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Probability theory course by Andronick Arutyunov at Free Moscow University, spring 2021, Moscow tutorials by Alex Shpilkin, tutorial 10 on 2021-04-25.

Probability theory course by Andronick Arutyunov at Free Moscow University, spring 2021, Moscow tutorials by Alex Shpilkin, tutorial 14 on 2021-05-23. Incomplete recording. Addendum: Уравнение 4 t 3 (1− t ) 2 − 2 t 4 (1− t ) = 0 выглядит страшно, но всё становится просто, если заметить общий множитель t 3 (1− t ) и вынести его. Получается t 3 (1− t )(4−4 t −2 t ) = 0, а произведение равно нулю тогда и только тогда, когда равен нулю один из его множителей, так что кандидатами на максимум получаются точки t = 0, t = 1 и t = 4/6. Первые две — это края отрезка [0,1]; в них правдоподобие равно нулю (они соответствуют тому, что монета всегда падает только одной стороной, а по нашим данным получается оба варианта), так что это минимумы, а не максимумы. Поскольку максимум хоть где-нибудь да будет, он должен быть в последней точке. Таким образом, оценка максимального правдоподобия совпала с очевидной оценкой 4 / (4+2) = (число успехов) / (число испытаний). Теперь, когда мы знаем, что делать, нетрудно сделать этот расчёт и в общем виде, для произвольных k и l.  Правдоподобие, как и раньше, есть p k (1− p ) l , так что изменение правдоподобия при небольшом отклонении ε от кандидата t на максимум равно ( t +ε) k [(1− t )−ε] l . Раскрывать скобки в общеми виде было бы грустно, но это и не нужно: как мы только что поняли в случае k = 4, l = 2, нас интересуют только слагаемые нулевой и первой степени по ε, а остальные можно даже не выписывать. [На самом деле и нулевой не нужны, потому что мы заранее знаем, что они в итоге дадут  t k (1− t ) l , но их проще оставить.] Первая скобка по биному Ньютона даёт C k 0 t k + C k 1 t k −1 ε + ⋯ = t k + t k −1 ε + ⋯ , вторая C l 0 (1− t ) l  − C l 1 (1− t ) l −1 ε + ⋯ = (1− t ) l + (1− t ) l −1 ε + ⋯ (если знаете производную степенной функции — заметьте, что это она), их произведение равно  t k (1− t ) l  + [ kt k −1 (1− t ) l  − lt k (1− t ) l −1 ]ε + ⋯ , так что остаётся, как и раньше, приравнять нулю коэффициент  kt k −1 (1− t ) l  −  lt k (1− t ) l −1  при ε (производная произведения!) и решить получившееся уравнение (случаи k = 0 и l = 0, когда максимум оказывается на краю отрезка, в принципе надо разобрать отдельно, но ответ в них совпадает с полученным по формуле ниже). Снова вынося общий множитель, получаем уравнение  t k −1 (1− t ) l −1 [ k (1− t ) −  l ] и корни t = 0 (если k > 1), t = 1 (если l > 1) и t = k / ( k + l ). При k , l ≠ 0 максимум на краю отрезка невозможен по той же причине, что и раньше, так что оценка максимального правдоподобия есть  k  / ( k + l ) и опять совпадает с очевидной (число успехов) / (число испытаний).

xvii, 746 pages 23 cm

This paper covers issues relating to the establishment of a sound and conditional probability-compatible rationale for generating linguistic-based inference rules concerning a population. By extending previous preliminary results, the authors detail, in a fully rigorous manner and within the confines of traditional probability theory, that a comprehensive technique can be derived that converts linguistic-based conditional information, couched only in fuzzy-logic terms, into naturally corresponding conditional probabilities. In turn, they demonstrate how such typically underconstrained conditional probabilities can be combined for suitable conclusions and decision making, via a new use of second-order probability logic. This research is part of the ongoing SSC San Diego In-house Laboratory Independent Research FY 01 project CRANOF (a Complexity-Reducing Algorithm for Near-Optimal Fusion).

