Finite Probability Research Articles

Prior selection method using likelihood confidence region and Dirichlet process Gaussian mixture model for Bayesian inference of building energy models

It is widely acknowledged that Bayesian inference is only beneficial when prior information is properly defined. However, there is no clear rule for prior selection, and it is apparently a matter of subjective selection by the domain expert(s). In other words, because the posterior inference results can vary depending on how the prior is set, a proper definition of the prior is important in terms of objectivity and accuracy for Bayesian inference of building energy models. Hence, the authors suggest a new prior selection method using Dirichlet process Gaussian mixture model (DPGMM) and the likelihood confidence region (hereafter referred to as likelihood CR). The DPGMM is a Bayesian nonparametric clustering technique that optimizes both the cluster shape and the number of clusters. Using the DPGMM, the finite probability distributions that make up the likelihood CR can be estimated, where the distribution with the highest maximum likelihood is applied as the informative prior. In this study, a reference office building of the United States Department of Energy was selected, and a surrogate model was generated using an artificial neural network. Based on a comparison between the authors’ suggestion and traditional informative (and/or non-informative) priors by domain experts, the proposed method requires only minimum information about the parameters (min and max) and performs better than the traditional approach

A Numerical Investigation of the Minimum Ignition Energy Requirement for Forced Ignition of Turbulent Droplet-laden Mixtures

ABSTRACT The effects of droplet diameter, overall (i.e. liquid and gaseous phases) equivalence ratio, and turbulence intensity on the variation of the Minimum Ignition Energy (MIE) for localized forced ignition of uniformly dispersed mono-sized n-heptane droplet-laden mixtures under homogeneous isotropic decaying turbulence have been analyzed based on Direct Numerical Simulation (DNS) data. The MIE was evaluated just for (i) obtaining thermal runway irrespective of the fate of the resulting flame kernel, and (ii) also for a successful self-sustained flame propagation without the assistance of an external energy source following the energy deposition by the ignitor. It has been found that the MIE requirement increases with increasing turbulence intensity and this trend for the MIE increase is especially significant for large values of turbulence intensity. The MIE requirement increases with increasing initial droplet diameter and with decreasing overall equivalence ratio. The MIE requirements for droplet-laden mixtures have been found to be greater than the corresponding value for homogeneous mixture with the same nominal values of initial turbulence intensity and equivalence ratio for the parameter range considered here. This behavior arises due to the deposited energy being partially utilized to supply the latent heat of evaporation and also due to the predominantly fuel-lean composition of the gaseous flammable mixture. This tendency of obtaining fuel-lean mixture strengthens with increasing (decreasing) initial droplet diameter (overall equivalence ratio). It has also been demonstrated that combustion takes place predominantly in the fuel-lean premixed mode although there is a finite probability of having non-premixed combustion in all cases. Moreover, there is a small probability of fuel-rich combustion occurring for small droplets, especially under fuel-rich overall equivalence ratios. The stochastic nature of the ignition event has been demonstrated by considering different realizations of statistically similar turbulent flow fields. The conditions giving rise to a successful thermal runaway/self-sustained flame propagation have been identified by a detailed analysis of the energy budget.

Lewis Number Effects on Flame Speed Statistics in Spherical Turbulent Premixed Flames

