Core Concepts

Sharp, nonasymptotic bounds are obtained for the relative entropy between the distributions of sampling with and without replacement from an urn with balls of c ≥ 2 colors. The bounds depend on the number of balls of each color in the urn.

Abstract

The content presents a study of the relationship between sampling with and without replacement from an urn containing n balls of c ≥ 2 different colors. The authors derive sharp, nonasymptotic bounds on the relative entropy (Kullback-Leibler divergence) between the distributions of sampling with and without replacement.
Key highlights:
The relative entropy D(n, k, ℓ) is bounded above by an expression that depends on the number of balls of each color ℓ = (ℓ1, ℓ2, ..., ℓc) in the urn.
The first term of the bound matches the asymptotic expression obtained in prior work, showing that the new bound is a nonasymptotic version of the earlier result.
The dependence on ℓ is via two quantities Σ1(n, c, ℓ) and Σ2(n, c, ℓ), which are bounded in 'balanced' cases but can grow large in 'unbalanced' cases.
An alternative bound is provided for the c = 2 case, which outperforms the earlier uniform bounds across a wide range of parameter values.
The connection between finite de Finetti theorems and sampling bounds is explored, leading to a sharp finite de Finetti bound.

Stats

The following sentences contain key metrics or figures:
The relative entropy D(n, k, ℓ) between H and B satisfies:
D(n, k, ℓ) ≤ (c-1)/2 * log(n/(n-k)) - k/(n-1) + k(2n+1)/(12n(n-1)(n-k)) * Σ1(n, c, ℓ) + 1/360 * (1/(n-k)^3 - 1/n^3) * Σ2(n, c, ℓ).
For c = 2 and ℓ ≤ n/2, D(n, k, (ℓ, n-ℓ)) ≤ ℓ * [1 - k/n * log(1-k/n) + k/n - k/(2n(n-1))] + kℓ/(n-1)(n-ℓ)(n-k) - k(k-1)/(2n(n-1)ℓ(ℓ-1)) * log(ℓ/(ℓ-1)) - k(k-1)(k-2)/(6n(n-1)(n-2)ℓ(ℓ-1)(ℓ-2)) * log((ℓ-1)^2/(ℓ(ℓ-2))).

Quotes

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Key Insights Distilled From

by Oliver Johns... at **arxiv.org** 04-11-2024

Deeper Inquiries

The relative entropy bounds in the context provided depend not only on the summary statistics Σ1 and Σ2 but also on the specific distribution of balls across the c colors. The bounds take into account the number of balls of each color in the urn, which provides more detailed information about the composition of the urn. By incorporating this detailed information, the bounds can be made tighter as they capture the nuances of the distribution of balls across colors. This allows for a more precise comparison between sampling with and without replacement, leading to more accurate relative entropy bounds.

In finite de Finetti theorems, the optimal dependence on the alphabet size c is linear, as shown in the context provided. The linear dependence on the alphabet size c is considered optimal for the sampling problem, indicating that any upper bound that holds for any c, k, and n must have at least linear dependence on c. While the linear dependence is optimal in this context, it is possible that future research may uncover ways to improve upon this linear dependence and potentially achieve tighter bounds with a different functional relationship with the alphabet size c.

The relative entropy distances D(Pk||Mk,μ) may not necessarily be monotone in the sample size n, unlike the total variation distances which are known to exhibit monotonicity properties. While the total variation distances are nonincreasing in the sample size n, it is not guaranteed that the relative entropy distances follow the same pattern. Establishing a monotonicity property for relative entropy distances in sample size n could provide further insights into the behavior of the distributions and their convergence rates. Further research and analysis would be needed to determine the monotonicity properties of relative entropy distances in relation to sample size.

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