  Interior characterization traditionally relies on individual planetary
properties, ignoring correlations between different planets of the same system.
For multi-planetary systems, planetary data are generally correlated. This is
because, the differential masses and radii are better constrained than absolute
planetary masses and radii. We explore such correlations and data specific to
the multiplanetary-system of TRAPPIST-1 and study their value for our
understanding of planet interiors. Furthermore, we demonstrate that the rocky
interior of planets in a multi-planetary system can be preferentially probed by
studying the most dense planet representing a rocky interior analogue. Our
methodology includes a Bayesian inference analysis that uses a Markov chain
Monte Carlo scheme. Our interior estimates account for the anticipated
variability in the compositions and layer thicknesses of core, mantle, water
oceans and ice layers, and a gas envelope. Our results show that (1) interior
estimates significantly depend on available abundance proxies and (2) that the
importance of inter-dependent planetary data for interior characterization is
comparable to changes in data precision by 30 %. For the interiors of
TRAPPIST-1 planets, we find that possible water mass fractions generally range
from 0-25 %. The lack of a clear trend of water budgets with orbital period or
planet mass challenges possible formation scenarios. While our estimates change
relatively little with data precision, they critically depend on data accuracy.
If planetary masses varied within ~24 %, interiors would be consistent with
uniform (~7 %) or an increasing water mass fractions with orbital period (~2-12
%).