The influences of characteristic Lewis number hbox{Le} on the statistics of density-weighted displacement speed and consumption speed in spherically expanding turbulent premixed flames have been analysed using three-dimensional direct numerical simulations data for hbox{Le} = 0.8 , 1.0 and 1.2 under statistically similar flow conditions. It has been found that the extents of flame wrinkling and burning increase with decreasing hbox{Le} , which is reflected in increasing trends of mean and most probable values of both density-weighted displacement and consumption speed. Moreover, in all cases the marginal probability density functions of density-weighted displacement speed show finite probabilities of obtaining negative values, whereas consumption speed remains deterministically positive. The strain rate and curvature dependences of scalar gradient and temperature have been found to be strongly dependent on hbox{Le} , and these statistics, along with the interrelation between strain rate and curvature, influence the local strain rate and curvature responses of consumption speed and both reaction and normal diffusion components of density-weighted displacement speed. Density-weighted displacement speed and curvature have been found to be negatively correlated, whereas positive correlations are obtained between density-weighted displacement speed and tangential strain rate for all flames considered here. The positive correlation between temperature and curvature arising from differential diffusion of heat and mass in the hbox{Le} = 0.8 case induces a positive correlation between consumption speed and curvature, whereas these correlations are negative in the hbox{Le} = 1.2 flame. The statistical behaviour of density-weighted displacement speed has been utilised to demonstrate that Damköhler’s first hypothesis does not strictly hold for spherically expanding turbulent premixed flames.

Nonuniversal entanglement level statistics in projection-driven quantum circuits

We study the level-spacing statistics in the entanglement spectrum of output states of random universal quantum circuits where qubits are subject to a finite probability of projection to the computational basis at each time step. We encounter two phase transitions with increasing projection rate: The first is the volume-to-area law transition observed in quantum circuits with projective measurements; The second separates the pure Poisson level statistics phase at large projective measurement rates from a regime of residual level repulsion in the entanglement spectrum within the area-law phase, characterized by non-universal level spacing statistics that interpolates between the Wigner-Dyson and Poisson distributions. By applying a tensor network contraction algorithm introduced in Ref. [1] to the circuit spacetime, we identify this second projective-measurement-driven transition as a percolation transition of entangled bonds. The same behavior is observed in both circuits of random two-qubit unitaries and circuits of universal gate sets, including the set implemented by Google in its Sycamore circuits.

Effect of the Optical Pumping and Magnetic Field on the States of Phase Separation Domains in Eu0.8Ce0.2Mn2O5

The effect of optical pumping and applied magnetic field on the characteristics of ferromagnetic layers in one-dimensional superlattices is studied. At low enough temperatures, these layers correspond to phase separation domains in RMn_2O_5 and R_{0.8}Ce_{0.2}Mn_2O_5 multiferroics. The formation of such domains occurs owing to the charge ordering of Mn^{3+} and Mn^{4+} ions and to the finite probability for e_g electrons to tunnel between these pairs of ions. The volume occupied by such superlattices is rather small, and they can be treated as isolated ferromagnetic semiconductor heterostructures, spontaneously formed in the host crystal. The sequences of ferromagnetic resonances related to the superlattice layers in Eu_{0.8}Ce_{0.2}Mn_2O_5 are studied. The characteristics of these resonances give information on the properties of such layers. For the first time, it is demonstrated that the optical pumping gives rise to a new metastable state of superlattices, which can be recovered by the magnetic field cycling to the state existing before the optical pumping. It is found that the superlattices recovered by the magnetic field exist up to temperatures higher than those in as-grown crystals.

A GENERALISED PROPERTY EXPOSURE RATING FRAMEWORK THAT INCORPORATES SCALE-INDEPENDENT LOSSES AND MAXIMUM POSSIBLE LOSS UNCERTAINTY

AbstractA generalised property exposure rating framework is presented here to address two issues arising in the standard approach to exposure rating, especially in the context of direct insurance and facultative reinsurance (D&F) property pricing: (a)What to do when the main assumption of exposure rating, scalability – that is, that the probability of a given damage ratio does not depend on the maximum possible loss (MPL) but only on the type of property – breaks down.(b)How to take account of the uncertainty around the MPL, that is, the fact that the MPL (unlike the insured value, IV) is an informed estimate rather than a contractual feature.The first difficulty is addressed by making the exposure rating framework more flexible by introducing exposure curves that are a mixture of scale-dependent and scale-independent losses, and where the weight given to the two components is a function of the MPL. This allows to model the impact on the expected losses of changes in the underlying deductibles, a classic problem in D&F pricing normally solved with the help of deductible impact tables. The second difficulty is addressed by working out the mathematical implications of having a finite probability of exceeding the MPL and extending the exposure curve up to the maximum of MPL and IV. A practical application of this framework – in which the scale-independent and scale-dependent losses are identified with attritional and large losses respectively – is described. A discussion of how this generalised framework can be calibrated based on actual data is included, and implementation code for the framework is made available.

