# Sequential Weiss-Weinstein Bounds

Sequential Bayesian estimation and detection [VB07] inferes temporal evolving states in contrast to non-evolving parameters. The sequential Weiss-Weinstein bound is a lower bound on the mean-square-error matrix of any Bayesian estimator. This article is based on  Bayesian Cramer-Rao Bounds and Weiss-Weinstein Bounds about non-sequential Bayesian inference and performance bounds. Note that here only a short overview is possible and hence I neglect many details that can be found in one of the references.

# Sequential Bayesian Inference

Compared to the non-sequential inference of parameters, in the sequential case the measurement model is added by a state-transition model. We obtain the state-space model

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

$\boldsymbol{x}_{k+1} = \Phi_k(\boldsymbol{x}_k)~,$

where a sensor modeled by a probabilistic mapping $C_k$ measures a random state $\boldsymbol{x}_k$ at time $k \in \mathbb{N}_0$. Initial state $\boldsymbol{x}_0$ evolves over time according the state-transition function $\Phi_k$.  The Bayesian modelling demands for  following probability densities (probability mass functions or probability density functions):

$v(\boldsymbol{x}_0), \quad v(\boldsymbol{x}_{k+1}|\boldsymbol{x}_k),\quad v(\boldsymbol{y}_k|\boldsymbol{x}_k) ~.$

The sequential estimator $\hat{\boldsymbol{x}}_k(\boldsymbol{y}_k)$ tries to infer state $\boldsymbol{x}_k$ using measurements $\boldsymbol{y}_k$.  The mean-square-error matrix $\mathrm{E}(\tilde{\boldsymbol{x}}_k\tilde{\boldsymbol{x}}_k^{\mathrm{T}})$ is defined as the estimation error $\tilde{\boldsymbol{x}}_k = \hat{\boldsymbol{x}}_k(\boldsymbol{y}_k) - \boldsymbol{x}_k$ at time $k$.

One prominant relative of the family of sequential Weiss-Weinstein bounds [RO04, Xav+13,Xav13] is the sequential Bayesian Cramer-Rao bound [Tic+98].

# Sequential Weiss-Weinstein Bound

The sequential Weiss-Weinstein bound $\boldsymbol{W}_k$ at $k$ is a lower bound of the mean-square-error matrix of any Bayesian estimator
$\mathrm{E}(\tilde{\boldsymbol{x}}\tilde{\boldsymbol{x}}^{\mathrm{T}}) \succcurlyeq \boldsymbol{W}_k = \boldsymbol{H}_k \boldsymbol{J}_{k}^{-1} \boldsymbol{H}_k^{\mathrm{T}}$
with following recursion
$\boldsymbol{A}_{k} = \boldsymbol{D}_{k+1}^{11} - \boldsymbol{D}_{k+1}^{10} \boldsymbol{A}_{k-1}^{-1} \boldsymbol{D}_{k+1}^{01}~,$
$\boldsymbol{J}_{k+1} = \boldsymbol{D}_{k+1}^{22} - \boldsymbol{D}_{k+1}^{21} \boldsymbol{A}_{k}^{-1} \boldsymbol{D}_{k+1}^{12}~.$

The relational operator $\succcurlyeq$ indicates that the difference between left and right hand sides is a positive semi-definite matrix. Every element of $\boldsymbol{D}_{k+1}^{ij}$ uses the Bhattacharyya coefficient $\rho$ (see elements of $\boldsymbol{J}$ in Bayesian Cramer-Rao Bounds and Weiss-Weinstein Bounds for details) that depends on the probability densities presented above. The columns of $\boldsymbol{H}_k$ are test points. They are abitrary with two exceptions:

1. Finite supports of probability distributions induces  box conditions on the test points.
2. If the states are in a finite alphabet, the test points are in a finite alphabet as well (sequential detection).

For linear state-space models, the bound simplifies to
$\boldsymbol{W}_k = \boldsymbol{H}_k \boldsymbol{J}_{k}^{-1} \boldsymbol{H}_k^{\mathrm{T}}$
with
$\boldsymbol{J}_{k+1} = \boldsymbol{D}_{k+1}^{22} - \boldsymbol{D}_{k+1}^{21} ( \boldsymbol{D}_{k+1}^{11} + \boldsymbol{J}_k - \boldsymbol{B}_k^{11} )^{-1} \boldsymbol{D}_{k+1}^{12} ~.$

# Facts

• The SWW bound is a generalization of the SCR bound [RO04,Xav+13,Xav13]
• SWW bounds support any type of probability distribution with density [Xav+13,Xav13], i.e. probability density function and probability mass function
• Finite supports of probability distributions induces  box conditions on test-points. Discrete random states induce test points that are in a finite alphabet [Xav+13,Xav13].
• Linear state-space model [Xav+13,Xav13]:
• Analytic solutions exist for various densities
• Solutions have the same structure
• The computational effort stays constant in time
• Non-linear state-space model [Xav+12]:
• There is no general analytic solution
• Models with continuous random states and discrete random states from categorical densities have closed-form solutions.
• The computational effort is quadratic in time

[RO04]  Rapoport, I. and Y. Oshman  ”A new estimation error lower bound for interruption indicators in systems with uncertain measurements”. In: IEEE Trans. Inf. Theory 50.12, pp. 3375–3384, 2004

[Tic+98] Tichavsky, P., C. H. Muravchik, and A. Nehorai . “Posterior Cramer-Rao bounds for discrete-time nonlinear filtering”. In: IEEE Trans. Signal Process. 46.5, pp. 1386–1396, 1998

[VB07] H. L. Van Trees et al. Bayesian bounds for parameter estimation and nonlinear filtering/tracking. IEEE Press, 2007.

[WW88] A. J. Weiss et al. “A general class of lower bounds in parameter estimation.” In: IEEE Trans. Inf. Theory 34.2 (1988), pp. 338–342.

[Xav+12]  Xaver, F., G. Matz, P. Gerstoft, and N. Görtz, ”Localization of acoustic sources utilizing a decentralized particle filter”, **Proc. 46th Asilomar Conf. Signals, Syst., Comput., Pacific Grove, CA, Nov., 2012.

[Xav13] Xaver, F., Decentralized localization based on wave fields — Particle filters and Weiss-Weinstein error bounds: Institute of Telecommunications, Vienna University of Technology, 2013

[Xav+13]  Xaver, F., P. Gerstoft, G. Matz, and C.F. Mecklenbräuker, ”Analytic Sequential Weiss-Weinstein Bounds”, IEEE Transactions on Signal Processing, vol. 61, no. 20, pp. 5049-5062, 2013.