A Tour Of Learned And Reference-Free Bayesian Filters

nodes, weights = hermgauss(num_points)
z = prev_mean[..., None] + sqrt(2.0 * pred_state_var[..., None]) * nodes
obs_mean = x[..., None] * sin(z)
log_likelihood = normal_log_prob(y[..., None], obs_mean, r)
log_prob_y = logsumexp(log_weights + log_likelihood, axis=-1) - 0.5 * log(pi)

This term is causal only if the update path has not already used \(y_t\) to construct the prediction. The reports treat that as a guardrail.

Technique 12: masked-y span training

Masked-y training withholds measurements from the update path for random points or spans. When \(y_t\) is masked, the filter should perform the exact transition prediction:

That behavior is tested in the nonlinear reference tests. The purpose is to make the carried belief robust during uninformative spans. In the promoted Gaussian objective, masked spans are paired with the predictive-y score and windowed ELBO.

Technique 13: Gaussian mixture filters

The sine likelihood is periodic, so a single Gaussian belief can be a poor description of the posterior. The mixture branch uses:

The direct mixture head emits weights, component means, component variances, and componentwise backward conditionals. In simplified code:

raw = hidden @ w2 + b2
raw = raw.reshape(..., num_components, 6)
filter_weights = softmax(raw[..., 0], axis=-1)
component_mean = prev_mean[..., None] + raw[..., 1]
component_var = softplus(raw[..., 2]) + min_var

Mixtures by themselves did not solve the problem. The useful result came from pairing a mixture family with a multi-sample objective.

Technique 14: IWAE window objectives

The IWAE objective changes how samples are combined. If \(w_k\) are importance weights for sampled paths, the ordinary ELBO averages log weights:

IWAE uses:

In code:

if objective_family == "iwae":
    return logsumexp(log_weights, axis=0) - log(num_samples)

The best K2 mixture IWAE row in the April 29 report was:

direct_mixture_k2_joint_iwae_h4_k32

It improved state density dramatically:

But predictive-y NLL regressed on nonzero-observation stressors. That means the row was not promotable as a fully solved filter:

This was the first major coupled-bottleneck result: posterior expressivity helps when paired with the right objective, but a state-density win can be an observation-prediction loss.

Technique 15: Renyi and alpha objectives

The Renyi objective generalizes the sample aggregation:

As \(\alpha\to 1\), it approaches the ELBO. Smaller values change the pressure on high-weight samples.

In the reported Gaussian branch, changing only the Gaussian objective to IWAE or Renyi did not solve calibration. The mixture plus IWAE pairing mattered more.

Technique 16: FIVO-style objectives

FIVO-style objectives estimate a sequence marginal likelihood with particles and resampling. The broad shape is:

where particles are propagated and optionally resampled over time.

The repo includes several proposal families:

• marginal filter proposal;
• transition-filter bridge proposal;
• learned transition-filter bridge proposal.

The bridge proposal combines a transition from the previous particle with the filter belief at the current time. In Gaussian form:

The latest short FIVO bridge resampling suite shows the method is sensitive to resampling and proposal details. Stop-gradient resampling improved coverage in some 250-step K4 runs, but the auxiliary learned bridge was poor in the same short-budget setting. This branch is still exploratory.

Technique 17: bootstrap particle filter reference

A bootstrap particle filter is also used as a diagnostic reference. It samples particles from the transition:

weights them by the observation likelihood:

then resamples.

The particle filter is not the headline reference because the grid filter is deterministic and scalar. But particle filtering is useful for sanity checking mixture and quadrature rows, especially when comparing state density and predictive-y behavior.

Technique 18: local tilted projection

The quadrature and ADF branches start from the local tilted distribution:

Then they project \(\tilde{p}\) back into the chosen belief family.

For a Gaussian ADF update, the projection is moment matching:

This is reference-free because it uses the known transition, known observation model, and current observations, not the grid posterior cache.

Technique 19: quadrature ADF

For the scalar sine model, the tilted distribution can be integrated with deterministic quadrature. This is a different class from learned amortized filters. It is closer to classic assumed-density filtering.

The April 30 quadrature suite found:

That was an important reversal. The best neural fully unsupervised filters were still far from the grid reference, but deterministic reference-free updates could get much closer.

Technique 20: Power-EP

Expectation propagation uses local factors and projections. Power-EP introduces a power parameter \(\alpha\), which changes how strongly the likelihood tilts the current belief before projection.

In these reports, Power-EP style updates are used as reference-free filtering rules. They are not trained on reference moments.

The K4 Power-EP row with \(2\pi\)-spread components was strong on weak and intermittent patterns. It still struggled on random-normal observations until alias-indexed components were introduced.

Technique 21: alias-indexed mixture components

The sine likelihood has a built-in alias structure:

Instead of asking a neural network to discover this from scratch, the alias-indexed quadrature branch places mixture components on \(2\pi\)-spaced aliases and weights them with the prior and likelihood.

