Reference-Free Quadrature Filters For The Sine Benchmark

The last part of the week stepped away from amortized training and asked a sharper question:

How far can a deterministic, reference-free filtering update go if it directly integrates the known nonlinear likelihood and projects the result back to a strict family?

That produced the quadrature ADF and Power-EP suite. These rows used only the known transition, known observation model, and observed \(x,y\). They did not train on grid posterior moments or latent states. The update locally formed a tilted distribution and projected it back to a Gaussian or Gaussian mixture.

The figure compares the deterministic quadrature rows from the committed sweep code against the grid-reference metrics. The left panel shows state-density quality; the right panel shows the pre-update predictive score, where the alias-heavy state update was less well calibrated.

Baseline Quadrature ADF

The first suite compared a Gaussian ADF update, K2 ADF, K4 spread ADF, and K4 Power-EP with alpha 0.5.

The result was uneven but important. K4 Power-EP closed most of the state-density gap in clean, weak, intermittent, and zero-observation settings. Random-normal observations remained difficult.

Alias-Indexed Components

The sine likelihood creates alias modes. The next variants indexed mixture components by \(2\pi\)-spaced aliases and then tested prior weighting, top-k pruning, moment shrinkage, and entropy gates.

The prior-weighted K5 alias Power-EP row improved random-normal state NLL from 5.002 to 2.717, close to the grid reference 2.698, but it inflated variance and damaged predictive-y NLL:

Shrink variants fixed some overdispersion. For weak, intermittent, and zero observations, shrink 0.85 brought variance ratio near one:

The tradeoff was that shrink hurt sinusoidal and random-normal state NLL. That made it a calibration variant, not the overall promotion row.

State-Predictive Pareto Split

The final quadrature Pareto suite separated a state-density leg from a predictive leg. It paired the prior-weighted K5 alias state update with the K4 Power-EP predictive scorer:

This was not a single perfect filter. It was a useful decomposition: the state-density update and predictive normalizer wanted different approximations. That matched the later predictive-decomposition report, where K4 state-density rows had strong state NLL but still lagged the particle-filter predictive likelihood on clean and random-normal patterns.

What This Says About The Week

The week started with an amortized VBF implementation question and ended with a stronger algorithmic diagnosis:

• local ELBO was too mode-seeking and under-dispersed;
• combined unsupervised objective repair helped but did not solve calibration;
• mixture IWAE, FIVO bridge, and local projection rows showed that reference-free updates could improve dramatically;
• deterministic quadrature and Power-EP made the alias structure explicit and reached grid-scale state NLL on several stressors.

The remaining research target is not just “make the posterior bigger.” It is to align state-density quality with the pre-update predictive normalizer, while preserving the strict online filter.

Source artifacts:

• quadrature ADF suite artifact
• alias entropy shrink suite artifact
• quadrature Pareto suite artifact
• 59f084f: quadrature ADF nonlinear baselines
• 78d5ba7: alias-indexed quadrature filter experiment
• 7d88e2e: prior-weighted alias quadrature experiment
• c1ff610: quadrature Pareto sweep
• c49c7fa: entropy-gated alias shrink variants