Inverse Bayesian Estimation of Gravitational Mass Density in Galaxies from Missing Kinematic Data ()

Dalia Chakrabarty, Prasenjit Saha

Department of Mathematics, University of Leicester, Leicester, UK.

Institute for Theoretical Physics, University of Zurich, Zurich, Switzerland.

**DOI: **10.4236/ajcm.2014.41002
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Department of Mathematics, University of Leicester, Leicester, UK.

Institute for Theoretical Physics, University of Zurich, Zurich, Switzerland.

In this paper, we focus on a type of inverse problem in which
the data are expressed as an unknown function of the sought and unknown model
function (or its discretised representation as a model parameter vector). In
particular, we deal with situations in which training data are not
available. Then we cannot model the unknown functional relationship between
data and the unknown model function (or parameter vector) with a Gaussian
Process of appropriate dimensionality. A Bayesian method based on state space
modelling is advanced instead. Within this framework, the likelihood is
expressed in terms of the probability density function (*pdf*) of the state space variable and the sought model parameter
vector is embedded within the domain of this *pdf*. As the measurable vector lives only inside an identified
sub-volume of the system state space, the *pdf* of the state space variable is projected onto the space of the measurables, and
it is in terms of the projected state space density that the likelihood is
written; the final form of the likelihood is achieved after convolution with
the distribution of measurement errors. Application motivated vague priors are
invoked and the posterior probability density of the model parameter vectors,
given the data are computed. Inference is performed by taking posterior samples with
adaptive MCMC. The method is illustrated on synthetic as well as real galactic
data.

Share and Cite:

D. Chakrabarty and P. Saha, "Inverse Bayesian Estimation of Gravitational Mass Density in Galaxies from Missing Kinematic Data," *American Journal of Computational Mathematics*, Vol. 4 No. 1, 2014, pp. 6-29. doi: 10.4236/ajcm.2014.41002.

Conflicts of Interest

The authors declare no conflicts of interest.

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