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 Incoming links: Effective sample size of importance sampling

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 219 / 2 / c / Solution 2026-10-03

For arbitrary nonnegative raw weights

W_{i}

, let

\overset{ˉ}{W} = K^{- 1} \sum_{i} W_{i}

. The empirical squared coefficient of variation, using variance divisor

K

, is

CV^{2} (W) = \frac{K ^{- 1} \sum _{i} ( W _{i} - W ˉ ) ^{2}}{W ˉ ^{2}} = \frac{K \sum _{i} W _{i}^{2}}{( \sum _{i} W _{i} ) ^{2}} - 1.

Substitution into the stated definition gives the usual effective sample size of importance sampling

ESS = \frac{( \sum _{i} W _{i} ) ^{2}}{\sum _{i} W _{i}^{2}} .

For normalized weights

w_{i} = W_{i} / \sum_{j} W_{j}

, this reduces to

ESS = 1/ i \sum w_{i}^{2}

.

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Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 219 / 2 / c / Solution Created 2026-09-24 Updated 2026-09-25

If equally informative independent draws have variance

s_{m}^{2}

, an ordinary mean of

K_{eff}

draws has variance

s_{m}^{2} / K_{eff}

. The weighted mean has the corresponding variance

s_{m}^{2} \sum_{i} w_{i}^{2}

. Equating them gives the effective sample size of importance sampling

K_{eff} = \frac{1}{\sum _{i = 1}^{K} w _{i}^{2}} = \frac{( \sum _{i} p ( d ∣ x _{i} ) ) ^{2}}{\sum _{i} p ( d ∣ x _{i} ) ^{2}} \leq K,

where the inequality follows from Cauchy-Schwarz inequality.

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Past exam of the MSc in advanced computer science course of the University of Oxford / 2021 / Bayesian statistical probabilistic programming / 3 / e / Solution 2026-10-25

I would first collapse the duration noises as in part (c), then use self-normalized importance sampling on the two shared parameters. Draw independent

(μ_{j}, σ_{j}) \sim ρ

, give each the importance weight

w_{j} = L (μ_{j}, σ_{j})

, and return

P_{N} = \frac{\sum _{j = 1}^{N} w _{j} π ( μ _{j} , σ _{j} ) ^{5}}{\sum _{j = 1}^{N} w _{j}} .

This is likelihood weighting: the observations affect the weights rather than forcing repeated simulation of twenty latent inequalities. Integrating out both the constraints' auxiliary randomness and the future Boolean query is Rao-Blackwellization; at fixed parameters it removes randomness without changing the relevant conditional expectations.

The weights and weighted queries lie in

[0, 1]

, and the evidence is positive. The strong law of large numbers therefore makes the numerator and denominator averages converge to their corresponding integrals, proving almost sure convergence of their ratio to part (d). The finite-

N

ratio is generally biased; the assertion concerns consistency rather than a finite-sample unbiased estimator. Use logarithms of importance weights to avoid numerical underflow:

def weighted_answer(n, rng):
    if n <= 0:
        raise ValueError("n must be positive")
    samples = []
    for _ in range(n):
        mu = rng.uniform(0.0, 60.0)
        sigma = rng.uniform(1.0, 20.0)
        samples.append((journey_loglik(mu, sigma),
                        math.exp(5.0 * normal_logcdf((mu - 30.0) / sigma))))
    offset = max(logw for logw, h in samples)
    if not math.isfinite(offset):
        raise ArithmeticError("all weights underflowed")
    scaled = [(math.exp(logw - offset), h) for logw, h in samples]
    den = math.fsum(w for w, h in scaled)
    answer = math.fsum(w * h for w, h in scaled) / den
    ess = den * den / math.fsum(w * w for w, h in scaled)
    return answer, ess

The returned effective sample size of importance sampling is a concentration diagnostic, not a proof of accuracy. Repeat runs and inspect the weights. If the prior is much broader than the posterior, improve the proposal with the corresponding importance sampling density correction, or use the Metropolis–Hastings algorithm targeting

ρ L

. Because this particular collapsed problem is only two-dimensional, deterministic numerical integration is also an excellent independent accuracy check; it was used for the figure and numerical illustration. Straight rejection of the original traces is correct but wastes most proposals here.

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