### Abstract

A random variable with distribution P is observed in Gaussian noise and is estimated by a mismatched minimum mean-square estimator that assumes that the distribution is Q, instead of P. This paper shows that the integral over all signal-to-noise ratios (SNRs) of the excess mean-square estimation error incurred by the mismatched estimator is twice the relative entropy D(P ∥ Q) (in nats). This representation of relative entropy can be generalized to nonreal-valued random variables, and can be particularized to give new general representations of mutual information in terms of conditional means. Inspired by the new representation, we also propose a definition of free relative entropy which fills a gap in, and is consistent with, the literature on free probability.

Original language | English (US) |
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Article number | 5508632 |

Pages (from-to) | 3712-3720 |

Number of pages | 9 |

Journal | IEEE Transactions on Information Theory |

Volume | 56 |

Issue number | 8 |

DOIs | |

State | Published - Aug 1 2010 |

### All Science Journal Classification (ASJC) codes

- Information Systems
- Computer Science Applications
- Library and Information Sciences

### Keywords

- Divergence
- Shannon theory
- free probability
- minimum mean- square error (MMSE) estimation
- mutual information
- relative entropy
- statistics

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## Cite this

*IEEE Transactions on Information Theory*,

*56*(8), 3712-3720. [5508632]. https://doi.org/10.1109/TIT.2010.2050800