  In order to handle large data sets omnipresent in modern science, efficient
compression algorithms are necessary. Here, a Bayesian data compression (BDC)
algorithm that adapts to the specific measurement situation is derived in the
context of signal reconstruction. BDC compresses a data set under conservation
of its posterior structure with minimal information loss given the prior
knowledge on the signal, the quantity of interest. Its basic form is valid for
Gaussian priors and likelihoods. For constant noise standard deviation, basic
BDC becomes equivalent to a Bayesian analog of principal component analysis.
Using Metric Gaussian Variational Inference, BDC generalizes to non-linear
settings. In its current form, BDC requires the storage of effective instrument
response functions for the compressed data and corresponding noise encoding the
posterior covariance structure. Their memory demand counteract the compression
gain. In order to improve this, sparsity of the compressed responses can be
obtained by separating the data into patches and compressing them separately.
The applicability of BDC is demonstrated by applying it to synthetic data and
radio astronomical data. Still the algorithm needs further improvement as the
computation time of the compression and subsequent inference exceeds the time
of the inference with the original data.
