# To appear in Trans. Inf. Theory

W. Yang, A. Collins, G. Durisi, Y. Polyanskiy, and H. V. Poor, “Beta-beta bounds: Finite-blocklength analog of the golden formula,” *IEEE Tran. Inf. Theory*, 2018, to appear [arXiv].

**Abstract**:
It is well known that the mutual information between two random variables can be expressed as the difference of two relative entropies that depend on an auxiliary distribution, a relation sometimes referred to as the golden formula. This paper is concerned with a finite-blocklength extension of this relation. This extension consists of two elements: 1) a finite-blocklength channel-coding converse bound by Polyanskiy and Verdú (2014), which involves the ratio of two Neyman-Pearson $\beta$ functions (beta-beta converse bound); and 2) a novel beta-beta channel-coding achievability bound, expressed again as the ratio of two Neyman-Pearson $\beta$ functions.
To demonstrate the usefulness of this finite-blocklength extension of the golden formula, the beta-beta achievability and converse bounds are used to obtain a finite-blocklength extension of Verdú’s (2002) wideband-slope approximation. The proof parallels the derivation of the latter, with the beta-beta bounds used in place of the golden formula.
The beta-beta (achievability) bound is also shown to be useful in cases where the capacity-achieving output distribution is not a product distribution due to, e.g., a cost constraint or structural constraints on the codebook, such as orthogonality or constant composition. As an example, the bound is used to characterize the channel dispersion of the additive exponential-noise channel and to obtain a finite-blocklength achievability bound (the tightest to date) for multiple-input multiple-output Rayleigh-fading channels with perfect channel state information at the receiver.