Likelihood functions and confidence bounds for total-least-squares problems

This paper addresses the derivation of likelihood functions and confidence bounds for problems involving over-determined linear systems with noise in all measurements, often referred to as total-least-squares (TLS). It has been shown previously that TLS provides maximum likelihood estimates. But rather than being a function solely of the variables of interest, the associated likelihood functions increase in dimensionality with the number of equations. This has made it difficult to derive suitable confidence bounds, and impractical to use these probability functions with Bayesian belief propagation or Bayesian tracking. This paper derives likelihood functions that are defined only on the parameters of interest. This has two main advantages: first, the likelihood functions are much easier to use within a Bayesian framework; and second it is straightforward to obtain a reliable confidence bound on the estimates. We demonstrate the accuracy of our confidence bound in relation to others that have been proposed. Also, we use our theoretical results to obtain likelihood functions for estimating the direction of 3d camera translation.