  This paper focuses on inverse problems to identify parameters by
incorporating information from measurements. These generally ill-posed problems
are formulated here in a probabilistic setting based on Bayes's theorem because
it leads to a unique solution of the updated distribution of parameters. Many
approaches build on Bayesian updating in terms of probability measures or their
densities. However, the uncertainty propagation problems and their
discretisation within the stochastic Galerkin or collocation method are
naturally formulated for random vectors which calls for updating of random
variables, i.e. a filter. Such filters typically build on some approximation to
conditional expectation (CE). Specifically, the approximation of the CE with
affine functions leads to the familiar Kalman filter which works best on linear
or close to linear problems only. Our approach builds on a reformulation, which
allows to localise the operator of the CE to the point of measured value. The
resulting conditioned expectation (CdE) predicts correctly the quantities of
interest, e.g. conditioned mean and covariance, even for general highly
non-linear problems. The novel CdE allows straight-forward numerical
integration; particularly, the approximated covariance matrix is always
positive definite for integration rules with positive weights. The theoretical
results are confirmed by numerical examples.
