References & Further Reading

References

1. S. M. Kay, Fundamentals of Statistical Signal Processing, Volume I: Estimation Theory, Prentice Hall, 1993

   Chapters 10–12 provide the most accessible treatment of Bayesian estimation, MMSE, and LMMSE in the signal-processing literature.

2. H. L. Van Trees, K. L. Bell, and Z. Tian, Detection, Estimation, and Modulation Theory, Part I, Wiley, 2nd ed., 2013

   The classical reference. The Van Trees inequality (Bayesian CRLB) is developed here.

3. H. V. Poor, An Introduction to Signal Detection and Estimation, Springer, 2nd ed., 1994

   Rigorous treatment of orthogonality, projections, and MMSE in Hilbert spaces. Chapter IV is directly relevant.

4. A. Gelman et al., Bayesian Data Analysis, Chapman & Hall/CRC, 3rd ed., 2013

   Comprehensive modern treatment of Bayesian inference. Chapters 2–3 cover priors, posteriors, and point estimates.

5. N. Wiener, Extrapolation, Interpolation, and Smoothing of Stationary Time Series, MIT Press, 1949

   The original LMMSE filter for WSS processes. Declassified 1949 from a 1942 classified report.

6. A. N. Kolmogorov, Interpolation und Extrapolation von stationären zufälligen Folgen, 1941

   Kolmogorov's independent discovery of the Wiener filter.

7. T. Bayes and R. Price, An Essay towards solving a Problem in the Doctrine of Chances, 1763

   The original (posthumous) essay introducing what is now called Bayes' theorem.

8. P.-S. Laplace, Théorie Analytique des Probabilités, Courcier (Paris), 1812

   Laplace's systematic development of inverse probability, rediscovering and vastly extending Bayes.

9. D. Guo, S. Shamai, and S. Verdú, Mutual information and minimum mean-square error in Gaussian channels, 2005

   The I-MMSE identity: the derivative of mutual information with respect to SNR is half the MMSE.

10. D. Neumann, T. Wiese, W. Utschick, and G. Caire, Learning the MMSE Channel Estimator, 2018

    Modern deep-learning approach to Bayesian channel estimation with unknown covariance. CommIT contribution.

11. M. Biguesh and A. B. Gershman, Training-based MIMO channel estimation: A study of estimator tradeoffs and optimal training signals, 2006

    Detailed LS vs. MMSE comparison for MIMO pilot design.

12. L. L. Scharf, Statistical Signal Processing: Detection, Estimation, and Time Series Analysis, Addison-Wesley, 1991

    Geometric treatment of LMMSE as projection. Chapter 8 is especially relevant.

Further Reading

• Bayesian nonparametrics and Gaussian processes

  Rasmussen & Williams, *Gaussian Processes for Machine Learning*, 2006

  Infinite-dimensional Bayesian regression; the LMMSE estimator of this chapter is the finite-dimensional restriction.

• Bayesian Cramér–Rao and Ziv–Zakai bounds

  Van Trees & Bell, *Bayesian Bounds for Parameter Estimation and Nonlinear Filtering/Tracking*, 2007

  Lower bounds tighter than the Van Trees inequality, including in the large-error regime.

• Variational Bayes

  Blei, Kucukelbir, McAuliffe, *Variational Inference: A Review for Statisticians*, 2017

  When closed-form posteriors don't exist (Chapter 8), variational methods approximate the posterior by optimization.

• Robust Bayesian estimation

  Berger, *Statistical Decision Theory and Bayesian Analysis*, 1985

  What to do when the prior is only approximately known — directly relevant to the channel-covariance mismatch issue.