Particle filter — implementation notes.

This note records a compact reference for the particle-filter interpretation of Iₜ = C(Σ(Eₜ) ± Δₜ). It is intentionally pragmatic: equations, pseudocode, and a tiny JS implementation you can drop into a demo.

Core steps (per timestep): (1) weight each particle by likelihood given new evidence; (2) normalize weights; (3) resample particles according to weights; (4) apply the constancy term C as a shrinkage toward prior; (5) apply Δ as a value-directed nudge.

Reference JS (minimal):

function updateParticles(particles, weights, evidenceLikelihood, C = 0.9, delta = 0) {
  // weights: array of positive numbers
  // evidenceLikelihood(p, e) -> likelihood
  const newWeights = particles.map((p, i) => weights[i] * evidenceLikelihood(p, evidence));
  const sum = newWeights.reduce((s, w) => s + w, 0) || 1;
  const norm = newWeights.map(w => w / sum);
  // resample
  const resampled = [];
  for (let i = 0; i < particles.length; i++) {
    const r = Math.random();
    let acc = 0;
    for (let j = 0; j < norm.length; j++) { acc += norm[j]; if (r <= acc) { resampled.push(clone(particles[j])); break; } }
  }
  // apply constancy C and delta
  return resampled.map(p => {
    // shrink toward prior (here: mean of previous particles) then nudge by delta
    const mean = particles.reduce((m, q) => ({ x: m.x + q.x, y: m.y + q.y }), { x:0, y:0 });
    mean.x /= particles.length; mean.y /= particles.length;
    return {
      x: mean.x * (1 - C) + p.x * C + delta,
      y: mean.y * (1 - C) + p.y * C + delta,
    };
  });
}

Notes: this is a pedagogical reference, not a production-optimized implementation. Use stratified resampling, systematic resampling, or low-variance resampling for stability in production.