Conditional Expectation Values Research Articles

Systematic lowering of the scaling of Monte Carlo calculations by partitioning and subsampling

We propose to compute physical properties by Monte Carlo calculations using conditional expectation values. The latter are obtained on top of the usual Monte Carlo sampling by partitioning the physical space in several subspaces or fragments, and subsampling each fragment (i.e., performing side walks) while freezing the environment. No bias is introduced and a zero-variance principle holds in the limit of separability, i.e., when the fragments are independent. In practice, the usual bottleneck of Monte Carlo calculations-the scaling of the statistical fluctuations as a function of the number of particles N-is relieved for extensive observables. We illustrate the method in variational Monte Carlo on the two-dimensional Hubbard model and on metallic hydrogen chains using Jastrow-Slater wave functions. A factor O(N) is gained in numerical efficiency.

Relational observables, reference frames, and conditional probabilities

We discuss the construction of relational observables in time-reparametrization invariant quantum mechanics and we argue that their physical interpretation can be understood in terms of conditional probabilities, which are defined from the solutions of the quantum constraint equation in a generalization of the Page-Wootters formalism. In this regard, we show how conditional expectation values of worldline tensor fields are related to quantum averages of suitably defined relational observables. We also comment on how the dynamics of these observables can be related to a notion of quantum reference frames. After presenting the general formalism, we analyze a recollapsing cosmological model, for which we construct unitarily evolving quantum relational observables. We conclude with some remarks about the relevance of these results for the construction and interpretation of diffeomorphism-invariant operators in quantum gravity.

Symmetry Constrained Decoherence of Conditional Expectation Values

Conditional expectation values of quantum mechanical observables reflect unique non-classical correlations, and are generally sensitive to decoherence. We consider the circumstances under which such sensitivity to decoherence is removed, namely, when the measurement process is subjected to conservation laws. Specifically, we address systems with additive conserved quantities and identify sufficient conditions for the system state such that its coherence plays no role in the conditional expectation values of observables that commute with the conserved quantity. We discuss our findings for a specific model where the system-detector coupling is given by the Jaynes-Cummings interaction, which is relevant to experiments tracking trajectories of qubits in cavities. Our results clarify, among others, the role of coherence in thermal measurements in current architectures for quantum thermodynamics experiments.

Reduction of multiscale stochastic biochemical reaction networks using exact moment derivation.

Biochemical reaction networks (BRNs) in a cell frequently consist of reactions with disparate timescales. The stochastic simulations of such multiscale BRNs are prohibitively slow due to high computational cost for the simulations of fast reactions. One way to resolve this problem uses the fact that fast species regulated by fast reactions quickly equilibrate to their stationary distribution while slow species are unlikely to be changed. Thus, on a slow timescale, fast species can be replaced by their quasi-steady state (QSS): their stationary conditional expectation values for given slow species. As the QSS are determined solely by the state of slow species, such replacement leads to a reduced model, where fast species are eliminated. However, it is challenging to derive the QSS in the presence of nonlinear reactions. While various approximation schemes for the QSS have been developed, they often lead to considerable errors. Here, we propose two classes of multiscale BRNs which can be reduced by deriving an exact QSS rather than approximations. Specifically, if fast species constitute either a feedforward network or a complex balanced network, the reduced model based on the exact QSS can be derived. Such BRNs are frequently observed in a cell as the feedforward network is one of fundamental motifs of gene or protein regulatory networks. Furthermore, complex balanced networks also include various types of fast reversible bindings such as bindings between transcriptional factors and gene regulatory sites. The reduced models based on exact QSS, which can be calculated by the computational packages provided in this work, accurately approximate the slow scale dynamics of the original full model with much lower computational cost.

Mapping multiplicative to additive noise

The Langevin formulation of a number of well-known stochastic processes involves multiplicative noise. In this work we present a systematic mapping of a process with multiplicative noise to a related process with additive noise, which may often be easier to analyse. The mapping is easily understood in the example of the branching process. In a second example we study the random neighbour (or infinite range) contact process which is mapped to an Ornstein–Uhlenbeck process with absorbing wall. The present work might shed some light on absorbing state phase transitions in general, such as the role of conditional expectation values and finite size scaling, and elucidate the meaning of the noise amplitude. While we focus on the physical interpretation of the mapping, we also provide a mathematical derivation.

Joint distributions, the uncertainty principle and positive distributions

We examine the construction of joint probabilities for non-commuting observables. We show that there are indications in standard quantum mechanics that imply the existence of conditional expectation values, which in turn implies the existence of a joint distribution. We also argue that the uncertainty principle has no bearing on the existence of joint distributions but only constrains the marginal distributions. In addition, we show that within classical probability theory there are mathematical quantities that are similar to quantum mechanical wave functions. This is shown by generalising a theorem of Khinchin on the necessary and sufficient conditions for a function to be a characteristic function.

