An original and additional mathematical model characterizing a Bayesian approach to decision theory

We propose an original mathematical model according to a Bayesian approach explaining uncertainty from a point of view connected with vector spaces. A parameter space can be represented by means of random quantities by accepting the principles of the theory of concordance into the domain of subjective probability. We observe that metric properties of the notion of -product mathematically fulfill the ones of a coherent prevision of a bivariate random quantity. We introduce fundamental metric expressions connected with transformed random quantities representing changes of origin. We obtain a posterior probability law by applying the Bayes’ theorem into a geometric context connected with a two-dimensional parameter space

A Quadratic and Linear Metric Characterizing the Sampling Design with Fixed Sample Size Considered From a Geometric Viewpoint

The first-order and second-order inclusion probabilities are chosen by the statistician. They are subjective probabilities. We innovatively define univariate and bivariate random quantities whose logically possible values are samples of a given size in order to obtain the first-order and second-order inclusion probabilities by means of their coherent previsions. We consider linear maps connected with univariate random quantities as well as bilinear maps connected with bivariate random quantities. The covariance of two univariate random quantities that are the components of a bivariate random quantity has been expressed by means of two bilinear maps. We show that a univariate random quantity denoted by S is complementary to the univariate Horvitz-Thompson estimator. We identify a quadratic and linear metric with regard to two univariate random quantities representing deviations that we innovatively define. We use the α-criterion of concordance introduced byGini in order to identify it. It is a statistical criterion that we innovatively apply to probability

Sull’ambito del logicamente possibile secondo la concezione probabilistica di Bruno de Finetti

The fundamental geometric characteristics of random numbers, random events, random structures, and random functions are noticed before the subjective probabilistic evaluation

Metric representations of a preference ordering

We prove that when we decompose the expected utility function inside of an mdimensional metric space we refer to a preference ordering based on the notion of distance. We prove that when we deal with a scale of measurable utilities we refer to a preference ordering based on the notion of distance. A contingent consumption plan is studied inside of an m-dimensional metric space because utility and probability are both subjective. The right closed structure in order to deal with utility and probability is a metric space in which we study coherent decisions under uncertainty having as their goal the maximization of the prevision of the utility associated with a contingent consumption plan

A reinterpretation of principal component analysis connected with linear manifolds identifying risky assets of a portfolio

We use the mean-variance model to study a portfolio problem characterized by an investment in two different types of asset. We consider m logically independent risky assets and a risk-free asset. We analyze m risky assets coinciding with m distributions of probability inside of a linear space. They generate a distribution of probability of a multivariate risky asset of order m. We show that an m-dimensional linear manifold is generated by m basic risky assets. They identify m finite partitions, where each of them is characterized by n incompatible and exhaustive elementary events. We suppose that it turns out to be n > m without loss of generality. Given m risky assets, we prove that all risky assets contained in an m-dimensional linear manifold are related. We prove that two any risky assets of them are conversely αorthogonal, so their covariance is equal to 0. We reinterpret principal component analysis by showing that the principal components are basic risky assets of an m-dimensional linear manifold. We consider a Bayesian adjustment of differences between prior distributions to posterior distributions existing with respect to a probabilistic and economic hypothesi

Aggregate Bound Choices about Random and Nonrandom Goods Studied via a Nonlinear Analysis

In this paper, bound choices are made after summarizing a finite number of alternatives. This means that each choice is always the barycenter of masses distributed over a finite set of alternatives. More than two marginal goods at a time are not handled. This is because a quadratic metric is used. In our models, two marginal goods give rise to a joint good, so aggregate bound choices are shown. The variability of choice for two marginal goods that are the components of a multiple good is studied. The weak axiom of revealed preference is checked and mean quadratic differences connected with multiple goods are proposed. In this paper, many differences from vast majority of current research about choices and preferences appear. First of all, conditions of certainty are viewed to be as an extreme simplification. In fact, in almost all circumstances, and at all times, we all find ourselves in a state of uncertainty. Secondly, the two notions, probability and utility, on which the correct criterion of decision-making depends, are treated inside linear spaces over R having a different dimension in accordance with the pure subjectivistic point of vie

The consumer’s demand functions defined to study contingent consumption plans. Summarized probability distributions: a mathematical application to contingent consumption choices