This paper covers issues relating to the establishment of a sound and conditional probability-compatible rationale for generating linguistic-based inference rules concerning a population. By extending previous preliminary results, the authors detail, in a fully rigorous manner and within the confines of traditional probability theory, that a comprehensive technique can be derived that converts linguistic-based conditional information, couched only in fuzzy-logic terms, into naturally corresponding conditional probabilities. In turn, they demonstrate how such typically underconstrained conditional probabilities can be combined for suitable conclusions and decision making, via a new use of second-order probability logic. This research is part of the ongoing SSC San Diego In-house Laboratory Independent Research FY 01 project CRANOF (a Complexity-Reducing Algorithm for Near-Optimal Fusion).

This paper covers issues relating to the establishment of a sound and conditional probability-compatible rationale for generating linguistic-based inference rules concerning a population. By extending previous preliminary results, the authors detail, in a fully rigorous manner and within the confines of traditional probability theory, that a comprehensive technique can be derived that converts linguistic-based conditional information, couched only in fuzzy-logic terms, into naturally corresponding conditional probabilities. In turn, they demonstrate how such typically underconstrained conditional probabilities can be combined for suitable conclusions and decision making, via a new use of second-order probability logic. This research is part of the ongoing SSC San Diego In-house Laboratory Independent Research FY 01 project CRANOF (a Complexity-Reducing Algorithm for Near-Optimal Fusion).

This paper covers issues relating to the establishment of a sound and conditional probability-compatible rationale for generating linguistic-based inference rules concerning a population. By extending previous preliminary results, the authors detail, in a fully rigorous manner and within the confines of traditional probability theory, that a comprehensive technique can be derived that converts linguistic-based conditional information, couched only in fuzzy-logic terms, into naturally corresponding conditional probabilities. In turn, they demonstrate how such typically underconstrained conditional probabilities can be combined for suitable conclusions and decision making, via a new use of second-order probability logic. This research is part of the ongoing SSC San Diego In-house Laboratory Independent Research FY 01 project CRANOF (a Complexity-Reducing Algorithm for Near-Optimal Fusion).

Book digitized by Google from the library of the New York Public Library and uploaded to the Internet Archive by user tpb.

2 p.l., 3-412 p. 21 cm

2 p.l., 3-412 p. 21 cm

2 p.l., 3-412 p. 21 cm

2 p.l., 3-412 p. 21 cm

2 p.l., 3-412 p. 21 cm

We develop a systematic approach to quantum probability as a theory of rational betting in quantum gambles. In these games of chance the agent is betting in advance on the outcomes of several (finitely many) incompatible measurements. One of the measurements is subsequently chosen and performed and the money placed on the other measurements is returned to the agent. We show how the rules of rational betting imply all the interesting features of quantum probability, even in such finite gambles. These include the uncertainty principle and the violation of Bell's inequality among others. Quantum gambles are closely related to quantum logic and provide a new semantics to it. We conclude with a philosophical discussion on the interpretation of quantum mechanics.

In this report, the author suggests changes in the treatment of overhead costs by hypothesizing that "the effectiveness of standard costing in planning and controlling overhead costs can be increased through the use of probability theory and associated statistical techniques." To test the hypothesis, the author (1) presents an overview of the problem, (2) reviews probability theory, (3) discusses the selection of the proper empirical probability distribution to be used in the study, (4) uses the normal distribution to develop a cost control chart for dollar costs of manufacturing supplies, and (5) discusses the use of simple linear and multilinear regression and correlation analysis for budgeting overhead costs. (Author)

