We consider the indirect source coding problem of a Gaussian random walk decimated by the factor M, where the non-decimated source is hidden.  In the case where M is known at the encoder, we derive the DRF of the optimal encoding-recovery scheme estimate-and-compress.  We then consider the case in which M is known only at the decoder.  In this case, estimate-and-compress is no longer achievable, and the optimal encoding-recovery scheme is unknown.  We characterize the DRF of a common yet not necessarily optimal scheme, compress-and-estimate, and compare this with the upper-bounding estimate-and-compress.