Vijay Balasubramanian

Statistical inference, Occam's razor, and statistical mechanics on the space of probability distributions

The task of parametric model selection is cast in terms of a statistical mechanics on the space of probability distributions. Using the techniques of low-temperature expansions, I arrive at a systematic series for the Bayesian posterior probability of a model family that significantly extends known results in the literature. In particular, I arrive at a precise understanding of how Occams razor, the principle that simpler models should be preferred until the data justify more complex models, is automatically embodied by probability theory. These results require a measure on the space of model parameters and I derive and discuss an interpretation of Jeffreys prior distribution as a uniform prior over the distributions indexed by a family. Finally, I derive a theoretical index of the complexity of a parametric family relative to some true distribution that I call the razor of the model. The form of the razor immediately suggests several interesting questions in the theory of learning that can be studied using the techniques of statistical mechanics.

On D-branes and black holes in four dimensions

Abstract We find extremal four-dimensional black holes with finite area constructed entirely from intersecting D-branes. We argue that the microscopic degeneracy of these configurations agrees with the Bekenstein-Hawking entropy formula. The absence of solitonic objects in these configurations may make them useful for dynamical studies of black holes.

D-branes, moduli, and supersymmetry

We study toroidal compactifications of type II string theory with D-branes and nontrivial antisymmetric tensor moduli and show that turning on these fields modifies the supersymmetry projections imposed by D-branes. These modifications are seen to be necessary for the consistency of T duality. We also show the existence of unusual BPS configurations of branes at angles that are supersymmetric because of conspiracies between moduli fields. Analysis of the problem from the point of view of the effective field theory of massless modes shows that the presence of a two-form background must modify the realization of supersymmetry on the brane. In particular, the appropriate supersymmetry variation of the physical gaugino vanishes in any constant field strength background. These considerations are relevant for the E{sub 7(7)}-symmetric counting of states of four-dimensional black holes in type II string theory compactified on T{sup 6}. {copyright} {ital 1997} {ital The American Physical Society}

Topology from cosmology

We show that cosmological observables can constrain the topology of the compact additional dimensions predicted by string theory. To do this, we develop a general strategy for relating cosmological observables to the microscopic parameters of the potentials and field-dependent kinetic terms of the multiple scalar fields that arise in the low-energy limit of string theory. We apply this formalism to the Large Volume Scenarios in Type IIB flux compactifications where analytical calculations are possible. Our methods generalize to other settings.

Relativistic brane scattering

Abstract We calculate relativistic phase shifts resulting from the large impact parameter scattering of 0-branes off p-branes within supergravity. Their full functional dependence on velocity agrees with that obtained by identifying the p-branes with D-branes in string theory. These processes are also described by 0-brane quantum mechanics, but only in the non-relativistic limit. We show that an improved 0-brane quantum mechanics based on a Born-Infeld-type Lagrangian also does not yield the relativistic results. The scattering of 0-branes off bound states of arbitrary numbers of 0-branes and 2-branes is analyzed in detail, and we find agreement between supergravity and string theory at large distances to all orders in velocity. Our careful treatment of this system, which embodies the 11-dimensional kinematics of 2-branes in M(atrix) theory, makes it evident that control of 1 n corrections will be necessary in order to understand our relativistic results within M(atrix) theory.

Typicality, Black Hole Microstates and Superconformal Field Theories

We analyze the structure of heavy multitrace BPS operators in N = 1 superconformal quiver gauge theories that arise on the worldvolume of D3-branes on an affine toric cone. We exhibit a geometric procedure for counting heavy mesonic operators with given U(1) charges. We show that for any fixed linear combination of the U(1) charges, the entropy is maximized when the charges are in certain ratios. This selects preferred directions in the charge space that can be determined with the help of a piece of string. We show that almost all heavy mesonic operators of fixed U(1) charges share a universal structure. This universality reflects the properties of the dual extremal black holes whose microstates they create. We also interpret our results in terms of typical configurations of dual giant gravitons in AdS space.

Some Aspects of Massive World-Brane Dynamics

We study the internal dynamics of Ramond-Ramond solitons excited far from the BPS limit by leading Regge trajectory open strings. The simplest world volume process for such strings is splitting into two smaller pieces, and we calculate the corresponding decay rates. Compared to the conventional open superstring, the splitting of states polarized parallel to the brane is suppressed by powers of logarithms of the energy. The rate for states polarized transverse to the brane often decreases with increasing energy. We also calculate the static force between a D-brane excited by a massive open string and an unexcited D-brane parallel to it. The result shows that transversely polarized massive open strings endow D-branes with a size of order the string scale.

Extremal branes as elementary particles

Abstract The supersymmetric p -branes of type II string theory can be interpreted after compactification as extremal black holes with zero entropy and infinite temperature. We show how the p -branes avoid this apparent, catastrophic instability by developing an infinite mass gap. Equivalently, these black holes behave like elementary particles: they are dressed by effective potentials that prevent absorption of impinging particles. In contrast, configurations with 2, 3, and 4 intersecting branes and their non-extremal extensions, behave increasingly like conventional black holes. These results extend and clarify earlier work by Holzhey and Wilczek in the context of four-dimensional dilaton gravity.

Back reaction and complementarity in (1+1) dilaton gravity

Abstract We study radiation from black holes in the effective theory produced by integrating gravity and the dilaton out of 1 + 1 dilaton gravity. The semiclassical wavefunctions for the dressed particles show that the self-interactions produce an unusual renormalization of the frequencies of outgoing states. Modes propagating in the dynamical background of an incoming quantum state are seen to acquire large scattering phases that nevertheless conspire, in the absence of self-interactions, to preserve the thermality of the Hawking radiation. However, the in-out scattering matrix does not commute with the self-interactions and this could lead to observable corrections to the final state. Finally, our calculations explicitly display the limited validity of the semiclassical theory of Hawking radiation and provide support for a formulation of black hole complementarity.

Occam’s Razor for Parametric Families and Priors on the Space of Distributions

I define the razor, a natural measure of the complexity of a parametric family of distributions relative to a given true distribution. I show that empirical approximations of this quantity may be used to implement parsimonious inference schemes that favour simple models. In particular, the razor is seen to give finer classifications of model families than the Minimum Description Length principle as advocated by Rissanen. In a certain strong sense it is shown that the logarithm of the Bayesian posterior probability of a model family given a collection of data converges in the large sample limit to the logarithm of the razor of the family. This provides the most accurate asymptotics to date for Bayesian parametric inference. These results are derived by treating parametric families as manifolds embedded in the space of probability distributions. In the course of deriving a suitable integration measure on such manifolds, it is shown that, in a certain sense, a uniform prior on the space of probability distributions would induce a Jeffreys’ Prior on the parameters of a parametric family of distributions.