We survey a number of models from physics, statistical mechanics, probability theory and combinatorics, which are each described in terms of an orthogonal polynomial ensemble. The most prominent example is apparently the Hermite ensemble, the eigenvalue distribution of the Gaussian Unitary Ensemble (GUE), and other well-known ensembles known in random matrix theory like the Laguerre ensemble for the spectrum of Wishart matrices. In recent years, a number of further interesting models were found to lead to orthogonal polynomial ensembles, among which the corner growth model, directed last passage percolation, the PNG droplet, non-colliding random processes, the length of the longest increasing subsequence of a random permutation, and others. Much attention has been paid to universal classes of asymptotic behaviors of these models in the limit of large particle numbers, in particular the spacings between the particles and the fluctuation behavior of the largest particle. Computer simulations suggest that the connections go even farther and also comprise the zeros of the Riemann zeta function. The existing proofs require a substantial technical machinery and heavy tools from various parts of mathematics, in particular complex analysis, combinatorics and variational analysis. Particularly in the last decade, a number of fine results have been achieved, but it is obvious that a comprehensive and thorough understanding of the matter is still lacking. Hence, it seems an appropriate time to provide a surveying text on this research area.

Nonlinear evolution can sometimes be modelled by a deterministic mapping from initial to final of the local smoothed overdensity. Perturbation theory methods base on this deterministic and local mapping and ignore the 'cloud-in-cloud' effect, while the excursion set approach methods take this nonlocality into account. We compared these methods using the spherical collapse mapping and showed that, on scales where the rms fluctuation is small, both models give similar results and they are in good agreement with numerical simulations. If the deterministic mapping depends on quantities other than overdensity, this will also manifest as stochasticity if the other quantities are ignored. We considered the Zeldovich approximation and Ellipsoidal Collapse model, both include the tidal field in the evolution. Our anaylsis shows that the change in cell shape effect should be included on scales where the rms is of order of unity or larger. On scales where the rms is less than 2 methods based on the spherical collapse model allow a rather accurate reconstruction of the shape of the initial distribution from the nonlinear field. This can be used as the basis for constraining the statistical properties of the initial fluctuation field. (Abridge)

An open problem arising in the statistical description of turbulence is related to the \textit{theoretical prediction based on first principles} of the so-called multi-point velocity probability density functions (PDFs) characterizing a Navier-Stokes fluid. In this paper it will be shown that - based on a suitable axiomatic approach - a solution to this problem can actually be achieved based on the so-called inverse kinetic theory (IKT), recently developed for incompressible fluids. More precisely, we intend to show, based on the requirement that \textit{the Boltzmann-Shannon entropy for the s-point velocity PDF ($f_s$) is independent of the order $s$ and is also maximal at all times}, that all multi-point PDFs are \textit{necessarily factorized in terms of the corresponding 1-point velocity PDF} ($f_1$). As a consequence the multi-point PDFs usually considered for the phenomenological description of turbulence can be theoretically predicted \textit{based on the knowledge of $% f_1$ achieved by means of IKT.

The systematic biases seen in people's probability judgments are typically taken as evidence that people do not reason about probability using the rules of probability theory, but instead use heuristics which sometimes yield reasonable judgments and sometimes severe and systematic errors. This `heuristics and biases' view has had a major impact in economics, law, medicine, and other fields; indeed, the idea that people cannot reason with probabilities has become a widespread truism. We present a simple alternative to this view, where people reason about probability according to probability theory but are subject to random variation or noise in the recall of items from memory. We show that this `noisy recall' leads to systematic deviations in probability estimates, and that these deviations explain four reliable biases in human probabilistic reasoning: conservatism, subadditivity, conjunction fallacies, and disjunction fallacies. Analysing experimental data on probability estimation we find that when deviation due to noisy recall is cancelled, people's probability judgments are strikingly close to those required by probability theory. This shows that people's probability judgments embody the rules of probability theory, and that biases in those estimates are due to the effects of random noise in recall.

