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In this paper, we analyze the operational information rate distortionfunction (RDF) ${R}_{S;Z|Y}(\Delta_X)$, introduced by Draper and Wornell, for atriple of jointly independent and identically distributed, multivariateGaussian random variables (RVs), $(X^n, S^n, Y^n)= \{(X_{t}, S_t, Y_{t}):t=1,2, \ldots,n\}$, where $X^n$ is the source, $S^n$ is a measurement of $X^n$,available to the encoder, $Y^n$ is side information available to the decoderonly, $Z^n$ is the auxiliary RV available to the decoder, with respect to thesquare-error fidelity, between the source $X^n$ and its reconstruction$\widehat{X}^n$. We also analyze the RDF ${R}_{S;\widehat{X}|Y}(\Delta_X)$ thatcorresponds to the above set up, when side information $Y^n$ is available tothe encoder and decoder. The main results include,  (1) Structural properties of test channel realizations that inducedistributions, which achieve the two RDFs,  (2) Water-filling solutions of the two RDFs, based on parallel channelrealizations of test channels,  (3) A proof of equality ${R}_{S;Z|Y}(\Delta_X) ={R}_{S;\widehat{X}|Y}(\Delta_X)$, i.e., side information $Y^n$ at both theencoder and decoder does not incur smaller compression, and  (4) Relations to other RDFs, as degenerate cases, which show past literature,contain oversights related to the optimal test channel realizations and valueof the RDF ${R}_{S;Z|Y}(\Delta_X)$.

Structural Properties of Optimal Test Channels for Gaussian Multivariate Partially Observable Distributed Sources