A time sequence of counts of the number of sensor detections is the sum of the number of detections of objects and the number of false alarms. We model the count sequence as an insertion-deletion process, and model the time-varying number of objects as a birth-death process. Under these modeling assumptions, we derive the optimal recursive Bayesian posterior distribution for the number of objects conditioned only on the count sequence. The method is potentially applicable in management science to detect changes in demand in decisionindependent observed data streams and in social media to estimate the number of users who abuse hashtags. A maximum a posteriori (MAP) algorithm for estimating the parameters of the birth-death and insertion-deletion processes is presented.

Multi-object filtering is a generalisation of stochastic filtering to deal with an unknown and time-varying number of targets, largely based on modelling with point processes. Some early works on this topic from the Soviet Union from 1960s-l980s appeared prior to well known results in the contemporary liter-ature. This article reviews some of these historical contributions.

The method of analytic combinatorics and labeling is shown to be a unifying framework in which to pose both high and low level data fusion problems. The method uses labeled generating functions. Several examples from high level fusion and multiple target tracking are given. Examples from high level fusion include natural language processing and noisy graph association problems. Examples from multitarget tracking include multidimensional assignment problems, unlabeled and labeled JPDA, labeled multiBernoulli filters, and multihypothesis tracking.

The single-object Bayesian filter for an interval, or batch, of data is extended to the multiple object case using the method of analytic combinatorics. The exact expression for the probability generating functional of the Bayes posterior process is derived. It is a nested composition of functions and functionals that is evaluated via a backward recursion. Branching and immigration processes are used to model the initial multiple object process and new object arrival processes, respectively. The exact Bayes posterior distribution and various summary statistics of the interval filter are derivatives of the generating functional. These derivatives are written in equivalent Cauchy integral form and approximated using the saddle point method.

Probabilistic multi-hypothesis tracking (PMHT) is an algorithm for tracking multiple targets when measurement-to- target assignments are unknown and must be jointly estimated with the target tracks. Multi-frame assignment PMHT (MF- PMHT) is an algorithm designed to mitigate some performance problems associated with PMHT. In MF-PMHT, the PMHT algorithm is applied to multi-frame sequences in the last L frames of data and considers the set of all possible measurement sequences. While effective in improving tracking performance compared to PMHT, performance of the original MF-PMHT degrades when the target single-frame detection probability is non-unity. This is because missed detections are not considered in the multi-frame sequences. A new MF-PMHT implementation is derived in this paper which explicitly considers missed detections in the multi-frame sequences. Performance of this MF-PMHT is compared to the original MF-PMHT algorithm as well as to a Homothetic PMHT. Simulation results indicate that the new MF- PMHT algorithm performs the same as the original algorithm when there are no missed detections and also performs better than the alternative algorithms considered when there are missed detections.

An alternating directions method is presented for joint maximum a posteriori estimation of target track and sensor field using bistatic range data. The algorithm cycles over two sub-algorithms: one improves the target state estimate conditioned on sensor field state, and the other improves the sensor field state estimate conditioned on target state. Nonlinearities in the sub-algorithms are mitigated by decomposing their likelihood functions using integral representations. The kernels of these integrals are linear-Gaussian densities in the states to be estimated, a fact that facilitates the use of missing data methods. The resulting sub-algorithms are equivalent to linear-Gaussian Kalman smoothers. The alternating directions algorithm is guaranteed to converge to (at least) a local maximum of the joint target-field likelihood function.

Probabilistic multi-hypothesis tracking (PMHT) is an algorithm for tracking multiple targets when measurement-to-target assignments are unknown and must be estimated jointly with the target tracks. PMHT is linear in the number of targets and the number of measurements; moreover, it is guaranteed to converge to locally optimal state estimates. However, it violates the rule that no target can be assigned more than one measurement. This hereby leads to a plethora of local maxima that cause performance problems. These problems are greatly reduced by applying the PMHT method to multi-frame data sequences, that is, to the set of all possible measurement sequences in the last L scans. The blend of PMHT and limited enumeration reduces the mismatch induced by violating the "at most one measurement per target" rule. Two new PMHT algorithms are presented. Both are linear in the number of targets and the number of enumerated sequences