Finite de Finetti bounds in relative entropy (2407.12921v1)

Abstract: We review old and recent finite de Finetti theorems in total variation distance and in relative entropy, and we highlight their connections with bounds on the difference between sampling with and without replacement. We also establish two new finite de Finetti theorems for exchangeable random vectors taking values in arbitrary spaces. These bounds are tight, and they are independent of the size and the dimension of the underlying space.

• The paper introduces two finite de Finetti theorems that provide tight relative entropy bounds for exchangeable sequences in arbitrary spaces.
• It improves classical results by deriving bounds, including a logarithmic expression, that refine measures beyond total variation.
• The findings offer near-optimal tools for probabilistic analysis, with significant implications for Bayesian statistics and sampling methodologies.

Finite De Finetti Bounds in Relative Entropy

The paper "Finite de Finetti bounds in relative entropy" by Lampros Gavalakis, Oliver Johnson, and Ioannis Kontoyiannis explores the field of exchangeable random variables, focusing on new results regarding finite de Finetti theorems in the context of relative entropy. The work is notable for establishing high-precision bounds for exchangeable random vectors in arbitrary spaces, enhancing our theoretical and practical understanding of de Finetti's representation theorem. Their results are particularly relevant for applications in sampling-based problems, Bayesian statistics, and information theory.

Overview and Background

Exchangeable random variables underpin various probabilistic models, providing a natural framework for extending i.i.d. sequences to more complex structures. De Finetti's representation theorem characterizes exchangeable sequences as mixtures of i.i.d. processes, a cornerstone result in probability theory. Classical de Finetti theorems typically involve total variation distance, but recent developments have focused on more refined measures such as relative entropy.

The authors bridge previous results with new, tight bounds on relative entropy for finite exchangeable sequences. This approach is motivated by the need for sharper asymptotic estimates beyond those provided by traditional total variation bounds. Their results also connect with practical considerations in sampling, such as the difference between sampling with and without replacement, which have direct implications for statistical and probabilistic methodologies.

Main Results

The paper's primary contributions can be summarized succinctly:

1. New Finite de Finetti Theorems: Two new finite de Finetti theorems for exchangeable random vectors in arbitrary spaces, demonstrating tight bounds on relative entropy:
   • The first theorem provides a bound of $\frac{k(k-1)}{2(n-k+1)}$, which is notable for its simplicity and direct connection to existing sampling bounds.
   • The second theorem improves this to $\log{(\frac{n^k(n-k)!}{n!})}$, which tightens the result further and is particularly sharp for all practical purposes.
2. Sampling Bounds in Relative Entropy: Building on Stam's classical bound, the authors present detailed calculations and proofs, demonstrating that the differences between sampling with and without replacement are bounded tightly in terms of relative entropy.
3. Tightness and Analyzed Bounds: The provided bounds are shown to be essentially tight, a significant result confirming that these are near-optimal under the given framework and constraints.

Implications and Future Directions

The implications of these results are both theoretical and practical:

• Theoretical Significance: The sharpened finite de Finetti bounds strengthen the connection between exchangeability and information-theoretic measures. This bridges gaps between different domains, providing a more unified view of probabilistic convergence and the behavior of exchangeable sequences.
• Practical Applications: These results have immediate applications to problems in Bayesian statistics, particularly in approximating distributions of partial sums and in the analysis of Bayesian estimators. They also serve as a robust foundation for algorithms in machine learning that rely on exchangeable data models, such as those found in topic modeling and collaborative filtering.

Conclusion

The paper by Gavalakis, Johnson, and Kontoyiannis makes a significant contribution to the field of probability theory and information theory by establishing new, tight finite de Finetti bounds in relative entropy. Their work not only refines our understanding of exchangeable random sequences but also lays the groundwork for future research in the effective application of these concepts in statistics and artificial intelligence. Future developments could explore extensions to continuous spaces and multidimensional scenarios, particularly how these relative entropy bounds can be employed in advanced AI models and probabilistic inference mechanisms.