Many aspects of signal processing and communication theory have proven to be limited by a lack of sufficient developments in the areas of probability theory and mathematical statistics. Our investigations attempted to overcome this deficiency by contributing both to the underlying theoretical basis of the area as well as to some applied aspects of the area, and we have obtained a large body of results. Some principal thrust areas of our research effort have been concerned with quantization theory, signal detection, and estimation theory. A brief topical overview of our research areas follows. Quantization - existence of optimal quantizers, convergence properties of sequences of quantizers, design of quantizers, Signal detection - effects of statistical dependence, relative efficiency between detectors, nonparametric detection, Estimation - effects of imperfect data, a general class of fidelity criteria leading to nonlinear estimators, convergence properties of sequences of estimators; and Additional areas - meadian filters; stability of differential equations with random coefficients; bandwidth properties of random processes; contention resolution in local area computer networks; image compression; counterexamples.

Many aspects of signal processing and communication theory have proven to be limited by a lack of sufficient developments in the areas of probability theory and mathematical statistics. Our investigations attempted to overcome this deficiency by contributing both to the underlying theoretical basis of the area as well as to some applied aspects of the area, and we have obtained a large body of results. Some principal thrust areas of our research effort have been concerned with quantization theory, signal detection, and estimation theory. A brief topical overview of our research areas follows. Quantization - existence of optimal quantizers, convergence properties of sequences of quantizers, design of quantizers, Signal detection - effects of statistical dependence, relative efficiency between detectors, nonparametric detection, Estimation - effects of imperfect data, a general class of fidelity criteria leading to nonlinear estimators, convergence properties of sequences of estimators; and Additional areas - meadian filters; stability of differential equations with random coefficients; bandwidth properties of random processes; contention resolution in local area computer networks; image compression; counterexamples.

The notion of context (complex of physical conditions) is basic in this paper. We show that the main structures of quantum theory (interference of probabilities, Born's rule, complex probabilistic amplitudes, Hilbert state space, representation of observables by operators) are present in a latent form in the classical Kolmogorov probability model. However, this model should be considered as a calculus of contextual probabilities. In our approach it is forbidden to consider abstract context independent probabilities: ``first context and then probability.'' We start with the conventional formula of total probability for contextual (conditional) probabilities and then we rewrite it by eliminating combinations of incompatible contexts from consideration. In this way we obtain interference of probabilities without to appeal to the Hilbert space formalism or wave mechanics. However, we did not just reconstruct the probabilistic formalism of conventional quantum mechanics. Our contextual probabilistic model is essentially more general and, besides the projection to the complex Hilbert space, it has other projections. The most important new prediction is the possibility (at least theoretical) of appearance of hyperbolic interference.

Within the Kolmogorov theory of probability, Bayes' rule allows one to perform statistical inference by relating conditional probabilities to unconditional probabilities. As we show here, however, there is a continuous set of alternative inference rules that yield the same results, and that may have computational or practical advantages for certain problems. We formulate generalized axioms for probability theory, according to which the reverse conditional probability distribution P(B|A) is not specified by the forward conditional probability distribution P(A|B) and the marginals P(A) and P(B). Thus, in order to perform statistical inference, one must specify an additional "inference axiom," which relates P(B|A) to P(A|B), P(A), and P(B). We show that when Bayes' rule is chosen as the inference axiom, the axioms are equivalent to the classical Kolmogorov axioms. We then derive consistency conditions on the inference axiom, and thereby characterize the set of all possible rules for inference. The set of "first-order" inference axioms, defined as the set of axioms in which P(B|A) depends on the first power of P(A|B), is found to be a 1-simplex, with Bayes' rule at one of the extreme points. The other extreme point, the "inversion rule," is studied in depth.

Within the Kolmogorov theory of probability, Bayes' rule allows one to perform statistical inference by relating conditional probabilities to unconditional probabilities. As we show here, however, there is a continuous set of alternative inference rules that yield the same results, and that may have computational or practical advantages for certain problems. We formulate generalized axioms for probability theory, according to which the reverse conditional probability distribution P(B|A) is not specified by the forward conditional probability distribution P(A|B) and the marginals P(A) and P(B). Thus, in order to perform statistical inference, one must specify an additional "inference axiom," which relates P(B|A) to P(A|B), P(A), and P(B). We show that when Bayes' rule is chosen as the inference axiom, the axioms are equivalent to the classical Kolmogorov axioms. We then derive consistency conditions on the inference axiom, and thereby characterize the set of all possible rules for inference. The set of "first-order" inference axioms, defined as the set of axioms in which P(B|A) depends on the first power of P(A|B), is found to be a 1-simplex, with Bayes' rule at one of the extreme points. The other extreme point, the "inversion rule," is studied in depth.

