Entropy of Artificial Intelligence

We describe a model of artificial intelligence systems based on the dimension of the probability space of the input set available for recognition. In this scenario, we can understand a subset, which means that we can decide whether an object is an element of a given subset or not in an efficient way. In the machine learning (ML) process we define appropriate features, in this way shrinking the defining bit-length of classified sets during the learning process. This can also be described in the language of entropy: while natural processes tend to increase the disorder, that is, increase the entropy, learning creates order, and we expect that it decreases a properly defined entropy.

Strong Law of Large Numbers for Iterates of Some Random-Valued Functions

Assume $$ (\Omega , {\mathscr {A}}, P) $$ ( Ω , A , P ) is a probability space, X is a compact metric space with the $$ \sigma $$ σ -algebra $$ {\mathscr {B}} $$ B of all its Borel subsets and $$ f: X \times \Omega \rightarrow X $$ f : X × Ω → X is $$ {\mathscr {B}} \otimes {\mathscr {A}} $$ B ⊗ A -measurable and contractive in mean. We consider the sequence of iterates of f defined on $$ X \times \Omega ^{{\mathbb {N}}}$$ X × Ω N by $$f^0(x, \omega ) = x$$ f 0 ( x , ω ) = x and $$ f^n(x, \omega ) = f\big (f^{n-1}(x, \omega ), \omega _n\big )$$ f n ( x , ω ) = f ( f n - 1 ( x , ω ) , ω n ) for $$n \in {\mathbb {N}}$$ n ∈ N , and its weak limit $$\pi $$ π . We show that if $$\psi :X \rightarrow {\mathbb {R}}$$ ψ : X → R is continuous, then for every $$ x \in X $$ x ∈ X the sequence $$\left( \frac{1}{n}\sum _{k=1}^n \psi \big (f^k(x,\cdot )\big )\right) _{n \in {\mathbb {N}}}$$ 1 n ∑ k = 1 n ψ ( f k ( x , · ) ) n ∈ N converges almost surely to $$\int _X\psi d\pi $$ ∫ X ψ d π . In fact, we are focusing on the case where the metric space is complete and separable.

Using SEEPS with a TRMM-derived climatology to assess global NWP precipitation forecast skill

Monitoring precipitation forecast skill in global Numerical Weather Prediction (NWP) models is an important yet challenging task. Rain gauges are inhomogeneously distributed, providing no information over large swathes of land and the oceans. Satellite-based products on the other hand provide near-global coverage at a resolution of ~10-25 km, but limitations on data quality (e.g. biases) must be accommodated. In this paper the Stable Equitable Error in Probability Space (SEEPS) is computed using a precipitation climatology derived from the Tropical Rainfall Measurement Mission (TRMM) TMPA 3B42 V7 product and a gauge-based climatology, and applied to two global configurations of the Met Office Unified Model (UM). The representativeness and resolution effects on an aggregated SEEPS is explored by comparing the gauge scores, based on extracting the nearest model grid point, to those computed by upscaling the model values to the TRMM grid and extracting the TRMM grid point nearest the gauge location. The sampling effect is explored by comparing the aggregate SEEPS for this subset of ~6000 locations (dictated by the number of gauges available globally) to all land points within the TRMM region of 50°N and 50°S. Finally, the forecast performance over the oceanic areas is compared to performance over land. Whilst the SEEPS computed using the two different climatologies should never be expected to be identical, using the TRMM climatology provides a means of evaluating near-global precipitation using an internally consistent dataset in a climatologically consistent way.

Ergodic Decomposition of Dirichlet Forms via Direct Integrals and Applications

We study direct integrals of quadratic and Dirichlet forms. We show that each quasi-regular Dirichlet space over a probability space admits a unique representation as a direct integral of irreducible Dirichlet spaces, quasi-regular for the same underlying topology. The same holds for each quasi-regular strongly local Dirichlet space over a metrizable Luzin σ-finite Radon measure space, and admitting carré du champ operator. In this case, the representation is only projectively unique.

Random Power Series in Q p , 0 Spaces

Aulaskari et al. proved if 0 < p < 1 and ε n is sequence of independent, identically distributed Rademacher random variables on a probability space, then the condition Σ n = 0 ∞ n 1 − p a n 2 < ∞ implies that the random power series R f z = ∑ n = 0 ∞ a n ε n z n ∈ Q p almost surely. In this paper, we improve this result showing that the condition Σ n = 0 ∞ n 1 − p a n 2 < ∞ actually implies R f ∈ Q p , 0 almost surely.

Simultaneous Joint Lower and Upper record values Probability Laws for Absolutely Continuous or Discrete Data

This paper investigates the probability density function (pdf) of the \((2n-1)\)-vector \((n\geq 1)\) of both lower and upper record values for a sequence of independent random variables with common \textit{pdf} \(f\) defined on the same probability space, provided that the lower and upper record times are finite up to \(n\). A lot is known about the lower or the upper record values when they are studied separately. When put together, the challenges are far complicated. The rare results in the literature still present some flaws. This paper begins a new and complete investigation with a few number of records: (n=2\) and \(n=3\). Lessons from these simple cases will allow addressing the general formulation of simultaneous joint lower-upper records.

On probabilistic argumentation and subargument-completeness

Probabilistic argumentation combines probability theory and formal models of argumentation. Given an argumentation graph where vertices are arguments and edges are attacks or supports between arguments, the approach of probabilistic labellings relies on a probability space where the sample space is any specific set of argument labellings of the graph, so that any labelling outcome can be associated with a probability value. Argument labellings can feature a label indicating that an argument is not expressed, and in previous work these labellings were constructed by exploiting the subargument-completeness postulate according to which if an argument is expressed then its subarguments are expressed and through the use of the concept of ‘subargument-complete subgraphs’. While the use of such subgraphs is interesting to compare probabilistic labellings with other works in the literature, it may also hinder the comprehension of a relatively simple framework. In this short communication, we revisit the construction of probabilistic labellings and demonstrate how labellings can be specified without reference to the concept of subargument-complete subgraphs. By doing so, the framework is simplified and yields a more natural model of argumentation.

Some Studies in Hemirings by the Falling Fuzzy k -Ideals

In this article, we establish the idea of falling fuzzy k -ideals in hemirings through the falling shadow theory and fuzzy sets. We shall express the relations between fuzzy k -ideals and falling fuzzy k -ideals in hemirings. In particular, we shall establish different characterizations of k -hemiregular hemirings in the perfect positive correlation and independent probability space by means of falling fuzzy k -ideals.