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This work presents sequential Bayesian detection and estimation methods for nonlinear dynamic stochastic systems using measurements affected by three sources of uncertainty: stochastic, set-theoretic and data association uncertainty. Following Mahler's framework for information fusion, the paper develops the optimal Bayes filter for this problem in the form of the Bernoulli filter for interval measurements. Two numerical implementations of the optimal filter are developed. The first is the Bernoulli particle filter (PF), which turns out to require a large number of particles in order to achieve a satisfactory performance. For the sake of reduction in the number of particles, the paper also develops an implementation based on box particles, referred to as the Bernoulli Box-PF. A box particle is a random sample that occupies a small and controllable rectangular region of nonzero volume in the target state space. Manipulation of boxes utilizes the methods of interval analysis. The two implementations are compared numerically and found to perform remarkably well: the target is reliably detected and the posterior probability density function of the target state is estimated accurately. The Bernoulli Box-PF, however, when designed carefully, is computationally more efficient.Index Terms-Bernoulli filter, box particle filters, interval measurements, particle filters, random sets, sequential Bayesian estimation.