Problem-based learning (PBL) has been widely applied as an alternative to improve learning outcomes, but it is still little studied in the context of the probability theory course. This study described how implementing the PBL model improves students’ problem-solving and critical thinking skills in probability theory course and evaluates its quality. This design research involved 58 undergraduate students and two probability theory course lecturers from two universities in Indonesia as participants. Data collection used observation to describe PBL implementation, then questionnaire and pretest-posttest to evaluate the quality of the model. This study produced a PBL model for the probability theory course which is implemented through five steps: i) Orienting students on problems; ii) Organizing students to study; iii) Assisting individual and group investigations; iv) Developing and presenting work or solutions; and v) Analyzing and evaluating problem solving processes. Lecturers and students consider that the implementation of PBL is practical. PBL implementation can also improve students’ problem-solving skills and critical thinking in the probability theory course. Thus, implementing PBL can be used as a solution to optimize learning outcomes in the probability theory course.

This book is based on many years of experience of teaching probability theory and its applications at higher educational establishments. It contains many of the problems we ourselves encountered in our research and consultative work. The problems are related to a variety of fields including electrical engineering, radio engineering, data transmission, computers, information systems, reliability of technical devices, preventive maintenance and repair, accuracy of apparatus, consumer service, transport, and the health service.  The text is divided into eleven chapters; each of winch begins with a short theoretical introduction which is followed by relevant formulas. The problems differ both in the fields of application and in difficulty. At the beginning of each chapter the reader will find comparatively simple problems whose purpose is to help the reader grasp the fundamental concepts and acquire and consolidate the experience of applying probabilistic methods. Then follow more complicated applied problems, which can be solved only after the requisite theoretical knowledge has been acquired and the necessary techniques mastered.

This book is based on many years of experience of teaching probability theory and its applications at higher educational establishments. It contains many of the problems we ourselves encountered in our research and consultative work. The problems are related to a variety of fields including electrical engineering, radio engineering, data transmission, computers, information systems, reliability of technical devices, preventive maintenance and repair, accuracy of apparatus, consumer service, transport, and the health service.  The text is divided into eleven chapters; each of winch begins with a short theoretical introduction which is followed by relevant formulas. The problems differ both in the fields of application and in difficulty. At the beginning of each chapter the reader will find comparatively simple problems whose purpose is to help the reader grasp the fundamental concepts and acquire and consolidate the experience of applying probabilistic methods. Then follow more complicated applied problems, which can be solved only after the requisite theoretical knowledge has been acquired and the necessary techniques mastered.

This book is based on many years of experience of teaching probability theory and its applications at higher educational establishments. It contains many of the problems we ourselves encountered in our research and consultative work. The problems are related to a variety of fields including electrical engineering, radio engineering, data transmission, computers, information systems, reliability of technical devices, preventive maintenance and repair, accuracy of apparatus, consumer service, transport, and the health service.  The text is divided into eleven chapters; each of winch begins with a short theoretical introduction which is followed by relevant formulas. The problems differ both in the fields of application and in difficulty. At the beginning of each chapter the reader will find comparatively simple problems whose purpose is to help the reader grasp the fundamental concepts and acquire and consolidate the experience of applying probabilistic methods. Then follow more complicated applied problems, which can be solved only after the requisite theoretical knowledge has been acquired and the necessary techniques mastered.

In the paper, we discuss orthogonal polynomials in free probability theory. Especially, we prove an analogue of of Szego's limit theorem in free probability theory.

In the paper, we discuss orthogonal polynomials in free probability theory. Especially, we prove an analogue of of Szego's limit theorem in free probability theory.

