Hi,

I'm doing a research on a variation of Recursive Bayesian Estimation- it's a Bayesian estimation with a 'time smoothing' term:

$\displaystyle { \hat { P } }_{ k+1 }(x)\quad =\quad \eta \cfrac { { \hat { P } }_{ k }(x)p({ z }_{ k+1 }|x) }{ \int _{ -\infty }^{ \infty }{ { \hat { P } }_{ k }(x)p({ z }_{ k+1 }|x)dx } } \quad +\quad (1-\eta ){ \hat { P } }_{ k }(x)\\ where:\\\\ { \hat { P } }_{ k }(x)\quad is\quad the\quad esimated-prior\quad function,\quad at\quad time\quad k\\\\ z_{ k }\quad \in \quad \Re ,\quad is\quad a\quad measurement\quad presented\quad at\quad time\quad k\\\\ \eta \quad \in \quad [0,1]\quad is\quad the\quad learning\quad rate\quad constant\quad \\\\ p(z|x)\quad is\quad the\quad likelihood\quad function:\quad \quad p(z|x)\quad =\quad \cfrac { 1 }{ \sqrt { 2\pi } \sigma } exp\{ \cfrac { { (z-x) }^{ 2 } }{ { 2\sigma }^{ 2 } } \} $

My question is not regarding the math itself, but regarding its context:

This recursive relationdoes notwork well as an estimator, but it does display some interesting behavior (clustering, bifurcations as function of sigma and eta)

Can anyone suggest a field or a case where this recursive relation is/could be manifested? maybe a natural or an artificial system where a recursive bayesian estimation is applied, but the system fails to completely "forget" its previous state, and therefor a time-smoothing term is present.

Thank you