Probabilistic Multilateration: Model and Inference

We study the problem of inferring the position of a target point in an unbounded D-dimensional Euclidean space, starting from observations of its distance from a set of other points whose locations are assumed to be fixed and known. The problem is modelled by the joint probability density over observed and latent random variables: the set of distances and the target position, respectively. Inference about the latent position of the target is expressed in terms of a prior belief distribution on its location coupled with the information provided by the distance observations through the likelihood function. The position of the target is thus statistically determined by its posterior density. As an application of the inverse probability principle, this reasoning leads to the complete and unique solution of the inferential problem, conditional to the modelling assumptions. Specifically, we assume that observed distances are non-negative random numbers with some named probability density function whose parameters can be expressed in terms of the target position. As the choice of the likelihood is inherently problem-specific, we illustrate our model using a generic Rayleigh model. For the prior position, we assume, subjectively and yet without loss of generality, that it is a known Gaussian random D-vector. Under these assumptions, the posterior density can be known analytically only up to a normalisation constant, we use a stochastic approximation to evaluate it numerically.