  The entropy bottleneck introduced by Ball\'e et al. is a common component
used in many learned compression models. It encodes a transformed latent
representation using a static distribution whose parameters are learned during
training. However, the actual distribution of the latent data may vary wildly
across different inputs. The static distribution attempts to encompass all
possible input distributions, thus fitting none of them particularly well. This
unfortunate phenomenon, sometimes known as the amortization gap, results in
suboptimal compression. To address this issue, we propose a method that
dynamically adapts the encoding distribution to match the latent data
distribution for a specific input. First, our model estimates a better encoding
distribution for a given input. This distribution is then compressed and
transmitted as an additional side-information bitstream. Finally, the decoder
reconstructs the encoding distribution and uses it to decompress the
corresponding latent data. Our method achieves a Bj{\o}ntegaard-Delta (BD)-rate
gain of -7.10% on the Kodak test dataset when applied to the standard
fully-factorized architecture. Furthermore, considering computational
complexity, the transform used by our method is an order of magnitude cheaper
in terms of Multiply-Accumulate (MAC) operations compared to related
side-information methods such as the scale hyperprior.
