Scientific Research

An Academic Publisher

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

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:

*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.

[1] | V. Jugnon and L. Demanet, “Interferometric Inversion: A Robust Approach to Linear Inverse Problems,” In: J. M. Bernardo, J. O. Berger, A. P. Dawid and A. F. M. Smith, Eds., Proceedings of SEG Annual Meeting, Houston, September 2013, pp. 5180-5184. |

[2] | P. Qui, “A Nonparametric Procedure for Blind Image Deblurring,” Computational Statistics and Data Analysis, Vol. 52, No. 10, 2008, pp. 4828-4842. http://dx.doi.org/10.1016/j.csda.2008.03.027 |

[3] |
M. Bertero and P. Boccacci, “Introduction to Inverse Problems in Imaging,” Taylor and Francis. 1998. http://dx.doi.org/10.1887/0750304359 |

[4] |
P. Kutchment, “Generalised Transforms of the Radon Type and Their Applications,” In: G. Olafsson and E. T. Quinto, Eds., The Radon Transform, Inverse Problems, and Tomography, Vol. 63, American Mathematical Society, 2006, p. 67. http://dx.doi.org/10.1090/psapm/063/2208237 |

[5] | T. E. Bishop, S. D. Babacan, B. Amizik, A. K. Katsaggelos, T. Chan and R. Molina, “Blind Image Deconvolution: Problem Formulation and Existing Approaches,” In: P. Campisi and K. Egiazarian, Eds., Blind Image Deconvolution: Theory and Applications, CRC Press, Boca Raton, 2007, pp. 1-41 |

[6] | R. L. Parker, “Geophysical Inverse Theory,” Princeton series in Geophysics, Princeton University Press, Princeton, 1994. |

[7] |
A. Tarantola, “Inverse Problem Theory and Methods for Model Parameter Estimation,” SIAM, Philadelphia. 2005. http://dx.doi.org/10.1137/1.9780898717921 |

[8] | A. M. Stuart, “Bayesian Approach to Inverse Problems,” provides an introduction to the forthcoming book Bayesian Inverse Problems in Differential Equations by M. Dashti, M. Hairer and A. M. Stuart, 2013. available at arXiv:math/1302.6989. |

[9] | A. M. Stuart, “Inverse Problems: A Bayesian Perspective,” Cambridge University Press, Acta Numerica, Vol. 19, 2010, pp. 451-559. http://dx.doi.org/10.1017/S0962492910000061 |

[10] |
D. Draper and B. Mendes, “Bayesian Environmetrics: Uncertainty and Sensitivity Analysis and Inverse Problems,” 2008. http://users.soe.ucsc.edu/~draper/draper-brisbane-2008.pdf |

[11] | A. F. Bennett and P. C. McIntosh, “Open Ocean Modeling as an Inverse Problem: Tidal Theory,” Journal of Physical Oceanography, Vol. 12, No. 10, 1982, pp. 1004-1018. http://dx.doi.org/10.1175/1520-0485(1982)012<1004:OOMAAI>2.0.CO;2 |

[12] | W. P. Gouveia and J. A. Scales, “Bayesian Seismic Waveform Inversion: Parameter Estimation and Uncertainty Analysis,” Journal of Geophysical Research, Vol. 130, No. B2, 1998, pp. 2759-2779. http://dx.doi.org/10.1029/97JB02933 |

[13] | S. Strebelle, “Conditional Simulation of Complex Geological Structures Using Multiple-Point Statistics,” Mathematical Geology, Vol. 34, No. 1, 2002, pp. 1-21. http://dx.doi.org/10.1023/A:1014009426274 |

[14] | J. Caers, “Geostatistical Reservoir Modelling Using Statistical Pattern Recognition,” Journal of Petroleum Science and Engineering, Vol. 29, No. 3-4, 2001, pp. 177-188. http://dx.doi.org/10.1016/S0920-4105(01)00088-2 |

[15] | M. J. Way, J. D. Scargle, K. M. Ali and A. N. Srivastava, “Advances in Machine Learning and Data Mining for Astronomy,” Data Mining and Knowledge Discovery Series, Chapman and Hall/CRC, Boca Raton, 2012. |

[16] | Y. Liu, A. Harding, W. Abriel and S. Strebelle, “Multiple-Point Simulation Integrating Wells, Three-Dimensional Seismic Data, And Geology,” AAPG Bulletin, Vol. 88, No. 7, 2004, pp. 905-921. http://dx.doi.org/10.1306/02170403078 |

[17] | M. Henrion, D. Mortlock, D. Hand and A. Gandy, “Classification and Anomaly Detection for Astronomical Survey Data,” In: J. M. Hilbe, Ed., Astrostatistical Challenges for the New Astronomy, Springer, Berlin, 2013, pp. 149-184. |

