independent observations from the same distribution, then associated minimum variance unbiased estimators are symmetric in the observations [@Rao1965]. Therefore when coding for estimation, the multiset of observations is all that need be represented, cf. [@HanA1998]. Moving beyond the point-to-point case, in distributed inference, often particle-based [@ArulampalamMGC2002; @IhlerFMW2005] and kernel-based [@Parzen1962; @IhlerFW2004] representations of densities must be communicated. As $$p(x) = \sum_{i} \phi\left(x - x_i\right)
     = \sum_{i} \phi\left(x - x_{\pi(i)}\right)$$ for any permutation $\pi(\cdot)$, the multiset of representation coefficients $\{x_i\}$ may be communicated rather than the sequence of these values $(x_i)$. This extends to any destination that performs permutation-invariant computations.

The aim of this paper is to develop ramifications of order irrelevance on source coding problems. We consider lossless coding, universal lossless coding, high-rate and low-rate quantization, and rate distortion theory. In all of these, the rate requirement is obviously reduced by making order irrelevant. The reduction can be dramatic: for lossless coding of $n$ symbols, the required rate is changed from $O(n)$ to $O(\log n)$; for lossy coding, with large enough $n$ arbitrarily small mean squared error (MSE) can be achieved with zero rate. These examples are made precise in the sequel.

Notation and Formalism
----------------------

We consider the encoding 