In the paper, we discuss orthogonal polynomials in free probability theory. Especially, we prove an analogue of of Szego's limit theorem in free probability theory.

In the paper, we discuss orthogonal polynomials in free probability theory. Especially, we prove an analogue of of Szego's limit theorem in free probability theory.

This study replicates Lagattuta, Sayfan, and Harvey (2014) in an online experiment with ~70 adults recruted on prolific. In this task, each trial begins with a simple picture (a cloud, a castle, or a panda) that is then covered so only a small 1.5"×1.5" fragment remains visible. Informed participants first view and label the full picture, then see only the fragment. Uninformed participants see only the fragment from the start. All participants are asked to judge what a naïve observer—who has never seen the full picture—would think is behind the cover. They first freely guess, then rate a set of seven possible images (the true picture, a basic shape, and various more or less plausible alternatives) on a 0–10 probability scale, removing the ones they believe the observer "definitely would not" choose. By comparing ratings from those who knew the whole image versus those who did not, the task reveals how privileged knowledge distorts our predictions of another person’s thoughts: informed participants overestimate the chance that the observer would pick the correct picture and downplay options that seem plausible when the fragment is viewed in isolation.

The 1992-1993 year in Control Theory focused on robotics, optimal control for nonsmooth geometric data, adaptive control and filtering, discrete event systems, and power systems. It brought together mathematicians, electrical engineers and practitioners, who identified current difficulties and challenges. The 1993-1994 year in Emerging Applications of Probability brought together computer scientists and probabilists; it also dealt with stochastic models in geosystems, hidden Markov processes in speech and vision, stochastic networks and branching processes. The 1994-1995 year in Waves and Scattering dealt with random waves, numerical methods including hybrid methods (FFT and wavelets), and inverse problems, such as nondestructive evaluation, acoustics, NMR, Electro-magnetic, and ground penetrating radar.

The 1992-1993 year in Control Theory focused on robotics, optimal control for nonsmooth geometric data, adaptive control and filtering, discrete event systems, and power systems. It brought together mathematicians, electrical engineers and practitioners, who identified current difficulties and challenges. The 1993-1994 year in Emerging Applications of Probability brought together computer scientists and probabilists; it also dealt with stochastic models in geosystems, hidden Markov processes in speech and vision, stochastic networks and branching processes. The 1994-1995 year in Waves and Scattering dealt with random waves, numerical methods including hybrid methods (FFT and wavelets), and inverse problems, such as nondestructive evaluation, acoustics, NMR, Electro-magnetic, and ground penetrating radar.

The 1992-1993 year in Control Theory focused on robotics, optimal control for nonsmooth geometric data, adaptive control and filtering, discrete event systems, and power systems. It brought together mathematicians, electrical engineers and practitioners, who identified current difficulties and challenges. The 1993-1994 year in Emerging Applications of Probability brought together computer scientists and probabilists; it also dealt with stochastic models in geosystems, hidden Markov processes in speech and vision, stochastic networks and branching processes. The 1994-1995 year in Waves and Scattering dealt with random waves, numerical methods including hybrid methods (FFT and wavelets), and inverse problems, such as nondestructive evaluation, acoustics, NMR, Electro-magnetic, and ground penetrating radar.

The paper deals with the order statistics and empirical mathematical expectation (which is also called the estimate of mathematical expectation in the literature) in the case of infinitely increasing random variables. The Kolmogorov concept which he used in the theory of complexity and the relationship with thermodynamics which was pointed out already by Poincar\'e are considered. The mathematical expectation (generalizing the notion of arithmetical mean, which is generally equal to infinity for any increasing sequence of random variables) is compared with the notion of temperature in thermodynamics by using an analog of nonstandard analysis. The relationship with the Van-der-Waals law of corresponding states is shown. Some applications of this concept in economics, in internet information network, and self-teaching systems are considered.