Given two probability distributions expressing returns on two single risky assets of a portfolio, we innovatively define two consumer’s demand functions connected with two contingent consumption plans. This thing is possible whenever we coherently summarize every probability distribution being chosen by the consumer. Since prevision choices are consumption choices being made by the consumer inside of a metric space, we show that prevision choices can be studied by means of the standard economic model of consumer behavior. Such a model implies that we consider all coherent previsions of a joint distribution. They are decomposed inside of a metric space. Such a space coincides with the consumer’s consumption space. In this paper, we do not consider a joint distribution only. It follows that we innovatively define a stand-alone and double risky asset. Different summary measures of it characterizing consumption choices being made by the consumer can then be studied inside of a linear space over ℝ. We show that it is possible to obtain different summary measures of probability distributions by using two different quadratic metrics. In this paper, our results are based on a particular approach to the origin of the variability of probability distributions. We realize that it is not standardized, but it always depends on the state of information and knowledge of the consumer

Tensors Associated with Mean Quadratic Differences Explaining the Riskiness of Portfolios of Financial Assets

Bound choices such as portfolio choices are studied in an aggregate fashion using an extension of the notion of barycenter of masses. This paper answers the question of whether such an extension is a natural fashion of studying bound choices or not. Given n risky assets, the question of why it is appropriate to treat only two risky assets at a time inside the budget set of the decision-maker is handled in this paper. Two risky assets are two goods. They are two marginal goods. The question of why they always give rise to a joint good inside the budget set of the decision-maker is addressed by this research work. A single risky asset is viewed as a double one using four nonparametric joint distributions of probability. The variability of a joint distribution of probability always depends on the state of information and knowledge associated with a given decision-maker. For this reason, two variability tensors are defined to identify the riskiness of the same risky asset. A multilinear version of the Sharpe ratio is shown. It is based on tensors. After computing the expected return on an n-risky asset portfolio, its riskiness is obtained using mean quadratic differences developed through tensor

Non-parametric probability distributions embedded inside of a linear space provided with a quadratic metric

There exist uncertain situations in which a random event is not a measurable set, but it is a point of a linear space inside of which it is possible to study different random quantities characterized by non-parametric probability distributions. We show that if an event is not a measurable set then it is contained in a closed structure which is not a σ-algebra but it is a linear space over R. We think of probability as being a mass. It is really a mass with respect to problems of statistical sampling. It is a mass with respect to problems of social sciences. In particular, it is a mass with regard to economic situations studied by means of the subjective notion of utility. We are able to decompose a random quantity meant as a geometric entity inside of a metric space. It is also possible to decompose its prevision and variance inside of it. We show a quadratic metric in order to obtain the variance of a random quantity. The origin of the notion of variability is not standardized within this context. It always depends on the state of information and knowledge of an individual. We study different intrinsic properties of non-parametric probability distributions as well as of probabilistic indices summarizing them. We define the notion of α-distance between two non-parametric probability distributio

Subjective probability and geometry: three metric theorems concerning random quantities

Affine properties are more general than metric ones because they are independent of the choice of a coordinate system. Nevertheless, a metric, that is to say, a scalar product which takes each pair of vectors and returns a real number, is meaningful when n vectors, which are all unit vectors and orthogonal to each other, constitute a basis for the n-dimensional vector space A. In such a space n events Ei, i = 1; : : : ; n, whose Cartesian coordinates turn out to be xi, are represented in a linear form. A metric is also meaningful when we transfer on a straight line the n-dimensional structure of A into which the constituents of the partition determined by E1; : : : ; En are visualized. The dot product of two vectors of the ndimensional real space Rn is invariant: of these two vectors the former represents the possible values for a given random quantity, while the latter represents the corresponding probabilities which are assigned to them in a subjective fashion. We deduce these original results, which are the foundation of our next and extensive study concerning the formulation of a geometric, well-organized and original theory of random quantities, from pioneering works which deal with a specific geometric interpretation of probability concept, unlike the most part of the current ones which are pleased to keep the real and deep meaning of probability notion a secret because they consider a success to give a uniquely determined answer to a problem even when it is indeterminate. Therefore, we believe that it is inevitable that our references limit themselves to these pioneering works