF	$0.0381\pm 0.0358$	$-4.8894\pm 0.1309$	$0.4589\pm 0.2336$	$694.8429\pm 43.7893$	$22.9959\pm 2.8219$	$0.1894\pm 0.1233$	$0.4931\pm 0.2152$
NuPF	$0.0236\pm 0.0109$	$-4.9030\pm 0.1287$	$0.3295\pm 0.0631$	$1556.0490\pm 67.3410$	$23.0519\pm 3.0155$	$0.1193\pm 0.0307$	$\mathbf{0.8362\pm 0.2027}$
ISPF	$\mathbf{0.0136\pm 0.0111}$	$\mathbf{-4.9203\pm 0.1202}$	$\mathbf{0.2664\pm 0.0714}$	$\mathbf{5399.6077\pm 172.8422}$	$64.7673\pm 0.5461$	$\mathbf{0.0805\pm 0.0124}$	$0.9810\pm 0.0275$

Appendix

Table 3: Mean

\pm

standard deviation across 10 seeds for the 2D robot localization benchmark with biased range-only observations (Section 4.3 ). Bold marks the top two per column (closest to

0.9

for Joint PICP).

Filter	NMSE $\downarrow$	PNLL $\downarrow$	ES $\downarrow$	ESS $\uparrow$	WT [s] $\downarrow$	MADE $\downarrow$	Joint PICP $\to 0.9$
EnKF	$0.0293\pm 0.0348$	$-5.2189\pm 0.5531$	$0.3537\pm 0.1214$	–	$\mathbf{4.9304\pm 0.0908}$	$\mathbf{0.2125\pm 0.0463}$	$\mathbf{0.8545\pm 0.3030}$
BPF	$0.0302\pm 0.0401$	$-5.9356\pm 0.3396$	$0.3854\pm 0.1351$	$1118.3518\pm 170.3681$	$\mathbf{71.7007\pm 0.2993}$	$0.3092\pm 0.1128$	$0.0000\pm 0.0000$
UPF	$0.1176\pm 0.1132$	$7.6478\pm 7.1106$	$0.9492\pm 0.3356$	$1.0644\pm 0.0272$	$4545.2538\pm 16.3267$	$0.4902\pm 0.2829$	$0.0000\pm 0.0000$
NuPF	$0.0323\pm 0.0381$	$-5.7771\pm 0.5003$	$0.4065\pm 0.1575$	$1078.1617\pm 213.1463$	$85.0738\pm 0.6555$	$0.3266\pm 0.1164$	$0.0000\pm 0.0000$
APF	$0.0291\pm 0.0259$	$-5.7518\pm 0.4722$	$0.4071\pm 0.1036$	$364.4146\pm 100.0679$	$74.0771\pm 0.6063$	$0.3053\pm 0.1011$	$0.0000\pm 0.0000$
ISPF	$\mathbf{0.0228\pm 0.0270}$	$\mathbf{-6.1685\pm 0.3328}$	$\mathbf{0.3259\pm 0.1042}$	$\mathbf{15050.6945\pm 1188.3015}$	$1476.2126\pm 37.9838$	$\mathbf{0.2359\pm 0.0549}$	$\mathbf{0.2667\pm 0.2824}$

Appendix

Table 4: Mean

\pm

standard deviation across 10 seeds for the 3D robot localization benchmark with unordered range-only observations (Section 4.4 ). Bold marks the top two per column (closest to

0.9

for Joint PICP).

Benchmark	Hidden $H$	Blocks	Batch size	Training epochs
Sensor bias system	32	4	20	500
Known-map localization	64	4	50	500
Range-only localization	128	6	50	500
3D unordered-range localization	128	6	50	500

Appendix

Table 5: Neural proposal hyperparameters used in the experiments.

Figure 9: Additional metrics (MADE and Joint PICP) for benchmarks. Each column presents the metrics for a benchmark.

Filter	Propagation	Importance Sampling	Weight Update	Total
BPF	$n_{p}\mathcal{O}(\mathbf{f})$	–	$n_{p}\mathcal{O}(\mathbf{h})$	$n_{p}(\mathcal{O}(\mathbf{f})+\mathcal{O}(\mathbf{h}))$
NuPF	$n_{p}\mathcal{O}(\mathbf{f})$	$n_{p}\mathcal{O}(\mathbf{h})$	$n_{p}\mathcal{O}(\mathbf{h})$	$n_{p}\mathcal{O}(\mathbf{f})+2n_{p}\mathcal{O}(\mathbf{h})$
UPF	$n_{p}(4d_{x}+2d_{y}+1)(\mathcal{O}(\mathbf{f})+\mathcal{O}(\mathbf{h}))+n_{p}\mathcal{O}((2d_{x}+d_{y})^{3})$		$n_{p}\mathcal{O}(\mathbf{h})$	$n_{p}(4d_{x}+2d_{y}+1)\mathcal{O}(\mathbf{f})+n_{p}(4d_{x}+2d_{y}+2)\mathcal{O}(\mathbf{h})+n_{p}\mathcal{O}((2d_{x}+d_{y})^{3})$
ISPF	$n_{p}n_{q}(\mathcal{O}(\mathbf{f})+\mathcal{O}(\mathbf{h}))$			$n_{p}n_{q}(\mathcal{O}(\mathbf{f})+\mathcal{O}(\mathbf{h}))$
APF	$2n_{p}\mathcal{O}(\mathbf{f})$	$n_{p}\mathcal{O}(\mathbf{h})$	$n_{p}\mathcal{O}(\mathbf{h})$	$2n_{p}\mathcal{O}(\mathbf{f})+2n_{p}\mathcal{O}(\mathbf{h})$
EnKF	$n_{p}(\mathcal{O}(\mathbf{f})+\mathcal{O}(\mathbf{h}))+\mathcal{O}(n_{p}d_{x}d_{y}+n_{p}d_{y}^{2}+d_{y}^{3})$			$n_{p}(\mathcal{O}(\mathbf{f})+\mathcal{O}(\mathbf{h}))+\mathcal{O}(n_{p}d_{x}d_{y}+n_{p}d_{y}^{2}+d_{y}^{3})$

Appendix

Table 6: Per-step computational complexity of the particle filters considered in this work.

Benchmark	$d_{x}$	$d_{y}$	Time points	Particles
Sensor bias system	3	2	60	2000
Known-map localization	7	10	51	10000
Range-only localization	8	3	60	10000
3D unordered-range localization	20	4	35	50000

Appendix

Table 7: Benchmark details.

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Generative Model Proposal based Particle Filtering for Data Assimilation

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Abstract

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