Weights, Estimate, and Effective Sample Size

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.

See the relationship before changing it

Set the input, apply the rule, then read the result with its unit.

Derive the baseline in four named moves

  1. 1

    Name the input. The chapter baseline is 50 % of 5 particles.

  2. 2

    Name the relationship. threshold = fraction x 5 particles

  3. 3

    Substitute with units. 50 / 100 x 5 particles = 2.50 particles

  4. 4

    Read the result. Keep the unit beside the value. Use it only inside the technical boundary on this page.

Predict, then change resampling threshold

Try Predict the direction of threshold = fraction x 5 particles. Test another resampling threshold, then compare threshold.

50 % of 5 particles
Chapter baseline
Threshold

Observe A higher threshold resamples sooner. It does not change the measured effective sample size. Reset resampling threshold to 50 and compare threshold.

Explain The resampling rule compares effective population with a chosen fraction of five particles, so a larger fraction raises the particle-count trigger proportionally.

Check yourself

What should you do before trusting a moved-control result?
Answer: Predict its direction, apply the shown relationship, keep the units, and reset to the worked baseline.
What does this small model leave out?
Answer: Only resampling threshold moves here. Field effects named in the technical boundary stay fixed.

Honesty note. This teaching model changes only resampling threshold. It is not a field prediction.

Try

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. Calculate this case.

Observe

This audit confirms the weights, the estimate, and the effective sample size that decides when to resample. Check shows this.

Explain

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. Check confirms it.

Technical boundaries

The threshold is a chosen fraction of a fixed set of five particles; changing it does not change their measured effective sample size. The calculation omits poor choices of candidate states, resampling variability, loss of particle diversity, nonlinear motion constraints, map error, and computation limits.

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.