Handling Exotic Positions for Counterparty Risk Management

This paper proposes a simple approach for incorporating the dicultto-value trades to the rest of counterparty portfolios using the existing and already-approved pricing models. It requires simulating only the risk factors that these pricers require for the population of Monte Carlo simulation paths. A smaller subset of paths can be used to generate a path of trade values using these pricers. An appropriate set of basis functions over the risk factors can be selected in order to model the conditional expectation values and use the model to map the conditional values to the population. However, it is assumed that the input and output values of the pricers are properly manipulated in order to create the appropriate path dependency. The approach is applied to the case of in the money American put option where the counterparty risk exposure proles

Non-Gaussian buoyancy statistics in fingering convection

We examine the statistics of active scalar fluctuations in high-Rayleigh number fingering convection with high-resolution three-dimensional numerical experiments. Marked non-Gaussian tails are found in the one-point distribution of buoyancy fluctuations. A modified theory based on an original approach by Yakhot (1989) is used to model the active scalar distributions as a function of the conditional expectation values of scalar dissipation and fluxes in the flow. Simple models for these two quantities highlight the role of blob-like coherent structures for scalar statistics in fingering convection.

Correlation Structures of Correlated Binomial Models and Implied Default Distribution

We show how to analyze and interpret the correlation structures, the conditional expectation values and correlation coefficients of exchangeable Bernoulli random variables. We study implied default distributions for the iTraxx-CJ tranches and some popular probabilistic models, including the Gaussian copula model, Beta binomial distribution model and long-range Ising model. We interpret the differences in their profiles in terms of the correlation structures. The implied default distribution has singular correlation structures, reflecting the credit market implications. We point out two possible origins of the singular behavior.

Conditional probabilities in Ponzano-Regge minisuperspace

We examine the Hartle-Hawking no-boundary initial state for the Ponzano-Regge formulation of gravity in three dimensions. We consider the behavior of conditional probabilities and expectation values for geometrical quantities in this initial state for a simple minisuperspace model consisting of a two-parameter set of anisotropic geometries on a $2$-sphere boundary. We find dependence on the cutoff used in the construction of Ponzano-Regge amplitudes for expectation values of edge lengths. However, these expectation values are cutoff independent when computed in certain, but not all, conditional probability distributions. Conditions that yield cutoff independent expectation values are those that constrain the boundary geometry to a finite range of edge lengths. We argue that such conditions have a correspondence to fixing a range of local time, as classically associated with the area of a surface for spatially closed cosmologies. Thus these results may hint at how classical spacetime emerges from quantum amplitudes.

Scale Invariance Properties in the Simulated Annealing Algorithm

The Boltzmann distribution used in the steady-state analysis of the simulated annealing algorithm gives rise to several scale invariant properties. Scale invariance is first presented in the context of parallel independent processors and then extended to an abstract form based on lumping states together to form new aggregate states. These lumped or aggre- gate states possess all of the mathematical characteristics, forms and relationships of states (solutions) in the original problem in both first and second moments. These scale invari- ance properties therefore permit new ways of relating objective function values, conditional expectation values, stationary probabilities, rates of change of stationary probabilities and conditional variances. Such properties therefore provide potential applications in analysis, statistical inference and optimization. Directions for future research that take advantage of scale invariance are also discussed.

Dynamical and spatial aspects of sandpile cellular automata

The Bak, Tang, and Wiesenfeld cellular automaton is simulated in 1, 2, 3, 4, and 5 dimensions. We define a (new) set of scaling exponents by introducing the concept of conditional expectation values. Scaling relations are derived and checked numerically and the critical dimension is discussed. We address the problem of the mass dimension of the avalanches and find that the avalanches are noncompact for dimensions larger than 2. The scaling of the power spectrum derives from the assumption that the instantaneous dissipation rate of the individual avalanches obeys a simple scaling relation. Primarily, the results of our work show that the flow of sand down the slope does not have a 1/f power spectrum in any dimension, although the model does show clear critical behavior with scaling exponents depending on the dimension. The power spectrum behaves as 1/f 2 in all the dimensions considered.

Some conditional distributions of the structure factors and related conditional expectation values

Article Some conditional distributions of the structure factors and related conditional expectation values was published on November 1, 1972 in the journal Zeitschrift für Kristallographie - Crystalline Materials (volume 135, issue 1-6).

Some conditional distributions of the structure factors and related conditional expectation values

Accurate probability distributions of several structure factors and their magnitudes which have been obtained recently are employed here to derive in a systematic way a number of conditional distributions and associated conditional expectation values. The latter have found application in the development of new methods of phase determination.