[18] | V. M. Krasnopolsky, M. Fox-Rabinovitz and D. V. Chalikov, “New Approach to Calculation of Atmospheric Model Physics: Accurate and Fast Neural Network Emulation of Longwave Radiation in a Climate Model,” Monthly Weather Review, Vol. 133, No. 5, 2004, pp. 1370-1383. http://dx.doi.org/10.1175/MWR2923.1 |

[19] | C. E. Rasmussen and C. K. I. Williams, “Gaussian Processes for Machine Learning,” The MIT Press, New York, 2006. |

[20] |
G. Tilmann, K. William and S. Martin, “Matern Cross-Covariance Functions for Multivariate Random Fields,” Journal of the American Statistical Association, Vol. 105, No. 491, 2010, pp. 1167-1177. http://dx.doi.org/10.1198/jasa.2010.tm09420 |

[21] | D. Chakrabarty, F. Rigat, N. Gabrielyan, R. Beanland and S. paul, “Bayesian Density Estimation via Multiple Sequential Inversions of 2-D Images with Application in Electron Microscopy,” Technical Report, Submitted 2013. |

[22] | C. J. Paciorek, M. J. Schervish, “Spatial Modelling Using a New Class of Nonstationary Covariance Functions,” Environmetrics, Vol. 17, No. 5, 2006, pp. 483-506. http://dx.doi.org/10.1002/env.785 |

[23] | D. Chakrabarty, M. Biswas, S. Bhattacharya, “Bayesian Learning of Milky Way Parameters Using New Matrix-Variate Gaussian Process-based Method,” Technical Report, Submitted 2013. |

[24] | M. West and P. Harrison, “Bayesian Forecasting and Dynamic Models,” Springer-Verlag, New York, 1997. |

[25] | A. Pole, M. West and P. Harrison, “Applied Bayesian Forecasting and Time Series Analysis,” Texts in Statistical Science, Taylor and Francis, New York, 1994. |

[26] | A. Harvey, S. J. Koopman and N. Shephard, “State Space and Unobserved Component Models: Theory and Applications,” Cambridge University Press, Cambridge, 2012. |

[27] |
B. P. Carlin, N. G. Polson and D. S. Stoffer, “A Monte Carlo Approach to Nonnormal and Nonlinear State-Space Modeling,” Journal of the American Statistical Association, Vol. 87, No. 418, 1992, pp. 493-500. http://dx.doi.org/10.1080/01621459.1992.10475231 |

[28] | A. J. Winship, S. J. Jorgensen, S. A. Shaffer, I. D. Jonsen, P. W. Robinson, D. P. Costa and B. A. Block, “State-Space Framework for Estimating Measurement Error from Double-Tagging Telemetry Experiments,” Methods in Ecology and Evolution, Vol. 3, No. 2, 2012, pp. 291-302. http://dx.doi.org/10.1111/j.2041-210X.2011.00161.x |

[29] | J. Knape, N. Jonzen, M. Skold and L. Sokolov, “Multivariate State Space Modelling of Bird Migration Count Data,” In: D. L. Thompson, E. G. Gooch and M. J. Conroy, Eds., Modeling Demographic Processes in Marked Populations, Springer, Berlin, 2009. |

[30] |
S. Pellegrini and L. Ciotti, “Reconciling Optical and X-Ray Mass Estimates: The Case of the Elliptical Galaxy NGC3379,” Monthly Notices of the Royal Astronomical Society, Vol. 370, No. 4, 2006, pp. 1797-1803. http://dx.doi.org/10.1111/j.1365-2966.2006.10590.x |

[31] |
L. Coccato, O. Gerhard, M. Arnaboldi, et al., “Kinematic Properties of Early-Type Galaxy Haloes Using Planetary Nebulae,” Monthly Notices of the Royal Astronomical Society, Vol. 394, No. 3, 2009, pp. 1249-1283. http://dx.doi.org/10.1111/j.1365-2966.2009.14417.x |

[32] | P. Coté, D. E. McLaughlin, J. G. Cohen and J. P. Blakeslee, “Dynamics of the Globular Cluster System Associated with M49 (NGC4472): Cluster Orbital Properties and the Distribution of Dark Matter,” Astrophysical Journal, Vol. 591, No. 2, 2003, pp. 850-877. http://dx.doi.org/10.1086/375488 |

