probability domain
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Author(s):  
Daria Tolstykh ◽  
Laurent Lemmens ◽  
Stijn De Baerdemacker ◽  
Dimitri Van Neck ◽  
Patrick Bultinck ◽  
...  

Molecules ◽  
2021 ◽  
Vol 26 (7) ◽  
pp. 1930
Author(s):  
Andreas Savin

Pauling described metallic bonds using resonance. The maximum probability domains in the Kronig–Penney model can show a picture of it. When the walls are opaque (and the band gap is large) the maximum probability domain for an electron pair essentially corresponds to the region between the walls: the electron pairs are localized within two consecutive walls. However, when the walls become transparent (and the band gaps closes), the maximum probability domain can be moved through the system without a significant loss in probability.


2020 ◽  
pp. 325-346
Author(s):  
Kenric P. Nelson

This chapter introduces a simple, intuitive approach to the assessment of probabilistic inferences. The Shannon information metrics are translated to the probability domain. The translation shows that the negative logarithmic score and the geometric mean are equivalent measures of the accuracy of a probabilistic inference. The geometric mean of forecasted probabilities is thus a measure of forecast accuracy and represents the central tendency of the forecasts. The reciprocal of the geometric mean is referred to as the perplexity and defines the number of independent choices needed to resolve the uncertainty. The assessment method introduced in this chapter is intended to reduce the ‘qualitative’ perplexity relative to the potpourri of scoring rules currently used to evaluate machine learning and other probabilistic algorithms. Utilization of this assessment will provide insight into designing algorithms with reduced the ‘quantitative’ perplexity and thus improved the accuracy of probabilistic forecasts. The translation of information metrics to the probability domain is incorporating the generalized entropy functions developed Rényi and Tsallis. Both generalizations translate to the weighted generalized mean. The generalized mean of probabilistic forecasts forms a spectrum of performance metrics referred to as a Risk Profile. The arithmetic mean is used to measure the decisiveness, while the –2/3 mean is used to measure the robustness.


2020 ◽  
Vol 26 ◽  
pp. 105
Author(s):  
Zhou Fang ◽  
Chuanhou Gao

In this paper we prove the time-domain boundedness for noise-to-state exponentially stable systems, and further make an estimation of its lower bound function, which allows to answer the question that how long the solution of a stochastic noise-to-state exponentially stable system stays in the domain of attraction and what happens with it if it escapes from this region for a while. The results will complement the probability-domain boundedness of noise-to-state exponentially stable systems, and provide a new insight into noise-to-state exponential stability.


Sadhana ◽  
2006 ◽  
Vol 31 (4) ◽  
pp. 325-342
Author(s):  
Deepak Kumar ◽  
T. K. Datta

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