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Joint parameter and state estimation is proposed for linear state-space model with uniform, entry-wise correlated, state and output noises (LSU model for short). The adopted Bayesian modelling and approximate estimation produce an estimator that (a) provides the maximum a posteriori estimate enriched by information on its precision, (b) respects correlated noise entries without demanding the user to tune noise covariances, and (c) respects bounded nature of real-life variables.L. PAVELKOVÁ AND M. KÁRNÝ KF, however, faces significant, extensively counteracted, troubles with respect to both (b) and (c) as seen in the following representative samples of existing solutions.As for requirements (a) and (b), KF and its extensions work well when the noise covariances are well chosen. Then, they also provide adequate information on the estimate precision. The covariances are predominantly taken as design parameters of KF as their estimation represents a highly nonlinear problem [8,9]. The number of covariance entries grow quadratically with the state and output dimensions, which soon makes their experimental tuning infeasible. The number of opted entries can be decreased by a nontrivial parsimonious parametrisation [10] or made linearly growing in a factorised version of the state-space model [11,12], but the sensitivity to covariances choice persists, and neither (a) nor (b) are met.The requirement (c) can be fulfilled by a projection of the estimates onto the constraint surface via quadratic programming [13]. Another way is proposed by the probability density function (pdf) truncation approach [14] where the pdf of the state estimate is computed by the standard KF and then truncated at the constraint edges. The constrained state estimate is equal to the mean of the truncated pdf. The induced high-computational demands limit the degree of meeting (c). More importantly, techniques employing a projection or truncation in conjunction with the system model having unbounded support and light tails respect the constraints during estimation but not during modelling. Consequently, the posterior pdf forming the outcome of the Bayesian treatment is a worse estimator than necessary. Indeed, this pdf is a product of prior pdf on parameters and initial state, and likelihood function as a product of pdfs describing the state-space model with observed data inserted. The system model with the constrained states has a complex state and parameter-dependent normalising factor entering the likelihood. This factor is neglected by the discussed techniques.The complexity of the support and form of the exact posterior pdf corresponding to the adequate modelling of boundedness limits feasibility of the Bayesian estimation. Nevertheless, it is desirable to address it as there is a broad range of problems in which explicit inclusion of constraints into the system model significantly improves the estimation quality.The correct modelling of state boundedness and elaboration of the corresponding joint estimator providing both point estimates and info...