By-passing fluctuation theorems

Fluctuation theorems impose constraints on possible work extraction probabilities in thermodynamical processes. These constraints are stronger than the usual second law, which is concerned only with average values. Here, we show that such constraints, expressed in the form of the Jarzysnki equality, can be by-passed if one allows for the use of catalysts---additional degrees of freedom that may become correlated with the system from which work is extracted, but whose reduced state remains unchanged so that they can be re-used. This violation can be achieved both for small systems but also for macroscopic many-body systems, and leads to positive work extraction per particle with finite probability from macroscopic states in equilibrium. In addition to studying such violations for a single system, we also discuss the scenario in which many parties use the same catalyst to induce local transitions. We show that there exist catalytic processes that lead to highly correlated work distributions, expected to have implications for stochastic and quantum thermodynamics.

Survival probabilities in biased random walks: To restart or not to restart? That is the question

The time-dependent survival probability function S(t;x0,q) of biased Sisyphus random walkers, who at each time step have a finite probability q to step towards an absorbing trap at the origin and a complementary probability 1−q to return to their initial position x0, is derived analytically. In particular, we explicitly prove that the survival probability function of the walkers decays exponentially at asymptotically late times. Interestingly, our analysis reveals the fact that, for a given value q of the biased jumping probability, the survival probability function S(t;x0,q) is characterized by a critical (marginal) value x0crit(q) of the initial gap between the walkers and the trap, above which the late-time survival probability of the biased Sisyphus random walkers is larger than the corresponding survival probability of standard random walkers.

Non-universal critical initial slip of parity conserving branching and annihilating random walkers with long-range diffusion

We consider a parity conserving model of branching and annihilating random walkers with long-range diffusion. We follow the short-time dynamics at the critical region to obtain the set of critical exponents associated to the growth in the number particles and its fluctuations, as well as the critical second-order moment ratio. Diffusion and branching processes are controlled by a diffusion probability p and the flight distance follows a Lévy distribution with exponent α. Three short-time scaling regimes are identified as a function of the Lévy exponent. For α≤5∕2 infinitesimal branching is relevant and leads to a finite density of walkers. An absorbing-state dynamic phase transition takes place at a finite branching probability for α>5∕2. Short-range power-law scaling occurs for α≥7∕2. In the intermediate regime, continuously varying exponents are obtained with the second-order cumulant depicting a non-monotonous behavior. The relative influence of Lévy diffusion and branching on the critical diffusion probability is also discussed.

Extracting unambiguous information from a single qubit by sequential observers

In a recent paper [Phys. Rev. Lett. 111, 100501 (2013)], a scheme was proposed where subsequent observers can extract unambiguous information about the initial state of a qubit, with finite joint probability of success. Here, we generalize the problem for arbitrary preparation probabilities (arbitrary priors). We discuss two different schemes: one where only the joint probability of success is maximized and another where, in addition, the joint probability of failure is also minimized. We also derive the mutual information for these schemes and show that there are some parameter regions for the scheme without minimizing the joint failure probability where, even though the joint success probability is maximum, no information is actually transmitted by Alice.

On the Ruin Problem with Investment When the Risky Asset Is a Semimartingale

In this paper, we study the ruin problem with investment in a general framework where the business part X is a Levy process and the return on investment R is a semimartingale. We obtain upper bounds on the finite and infinite time ruin probabilities that decrease as a power function when the initial capital increases. When R is a Levy process, we retrieve the well-known results. Then, we show that these bounds are asymptotically optimal in the finite time case, under some simple conditions on the characteristics of X. Finally, we obtain a condition for ruin with probability one when X is a Brownian motion with negative drift and express it explicitly using the characteristics of R.