The probability for quantum creation of an inflationary universe with a pair of black holes is computed in a modified gravitational theory. Considering a gravitational action which includes a cosmological constant ($\Lambda$) in addition to $ \alpha R^{2} $ and $ \delta R^{-1}$ terms, the probabilities have been evaluated for two different kinds of spatial sections, one accommodating a pair of black holes and the other without black hole. We adopt a technique prescribed by Bousso and Hawking to calculate the above creation probability in a semiclassical approximation with Hartle-Hawking boundary condition. Depending on the parameters in the action some new and physically interesting instanton solutions are presented here which may play an important role in the creation of the early universe. We note that the probability of creation of a universe with a pair of black holes is strongly suppressed with a positive cosmological constant when $\delta = \frac{4 \Lambda^{2}}{3}$ for $\alpha > 0$ but it is more probable for $\alpha < - \frac{1}{6 \Lambda}$. It is also found that instanton solutions are allowed without a cosmological constant in the theory provided $\delta < 0$.

A theory of measurement uncertainty is presented, which, since it is based exclusively on the Bayesian approach and on the subjective concept of conditional probability, is applicable in the most general cases. The recent International Organization for Standardization (ISO) recommendation on measurement uncertainty is reobtained as the limit case in which linearization is meaningful and one is interested only in the best estimates of the quantities and in their variances.

We interpret the Central Limit Theorem as a fixed point theorem for a certain operator, and consider the problem of linearizing this operator. In classical as well as in free probability theory, we consider two methods giving such a linearization, and interpret the result as a weak form of the CLT. In the classical case the analysis involves dilation operators; in the free case more general composition operators appear.

This is the first installment of a series of papers whose aim is to lay a foundation for homotopy probability theory by establishing its basic principles and practices. The notion of a homotopy probability space is an enrichment of the notion of an algebraic probability space with ideas from algebraic homotopy theory. This enrichment uses a characterization of the laws of random variables in a probability space in terms of symmetries of the expectation. The laws of random variables are reinterpreted as invariants of the homotopy types of infinity morphisms between certain homotopy algebras. The relevant category of homotopy algebras is determined by the appropriate notion of independence for the underlying probability theory. This theory will be both a natural generalization and an effective computational tool for the study of classical algebraic probability spaces, while keeping the same central limit. This article is focused on the commutative case, where the laws of random variables are also described in terms of certain affinely flat structures on the formal moduli space of a naturally defined family attached to the given algebraic probability space. Non-commutative probability theories will be the main subject of the sequels. (This work is a spin-off from the author's program to characterize path integrals of quantum field theory in terms of the symmetries of the quantum expectation which should satisfy a certain coherence with a particular weight filtration generated by the Planck constant $\hbar$. A similar idea is adopted here in a simplified form, without the $\hbar$-conditions, and, in return, many results in this paper will be used as background materials in the forthcoming work on homotopy theory of quantum fields).

This is the first installment of a series of papers whose aim is to lay a foundation for homotopy probability theory by establishing its basic principles and practices. The notion of a homotopy probability space is an enrichment of the notion of an algebraic probability space with ideas from algebraic homotopy theory. This enrichment uses a characterization of the laws of random variables in a probability space in terms of symmetries of the expectation. The laws of random variables are reinterpreted as invariants of the homotopy types of infinity morphisms between certain homotopy algebras. The relevant category of homotopy algebras is determined by the appropriate notion of independence for the underlying probability theory. This theory will be both a natural generalization and an effective computational tool for the study of classical algebraic probability spaces, while keeping the same central limit. This article is focused on the commutative case, where the laws of random variables are also described in terms of certain affinely flat structures on the formal moduli space of a naturally defined family attached to the given algebraic probability space. Non-commutative probability theories will be the main subject of the sequels. (This work is a spin-off from the author's program to characterize path integrals of quantum field theory in terms of the symmetries of the quantum expectation which should satisfy a certain coherence with a particular weight filtration generated by the Planck constant $\hbar$. A similar idea is adopted here in a simplified form, without the $\hbar$-conditions, and, in return, many results in this paper will be used as background materials in the forthcoming work on homotopy theory of quantum fields).

