Weights, Estimate, and Effective Sample Size

Ada re-derives this chapter’s own numbers step by step, at full precision

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Ada ADA · CALCULATION AUDIT

Weights, Estimate, and Effective Sample Size

The chapter runs a five-particle filter against a 10.0 m measurement and produces a weighted estimate of 10.02 m, then tests whether that vote has collapsed with an effective sample size of 2.31 particles against a 2.5 resampling threshold. A particle filter is a weighted vote, and every step is auditable arithmetic. This audit confirms the weights, the estimate, and the effective sample size that decides when to resample.

Companion to the chapter Particle Filters for Localization — every number here comes from that chapter.

Ada: A particle filter is a weighted vote, and the whole thing is auditable arithmetic. Let me confirm the weights, the estimate, and the resampling trigger using only the stated values.

The measurement is 10.0 m with sigma 1.0 m, and the five predicted particles are at 8.8, 9.5, 10.2, 11.0, and 12.0 m. Each relative likelihood is exp(-0.5 x error^2 / sigma^2) with sigma^2 = 1.0:

  • x = 8.8, error -1.2: exp(-0.5 x 1.44) = exp(-0.72) = 0.486752, about 0.487.
  • x = 9.5, error -0.5: exp(-0.5 x 0.25) = exp(-0.125) = 0.882497, about 0.882.
  • x = 10.2, error +0.2: exp(-0.5 x 0.04) = exp(-0.02) = 0.980199, about 0.980.
  • x = 11.0, error +1.0: exp(-0.5 x 1.00) = exp(-0.5) = 0.606531, about 0.607.
  • x = 12.0, error +2.0: exp(-0.5 x 4.00) = exp(-2.0) = 0.135335, about 0.135.

Sum of likelihoods: 0.486752 + 0.882497 + 0.980199 + 0.606531 + 0.135335 = 3.091314, rounded to 3.091. Dividing each likelihood by that sum gives the normalized weights 0.157, 0.285, 0.317, 0.196, 0.044. The weighted estimate is:

  • 8.8 x 0.157458 + 9.5 x 0.285476 + 10.2 x 0.317082 + 11.0 x 0.196205 + 12.0 x 0.043779 = 10.015492 m, about 10.02 m.

Effective sample size then tests whether that vote has collapsed. For weights 0.62, 0.18, 0.10, 0.06, 0.04:

  • sum of squares = 0.62^2 + 0.18^2 + 0.10^2 + 0.06^2 + 0.04^2 = 0.3844 + 0.0324 + 0.0100 + 0.0036 + 0.0016 = 0.432000
  • ESS = 1 / 0.432000 = 2.314815, about 2.31 particles.
  • Threshold 0.5 x N = 0.5 x 5 = 2.5. Since 2.31 < 2.5, resample.

The audit conclusion is narrow: the estimate 10.02 m is trustworthy here because ESS 2.31 out of 5 is only just below the trigger, so several particles still carry weight. Report ESS with every estimate — when it falls toward 1, the same weighted-mean formula is quietly speaking for a single surviving hypothesis.

Every number above is taken from the chapter’s own material and re-derived step by step.