Влияние редкоземельных ионов на электрическую поляризацию, индуцированную областями фазового расслоения в RMn-=SUB=-2-=/SUB=-O-=SUB=-5-=/SUB=- (R=Er, Tb)

The effect of rare-earth ions (R = Er^{3+}, Tb^{3+}) with a strong spin-orbit coupling on the dielectric properties and electric polarization induced by the local polar phase separation domains in RMn_2O_5 multiferroics is studied. A qualitative difference between these parameters and those studied previously in GdMn_2O_5 was discovered, in which the Gd^{3+} ion in the ^8S_{7/2} ground state is weakly coupled to the lattice. It has been shown that the properties of the polar phase separation that arise in the subsystem of Mn^{3+} and Mn^{4+} ions due to the finite probability of electron tunneling between these ions of different valencies substantially depend on the magnitudes of the crystal fields in which these domains are located. Authors on English: B.Kh. Khannanov, E.I. Golovenchits, V.A. Sanina

Фрустрация обменно-стрикционного сегнетоэлектрического упорядочения и рост электрической поляризации областей фазового расслоения в твердых растворах R-=SUB=-0.8-=/SUB=-Ce-=SUB=-0.2-=/SUB=-Mn-=SUB=-2-=/SUB=-O-=SUB=-5-=/SUB=- (Er, Tb)

The effect of rare-earth ions (R = Er^{3+}, Tb^{3+}, Ce^{3.75+}) on the dielectric properties and electric polarization induced by the local polar phase separation domains in solid solutions of multiferroics R_{0.8}Ce_{0.2}Mn_2O_5 (R = Er, Tb) is studied. A qualitative difference between these parameters and the previously studied in initial RMn_2O_5 crystals (R = Er, Tb) was found. It has been shown that the properties of the polar phase separation domains that arise in the subsystem of Mn^{3+} and Mn^{4+} ions due to the finite probability of tunneling of electrons between these ions substantially depend on the magnitudes of the crystal fields in which these domains are located. It was shown that the combined effect of E^{r3+}, Tb^{3+}, and Ce^{3.75+} ions significantly changes the crystal field in R_{0.8}Ce_{0.2}Mn_2O_5 (R = Er, Tb) compared to RMn_2O_5 (R = Er, Tb).

Probability Theory with Superposition Events: A Classical Generalization in the Direction of Quantum Mechanics

In finite probability theory, events are subsets S⊆U of the outcome set. Subsets can be represented by 1-dimensional column vectors. By extending the representation of events to two dimensional matrices, we can introduce events. Probabilities are introduced for classical events, superposition events, and their mixtures by using density matrices. Then probabilities for experiments or `measurements' of all these events can be determined in a manner exactly like in quantum mechanics (QM) using density matrices. Moreover the transformation of the density matrices induced by the experiments or `measurements' is the Luders mixture operation as in QM. And finally by moving the machinery into the n-dimensional vector space over ℤ₂, different basis sets become different outcome sets. That `non-commutative' extension of finite probability theory yields the pedagogical model of quantum mechanics over ℤ₂ that can model many characteristic non-classical results of QM.

Lifetime of a greedy forager with long-range smell

We study a greedy forager who consumes food throughout a region. If the forager does not eat any food for S time steps it dies. We assume that the forager moves preferentially in the direction of greatest smell of food. Each food item in a given direction contributes towards the total smell of food in that direction, however the smell of any individual food item decays with its distance from the forager. We study both power-law decay and exponential decay of the smell with the distance of the food from the forager. For power-law decay, we vary the exponent α governing this decay, while for exponential decay we vary λ also governing the rate of the decay. For power-law decay we find, both analytically and through simulations, that for a forager living in one dimension, there is a critical value of α, namely , where for the forager will die in finite time, however for the forager has a nonzero probability to live infinite time. We calculate analytically the critical value, , separating these two behaviors and find that depends on S as . We find analytically that at the system has an essential singularity. For exponential decay we find analytically that for all λ, the forager has a finite probability to live for infinite time. We also study, using simulations, a forager with long-range decaying smell in two dimensions (2D) and find that for this case the forager always dies within finite time. However, in 2D we observe indications of an optimal α (and λ) for which the forager has the longest lifetime.