This is the first installment of a series of papers whose aim is to lay a foundation for homotopy probability theory by establishing its basic principles and practices. The notion of a homotopy probability space is an enrichment of the notion of an algebraic probability space with ideas from algebraic homotopy theory. This enrichment uses a characterization of the laws of random variables in a probability space in terms of symmetries of the expectation. The laws of random variables are reinterpreted as invariants of the homotopy types of infinity morphisms between certain homotopy algebras. The relevant category of homotopy algebras is determined by the appropriate notion of independence for the underlying probability theory. This theory will be both a natural generalization and an effective computational tool for the study of classical algebraic probability spaces, while keeping the same central limit. This article is focused on the commutative case, where the laws of random variables are also described in terms of certain affinely flat structures on the formal moduli space of a naturally defined family attached to the given algebraic probability space. Non-commutative probability theories will be the main subject of the sequels. (This work is a spin-off from the author's program to characterize path integrals of quantum field theory in terms of the symmetries of the quantum expectation which should satisfy a certain coherence with a particular weight filtration generated by the Planck constant $\hbar$. A similar idea is adopted here in a simplified form, without the $\hbar$-conditions, and, in return, many results in this paper will be used as background materials in the forthcoming work on homotopy theory of quantum fields).

This is the first installment of a series of papers whose aim is to lay a foundation for homotopy probability theory by establishing its basic principles and practices. The notion of a homotopy probability space is an enrichment of the notion of an algebraic probability space with ideas from algebraic homotopy theory. This enrichment uses a characterization of the laws of random variables in a probability space in terms of symmetries of the expectation. The laws of random variables are reinterpreted as invariants of the homotopy types of infinity morphisms between certain homotopy algebras. The relevant category of homotopy algebras is determined by the appropriate notion of independence for the underlying probability theory. This theory will be both a natural generalization and an effective computational tool for the study of classical algebraic probability spaces, while keeping the same central limit. This article is focused on the commutative case, where the laws of random variables are also described in terms of certain affinely flat structures on the formal moduli space of a naturally defined family attached to the given algebraic probability space. Non-commutative probability theories will be the main subject of the sequels. (This work is a spin-off from the author's program to characterize path integrals of quantum field theory in terms of the symmetries of the quantum expectation which should satisfy a certain coherence with a particular weight filtration generated by the Planck constant $\hbar$. A similar idea is adopted here in a simplified form, without the $\hbar$-conditions, and, in return, many results in this paper will be used as background materials in the forthcoming work on homotopy theory of quantum fields).

This is the first installment of a series of papers whose aim is to lay a foundation for homotopy probability theory by establishing its basic principles and practices. The notion of a homotopy probability space is an enrichment of the notion of an algebraic probability space with ideas from algebraic homotopy theory. This enrichment uses a characterization of the laws of random variables in a probability space in terms of symmetries of the expectation. The laws of random variables are reinterpreted as invariants of the homotopy types of infinity morphisms between certain homotopy algebras. The relevant category of homotopy algebras is determined by the appropriate notion of independence for the underlying probability theory. This theory will be both a natural generalization and an effective computational tool for the study of classical algebraic probability spaces, while keeping the same central limit. This article is focused on the commutative case, where the laws of random variables are also described in terms of certain affinely flat structures on the formal moduli space of a naturally defined family attached to the given algebraic probability space. Non-commutative probability theories will be the main subject of the sequels. (This work is a spin-off from the author's program to characterize path integrals of quantum field theory in terms of the symmetries of the quantum expectation which should satisfy a certain coherence with a particular weight filtration generated by the Planck constant $\hbar$. A similar idea is adopted here in a simplified form, without the $\hbar$-conditions, and, in return, many results in this paper will be used as background materials in the forthcoming work on homotopy theory of quantum fields).

46 p. : 21 cm

46 p. : 21 cm

46 p. : 21 cm

106 p

106 p