ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedjudge pass (gpt-6-luna)

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Gaussian barrier under the prescribed exponential tilt

Example

Assume AC. Take $P = N (0, 1)$ , target $A = [2, \infty)$ , and proposal $Q = N (2, 1)$ . The likelihood contribution is $Z = 1_{{X \geq 2}} \exp (- 2 X + 2)$ . The one-draw estimator is unbiased with finite second moment, with $E_{Q} Z = \overset{ˉ}{Φ} (2)$ and $E_{Q} Z^{2} = e^{4} \overset{ˉ}{Φ} (4)$ .

Facts & Assumptions

Given: The standard-normal target, barrier $2$ , translated-normal proposal, and likelihood contribution above.

[F1]

Gaussian tail tilting is logarithmically efficient but not bounded-relative-error: For $a > 0$ and $Q_{a} = N (a, 1)$ , the event contribution has mean $\overset{ˉ}{Φ} (a)$ and second moment $e^{a^{2}} \overset{ˉ}{Φ} (2 a)$ .

Verification

technique · direct

1.1F1givenalgebra

Set $a = 2$ in [F1]. Then $A_{a} = [2, \infty)$ , $Q_{a} = N (2, 1)$ , and the density ratio on the event is $\exp (- 2 X + 2)$ , giving the displayed contribution and moments.

2.1F1step 1.1algebra∎

Since $\overset{ˉ}{Φ} (2) > 0$ and $\overset{ˉ}{Φ} (4) < \infty$ , the mean is positive and finite and the second moment is finite. Thus the example has finite variance, while the asymptotic divergence in relative error concerns the family $a \to \infty$ , not this single barrier.

Depends on

Used by

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Dependency tree · two levels

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Sources

Gino Biondini, An Introduction to Rare Event Simulation and Importance Sampling, Handbook of Statistics 33, Chapter 2, §§3.1–3.4 (standard reference, not scraped)