Common Knowledge and Sequential Team Problems

We consider a general sequential team problem based on Witsenhausen's intrinsic model. Our formulation encompasses all teams in which the uncontrolled inputs can be viewed as random variables on a finite probability space, the number of control inputs/decisions is finite and the decisions take values in finite spaces. We define the concept of common knowledge in such teams and use it to construct a sequential decomposition of the problem of optimizing the team strategy profile. If the information structure is classical, our common knowledge based decomposition is identical to classical dynamic program. If the information structure is such that the common knowledge is trivial, our decomposition is similar in spirit to Witsenhausen's standard form based decomposition [17] . In this case, the sequential decomposition is essentially a sequential reformulation of the strategy optimization problem and appears to have limited value. For information structures with nontrivial common knowledge, our sequential decomposition differs from Witsenhausen's standard form based decomposition because of its dependence on common knowledge. Our common knowledge based approach generalizes the common information based methods of [12] – [14] .

Asymptotic Majorization of Finite Probability Distributions

This paper studies majorization of high tensor powers of finitely supported probability distributions. Viewing probability distributions as a resource with majorization as a means of transformation corresponds to the resource theory of pure bipartite quantum states under LOCC transformations vis-\`a-vis Nielsen's Theorem. In [T. Fritz (2017)] a formula for the asymptotic exchange rate between any two finitely supported probability distributions was conjectured. The main result of the present paper is Theorem 3.11, which resolves this conjecture.

On the Unfortunate Ignorance of Academics regarding the Technical and Mathematical Tools for Dealing with Uncertainty Developed by J M Keynes in the A Treatise on Probability in 1921 and in His Earlier Fellowship Dissertations in 1907 and 1908: Interval Valued Probability, Decision Weights, and Safety First Analysis

Keynes developed a number of technical, mathematical tools for dealing with the problems of uncertainty (ambiguity, vagueness, indeterminate probabilities, imprecise probabilities) in his A Treatise on Probability in 1921 (and in his earlier Fellowship Dissertations of 1907 and 1908) that continue to be overlooked by practically all academics in the nearly 100 years since its publication in 1921. Keynes’s technical and mathematical modeling developments took place in chapters 15-17 of Part II of the A Treatise on Probability, chapters 20 and 22 of Part III of the A Treatise on Probability, Chapter 26 of Part IV of the A Treatise on Probability, and chapters 29 and 30 of Part V of the A Treatise on Probability. Part II of the A Treatise on Probability included Keynes’s original work dealing with non –additive probabilities that was based on the original work of George Boole in his 1854 The Laws of Thought. Keynes’s Boolean upper and lower bounded probabilities, which Keynes categorized as inexact measurement and approximation, established his complete grasp of the concept of interval valued probability. The American Theodore Hailperin showed in 1986 that both Boole and Keynes were using early versions of linear programming techniques to solve both linear and non linear systems of equations and inequations. Keynes used the name “non –numerical probability“ when discussing his no additive, non linear interval valued probability approach. Keynes’s approach has been erroneously identified as an ordinal probability approach by all Post Keynesians, Institutional and heterodox economists, who were greatly influenced by the 1975 paper on Keynes and uncertainty in HOPE by E R Weintraub, which misinterpreted Keynes in terms of G L S Shackle’s theory of possibility. Part III of the A Treatise on Probability, building on Part II’s interval valued probability concept, developed the concept of finite probability, applicable to both numerical and non numerical (interval valued) probabilities, in order to deal with applications of probability to Keynes’s concepts of induction, based on degrees of analogy, involving degrees of similarity and dissimilarity using human memory ,intuition and pattern recognition. Part IV of the A Treatise on Probability provided the first decision weight approach to decision making, Keynes’s conventional coefficient of weight and risk, c, that incorporated both non linear probability preferences and non additivity by the use of decision weights .These decision weights incorporated Keynes’s logical analysis, contained in chapter 6 of the A Treatise on Probability of his Evidential weight of the argument approach, V(a/h),which was a logical relation like Keynes P(a/h)=α, 0≤α≤1,but which Keynes waited to provide the comparable mathematical analysis for in chapter 26.In chapter 26,Keynes set V(a/h) equal to w,where w was normalized on the unit interval, so that V(a/h) =w, 0≤w≤1, holds. Keynes then defined uncertainty in the General Theory as an inverse function of V in chapter 12 on page 148 in footnote 1. Keynes then defined his conventional coefficient of weight and risk, c, to equal the standard EMV and SEU rules multiplied by the decision weights [(2w/1+w)] for non additivity and [(1/(1+q)] for non linearity. F Y Edgeworth recognized Keynes’s approach as breaking new ground. However, his plea for reader assistance from subscribers to Mind was ignored. Part V of the A Treatise on Probability provided the first “safety first” approach to decision making. As acknowledged by Edgeworth, Keynes used Chebyshev’s Inequality to establish lower bounds for decision problems in order to minimize risk (Keynes, 1921, pp. 353-358). Samuelson, while analyzing Keynes‘s simplified Least Risk analysis on p. 315 of the A Treatise on Probability, unfortunately overlooked Keynes’s generalized approach using Chebyshev’s Inequality in chapter 29. Samuelson also overlooked on this same page Keynes’s development of the logical and mathematical tools needed to specify the investment multiplier in the General Theory, as well as Kahn’s employment multiplier on p.183 of his June, 1931 article in the Economic Journal.