[33] | A. J. Romanowsky, N. G. Douglas, M. Arnaboldi, K. Kuijken, M. R. Merrifield, N. R. Napolitano, M. Capaccioli and K. C. Freeman, “A Dearth of Dark Matter in Ordinary Elliptical Galaxies,” Science, Vol. 301, No. 5640, 2003, pp. 1696-1698. http://dx.doi.org/10.1126/science.1087441 |

[34] |
D. Chakrabarty, “Inverse Look at the Center of M15,” Astronomical Journal, Vol. 131, No. 5, 2006, pp. 2561-2570. http://dx.doi.org/10.1086/501433 |

[35] | D. Chakrabarty and S. Raychaudhury, “The Distribution of Dark Matter in the Halo of the Early-Type Galaxy NGC 4636,” Astronomical Journal, Vol. 135, No. 6, 2008, pp. 2350-2357. http://dx.doi.org/10.1088/0004-6256/135/6/2350 |

[36] | L. V. E. Koopmans, “Gravitational Lensing & Stellar Dynamics,” European Astronomical Society Publications Series, Vol. 20, 2006, pp. 161-166. http://dx.doi.org/10.1051/eas:2006064 |

[37] | D. Chakrabarty and B. Jackson, “Total Mass Distributions of Sersic Galaxies from Photometry and Central Velocity Dispersion,” Astronomy and Astrophysics, Vol. 498, No. 2, 2009, pp. 615-626. http://dx.doi.org/10.1051/0004-6361/200809965 |

[38] | J. R. Binney and S. Tremaine, “Galactic Dynamics,” Princeton University Press, Princeton, 1987. |

[39] |
G. Contopoulos, “A Classification of the Integrals of Motion,” Astrophysical Journal, Vol. 138, 1963, pp. 1297-1305. http://dx.doi.org/10.1086/147724 |

[40] | J. R. Binney, “Dynamics of Elliptical Galaxies and Other Spheroidal Components,” Annual Review of Astronomy and Astrophysics, Vol. 20, 1982, pp. 399-429. http://dx.doi.org/10.1146/annurev.aa.20.090182.002151 |

[41] | I. S. Liu, “Continuum Mechanics,” Springer-Verlag, New York, 2002. |

[42] | C. Truesdell, W. Noll and S. S. Antman, “The Non-Linear Field Theories of Mechanics,” Volume 3, Springer-Verlag, New York, 2004. http://dx.doi.org/10.1007/978-3-662-10388-3 |

[43] | C. C. Wang, “On Representations for Isotropic Functions,” Archive for Rational Mechanics and Analysis, Vol. 33, No. 4, 1969, pp. 249-267. http://dx.doi.org/10.1007/BF00281278 |

[44] | J. F. Navarro, C. S. Frenk and S. D. M. White, “The Structure of Cold Dark Matter Halos,” Astrophysical Journal, Vol. 462, 1996, p. 563. http://dx.doi.org/10.1086/177173 |

[45] |
W. J. G. de Blok, A. Bosma and S. McGaugh, “Simulating Observations of Dark Matter Dominated Galaxies: Towards the Optimal Halo Profile,” Monthly Notices of the Royal Astronomical Soc, Vol. 340, No. 2, 2003, pp. 657-678. http://dx.doi.org/10.1046/j.1365-8711.2003.06330.x |

[46] | H. Haario, M. Laine, A. Mira and E. Saksman, “DRAM: Efficient Adaptive MCMC,” Statistics and Computing Vol. 16, No. 4, 2006, pp. 339-354. http://dx.doi.org/10.1007/s11222-006-9438-0 |

[47] | F. C. Leone, R. B. Nottingham and L. S. Nelson, “The Folded Normal Distribution,” Technometrics, Vol. 3, No. 4, 1961, pp. 543. http://dx.doi.org/10.1080/00401706.1961.10489974 |

[48] |
N. G. Douglas, N. R. Napolitano, A. J. Romanowsky, L. Coccato, K. Kuijken, M. R. Merrifield, M. Arnaboldi, O. Gerhard, K. C. Freeman, H. Merrett, E. Noordermeer and M. Capaccioli, “The PN.S Elliptical Galaxy Survey: Data Reduction, Planetary Nebula Catalog, and Basic Dynamics for NGC 3379,” Astrophysical Journal, Vol. 664, No. 1, 2007, pp. 257-276. http://dx.doi.org/10.1086/518358 |

[49] |
G. Bergond, S. E. Zepf, A. J. Romanowsky, R. M. Sharples and K. L. Rhode, “Wide-Field Kinematics of Globular Clusters in the Leo I Group,” Astronomy & Astrophysics, Vol. 448, No. 1, 2006, pp. 155-164. http://dx.doi.org/10.1051/0004-6361:20053697 |

Copyright © 2020 by authors and Scientific Research Publishing Inc.

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.