The prior-weighted K5 alias Power-EP row reached near-reference state NLL across all five stressors:

The state-density result is excellent, but the row is over-dispersed and predictive-y NLL is weak in clean and random-normal settings.

Shrink variants reduce variance:

But shrink hurts other patterns. This is now a Pareto problem, not a single-row victory.

What has been learned

The research arc can be summarized as a sequence of increasingly specific diagnoses.

First, the linear-Gaussian case established the plumbing. Exact Kalman, frozen-marginal controls, supervised edge learning, and ELBO metrics all work. The scalar MC ELBO is already under-dispersed in weak and zero-observation regimes.

Second, the nonlinear learned Gaussian filter exposed a much larger under-dispersion problem. More steps, simple resampling, and variance penalties did not fix the fully unsupervised objective.

Third, reference-distilled diagnostics showed that good strict filters exist. The issue is not simply that online Gaussian filtering is impossible. The issue is finding the right reference-free objective or update rule.

Fourth, a combined unsupervised Gaussian objective - windowed ELBO, predictive-y, and masked-y spans - improved robustness but remained badly under-dispersed.

Fifth, direct K2 mixture IWAE improved state density substantially. This showed that posterior family and objective are coupled. It also introduced a new failure: state-density improvement can regress predictive-y NLL.

Sixth, deterministic quadrature ADF and Power-EP baselines showed that the known nonlinear likelihood contains enough structure to get close to the grid reference without reference targets. The current strongest state-density rows are algorithmic reference-free filters, not learned amortized ELBO filters.

A rough scorecard

This table compresses the main techniques into one view. Values are indicative from the reports, not a single unified sweep.

The most important caveat

The latest generated report artifacts are not all equally reliable as final summaries. One generated final report had empty robustness and diagnostic tables while still asserting a fixed best row. The source metrics and narrower summary files are more reliable than that incomplete aggregate. Until the aggregate report is regenerated with the full intended input set, I would treat the following as the clean state:

• linear-Gaussian benchmark: report-ready;
• nonlinear learned Gaussian objective repair: partial success, not solved;
• nonlinear K2 mixture IWAE: best learned fully unsupervised state-density row in the April 29 branch, blocked by predictive-y regression;
• quadrature ADF / Power-EP: strongest current reference-free algorithmic baselines, with a remaining state-density versus predictive-y tradeoff.

How to reproduce pieces

The repo uses uv and JAX. The common commands are:

uv sync --dev
make test

Run the scalar linear-Gaussian aggregate:

make aggregate-linear-gaussian-reports

Run nonlinear learned sweeps from the sweep script:

uv run python scripts/sweep_nonlinear_learned.py \
  --models structured_joint_elbo_h4_w005_predictive_y_masked_y_spans_h4 \
  --steps 1000

Run the direct nonlinear training script from a config:

uv run python scripts/train_nonlinear.py \
  --config experiments/nonlinear/09_structured_elbo_sine_mlp.yaml

The exact command surface changes as the research harness evolves, so the most stable entry points are the scripts and committed configs in:

• experiments/nonlinear/
• scripts/
• docs/results/

Where I would go next

I would split the next work into two tracks.

The first track is report hygiene. The latest findings should be consolidated into one committed status note that separates:

• learned neural filters;
• deterministic reference-free filters;
• reference-distilled diagnostics;
• oracle-calibrated diagnostics.

The second track is methodological. The quadrature results are now too strong to ignore. The next learned filters should probably learn from the structure that quadrature exposed, not just run another local ELBO sweep. Good next questions:

• Can an amortized filter learn the alias-aware Power-EP update?
• Can the state-density and predictive-y objectives be separated into a principled two-head selection rule?
• Can mixture components be initialized or constrained around known periodic aliases without using reference moments?
• Can the predictive-y regression be fixed without giving back the state-density gains?

The main lesson is not that one technique won. It is that filtering quality is multi-dimensional. A filter must carry calibrated state uncertainty, predict future measurements, survive weak observations, and remain online. The Kalman filter gets all of that for free in the linear-Gaussian case. In the nonlinear case, every approximation decides which part of that contract it is willing to pay for.

Source artifacts

The most useful source artifacts for this synthesis are linked below. Stable committed source files are linked to GitHub; generated summary snapshots used for the April 30 interpretation are bundled with this post under artifacts/.

• linear Gaussian scalar report
• nonlinear strict-filter final summary
• docs/results/nonlinear_unsupervised_elbo_t11_status_2026-04-28.md
• docs/results/nonlinear_divergence_family_status_2026-04-29.md
• nonlinear quadrature ADF suite summary
• nonlinear alias entropy/shrink suite summary
• nonlinear K4 preupdate predictive comparison summary