EXTENDING RELATIONAL DATABASE MODEL FOR UNCERTAIN INFORMATION

In this paper, we propose a new probabilistic relational database model, denote by PRDB, as an extension of the classical relational database model where the uncertainty of relational attribute values and tuples are respectively represented by finite sets and probability intervals. A probabilistic interpretation of binary relations on finite sets is proposed for the computation of their probability measures. The combination strategies on probability intervals are employed to combine attribute values and compute uncertain membership degrees of tuples in a relation. The fundamental concepts of the classical relational database model are extended and generalized for PRDB. Then, the probabilistic relational algebraic operations are formally defined accordingly in PRDB. In addition, a set of the properties of the algebraic operations in this new model also are formulated and proven.

Zero-temperature ordering dynamics in a two-dimensional biaxial next-nearest-neighbor Ising model.

We investigate the dynamics of a two-dimensional biaxial next-nearest-neighbor Ising model following a quench to zero temperature. The Hamiltonian is given by H=-J_{0}∑_{i,j=1}^{L}[(S_{i,j}S_{i+1,j}+S_{i,j}S_{i,j+1})-κ(S_{i,j}S_{i+2,j}+S_{i,j}S_{i,j+2})]. For κ<1, the system does not reach the equilibrium ground state and keeps evolving in active states forever. For κ≥1, though, the system reaches a final state, but it does not reach the ground state always and freezes to a striped state with a finite probability like a two-dimensional ferromagnetic Ising model and an axial next-nearest-neighbor Ising (ANNNI) model. The overall dynamical behavior for κ>1 and κ=1 is quite different. The residual energy decays in a power law for both κ>1 and κ=1 from which the dynamical exponent z has been estimated. The persistence probability shows algebraic decay for κ>1 with an exponent θ=0.22±0.002 while the dynamical exponent for ordering z=2.33±0.01. For κ=1, the system belongs to a completely different dynamical class with θ=0.332±0.002 and z=2.47±0.04. We have computed the freezing probability for different values of κ. We have also studied the decay of autocorrelation function with time for a different regime of κ values. The results have been compared with those of the two-dimensional ANNNI model.