A twin whose sensor never produces outliers: how many particles does a bootstrap filter need, and is the average effective sample size the right thing to size on?

What it found

For a bootstrap particle filter the large-N effective-sample-size fraction is exactly c₀ e^−ay² in the measurement y (c₀ = 0.866, a = 1/6 at P = R = 1). With 5% real outliers at 10× the noise scale the mean ESS fraction barely moves (0.671 in the twin, 0.645 real, at every dimension), but P(ESS < 1% of N) goes from 2.5×10⁻⁴ to 3.1×10⁻² (120×). Sizing N = 528 for ESS ≥ 50 at 1% failure in the twin fails 4.4% of real steps, and the particle posterior mean's error against the exact Kalman update is 45× to 421× the twin's for N = 50 to 1000, falling barely at all with N. The large-N formula is checked against quadrature and simulation and fails once Nρ is small (reported). Stylised: scalar update, simulated outliers, no resampling dynamics.

A figure from its own output

Paper, code and results

• README and method
• White paper
• Experiments and results
• Code
• Tests

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Produced with AI assistance (Claude). Results are from the code in this repository and are simulated or stylised unless the abstract says otherwise; nothing here implies affiliation with or endorsement by any lab, researcher or company cited. MIT licensed.