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

References

[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.. MatzP.. 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. GerstoftG. Matz, and C.F. Mecklenbräuker"Analytic Sequential Weiss-Weinstein Bounds", IEEE Transactions on Signal Processing, vol. 61, no. 20, pp. 5049-5062, 2013.