# Timeline of Blog

## Performance Bounds for Bayesian Estimators: Cramer-Rao Bounds and Weiss-Weinstein Bounds

Bayesian performance bounds are supposed to benchmark Bayesian estimators and detectors, which infer random parameters of interest from noisy measurements. These parameters are usually physical values as temperature, position, etc.

Consider a measurement model

$\boldsymbol{y} = C(\boldsymbol{x})~,$

where a sensor modeled by a probabilistic mapping $C$ measures a random parameter vector  $\boldsymbol{x}$. An vector-valued estimator  $\hat{\boldsymbol{x}}(\boldsymbol{y})$ infers the parameter vector using random measurements $\boldsymbol{y}$. A simple example adds noise to the paramter, i.e.

$\boldsymbol{y} = \boldsymbol{x} + \boldsymbol{v}~,$

where the random vector $\boldsymbol{v}$ models measurement noise.

A performance bound [VB07] is a lower bound on the mean-square-error matrix $\mathrm{E}(\tilde{\boldsymbol{x}}\tilde{\boldsymbol{x}}^{\mathrm{T}})$ for the estimation error $\tilde{\boldsymbol{x}} = \hat{\boldsymbol{x}}(\boldsymbol{y}) - \boldsymbol{x}$ of any Bayesian estimator. This "any" is in contrast to the traditional frequentist Cramer-Rao bound.

A popular performance bound is the van-Tree-Cramer-Rao (Bayesian Cramer-Rao) bound which is the Bayesian version of the traditional Cramer-Rao bound. It is a relative of the family of Weiss-Weinstein bounds, which in turn is a subclass of the family of Bayesian lower bounds. With these bounds it is possible to compare different Bayesian estimators. Note that $\tilde{\boldsymbol{x}}\tilde{\boldsymbol{x}}^{\mathrm{T}}$ is the square-error loss  that provides the minimum-mean-square-error (MMSE) estimator. Hence, all other Bayesian estimators are worse with respect to  loss $\tilde{\boldsymbol{x}}\tilde{\boldsymbol{x}}^{\mathrm{T}}$ than the MMSE estimator (cf. Algebraic vs. Frequentist vs. Bayesian Inference).