:: Workshop – Bayesianism II: Foundations and Applications ::

Invited Talk 3 (wednesday – 16h30)
Bayesian Non–Parametric Modelling Of Dark Matter In Galaxies.
Dalia Chakrabarty (The University of Warwick).

Abstract:
The estimation of the amount and distribution of dark matter in observed galaxies is of crucial importance in Cosmology. At the same time, such an exercise is plagued by multiple implementational difficulties which partly arise from partially available measurements and partly due to the essential impossibility of analytically connecting the observables to the total gravitational mass of the system. Adopting the view that galaxies are dynamical systems, I present a Bayesian method that performs non–parametric reconstruction of the dynamical rule that dictates the evolution in phase space W of a galaxy while estimating the phase space pdf f(w). This dynamical rule, for the gravitational dynamical system at hand, is nothing but the gravitational potential which is completely defined in terms of the density of the gravitational mass of all matter (ρ(x)) in the system. We estimate f(w) and ρ(x) under the assumptions that (1) the system spatial geometry is spherical, i.e. ρ = ρ(r) where r² = x₁² + x₂² + x₃² (2) f = f(E, L²), where E = v²/2 + ∇²ρ(r) is the energy of a galactic particle and L is its angular momentum. Here v is the particle velocity. The measured data comprises the component of v along the line-of-sight (v₃) and particle spatial location on the image plane (x₁, x₂). The marginal of f = f(E, L²) is computed by integrating f = f( . , . ) over the unobserved coordinates – v₁, v₂, x₃. The computation of the marginal is non-trivial and is accomplished by discretising the domains of our unknown functions and invoking the mapping between the i – j^th E – L² cell and the space of X₃ – V₁ – V₂. Given that the data of individual particles is iid, the likelihood is the product of the marginals over the whole data set. As nothing is known apriori about either ρ(r) or f = f(E, L²), we impose constraints of positivity and monotonicity but use uniform priors. Different proposals are tested, including those in which the updating is performed with a Beta-Stacey random variable, Dirichlet variable and a Gaussian variable. I find the first two proposals to be numerically unstable for #r-bin > 50; the last proposal gives robust results with typically sized data samples in test runs. Metropolis–Hastings is used to sample from the posterior and a typical acceptance rate of about 15% is recovered. When the data set is much smaller – ∼ 30 – a simpler form of f(.) is implemented: f = f(E). This tacitly involves assuming isotropy to prevail in W. The support in the data for such an assumption can be estimated using a newly adapted non-parametric implementation of Stern and Pereira's Fully Bayesian Significance Test. The method has been applied to analyse data of two different kinds of particles in an example galaxy; these samples indicate significantly different ρ(r) distributions – for the same system! This seemingly fallacious result is understood as the system phase space being characterised by multiple attractors in contrast to the hitherto prescribed picture of smooth phase space distributions. The new result indicates that it is potentially risky to estimate dark matter density distributions from particle samples, and particularly so, if data on only one kind of particle is available.

Promotion and Support:
			CAPES-Proex Pós–graduação em Estatística-USP