Prerequisites & Notation

Before You Begin

This chapter is research-level. It synthesises ideas from Bayesian estimation, hypothesis testing, information theory, and wireless signal processing. If any of the following feels rusty, the chapter will feel faster than its prerequisites allow.

Fisher information and the Cramer-Rao bound(Review FSI Ch. 18)
Self-check: Can you state the CRLB for a scalar parameter and explain why it may be loose at low SNR?
Bayesian MMSE estimation and the posterior mean(Review FSI Ch. 7-8)
Self-check: Can you derive the MMSE estimator for a Gaussian prior and linear Gaussian observation?
Binary hypothesis testing, minimum probability of error(Review FSI Ch. 1)
Self-check: Can you write the minimum error probability between two equi-prior Gaussians with means $\theta_0, \theta_1$ and common variance $\sigma^2$ ?
Mutual information and differential entropy(Review ITA Ch. 2-3)
Self-check: Can you compute $I(X; Y)$ when $Y = \sqrt{\text{snr}}\,X + N$ with $X \sim \mathcal{N}(0,1)$ ?
Convex optimization and KKT conditions(Review Telecom Ch. 3)
Self-check: Can you set up a convex problem with linear constraints and recognise when KKT is sufficient?

Notation for This Chapter

Symbols introduced in this chapter. Van Trees, Ziv-Zakai, and I-MMSE use overlapping notation in the literature; we follow the conventions that minimise collisions with the rest of the book.

Symbol	Meaning	Introduced
$\theta$	Scalar (or vector) parameter to be estimated, treated as random under a Bayesian prior	s01
$\pi(\theta)$	Prior density on $\theta$ (must vanish at the boundary of its support for Van Trees)	s01
$I_F(\theta)$	(Classical, frequentist) Fisher information at parameter value $\theta$	s01
$I_P$	Prior information: $I_P = \mathbb{E}_\pi\!\left[\left(\partial_\theta \log \pi(\theta)\right)^2\right]$	s01
$I_B$	Bayesian information: $I_B = \mathbb{E}_\pi[I_F(\theta)] + I_P$	s01
$P_{\min}(\theta_0,\theta_1)$	Minimum probability of error between two equi-prior hypotheses $\theta_0,\theta_1$	s02
$\text{ZZB}$	Ziv-Zakai bound on MSE	s02
$\text{snr}$	Scalar signal-to-noise ratio parameter in the I-MMSE identity	s03
$\text{mmse}(\text{snr})$	MMSE of estimating $X$ from $Y=\sqrt{\text{snr}}\,X+N$ where $N\sim\mathcal{N}(0,1)$	s03
$\mathbf{R}_s$	Transmit sample covariance matrix in ISAC, $\mathbf{R}_s = \mathbb{E}[\mathbf{s}\mathbf{s}^H]$	s04

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