Model Selection in Ultrasonic Measurements on Trabecular Bone

Previous work from our laboratory showed that the widely reported decrease in phase velocity with frequency (negative dispersion) for ultrasonic waves propagating through trabecular bone can arise from the interference of two compressional waves, each of which exhibits a positive dispersion. Previous simulations suggest that Bayesian probability theory can be employed to recover the material properties linked to these two interfering waves, even when the waves overlap sufficiently that visual inspection cannot distinguish two modes. In the present study, Bayesian probability theory is applied first to simulated data and then to representative experimental bone data to determine whether one or two compressional wave modes are present. Model selection is implemented by evaluating the posterior probability for each model. The calculation is implemented by defining a model indicator and then using Markov chain Monte Carlo with simulated annealing to draw samples from the joint posterior probability for the ultrasonic parameters and the model indicator. Monte Carlo integration is used to evaluate the marginal posterior probability for each parameter given the model indicator.