Institut des
NanoSciences de Paris

Probability : from Bayesian statistics for data analysis to the foundations of quantum mechanics

The very different subjects of Bayesian data analysis and foundation of quantum mechanics are related each other by a common definition of probability based on logic. The foundations of Bayesian statistics are based on this approach and some assumptions. At the same time, in the framework of Relational Quantum Mechanics, the definition of the probability by logic allows to reduce from three to two the number of postulates necessary for the formulation of quantum mechanics.

Nested_fit : a data analysis program based on Bayesian statistics

Nested_fit is a program based on the Bayesian statistics [1–3]. It provides not only commonly used outputs of standard fitting programs based on the maximization of the likelihood function or the minimization of the chi-square, but it also determines the complete probability distribution for each parameter and conjunct probability of pairs of parameters. More important, it provides the Bayesian evidence, a quantity required to compare different models (i.e. hypotheses, like the presence or not of additional peaks or the choice of the peak shape). In the case of several equiprobable models, Nested_fit outputs can be used to extract the probability distribution of one parameter common to the different models (the position of a main spectral component as e.g.) without having to determine uniquely the spectrum modeling. The evidence calculation is based on the nested algorithm presented in the literature (Sivia and J. Skilling, Data analysis : a Bayesian tutorial, 2006 Oxford University Press), which reduces a n-dimensional integral (the integral of the likelihood function in the n parameters space) to a one-dimensional integral. The Nested_fit code is written in Fortran90 with some Python complementary routines for visualizing the output results and for doing automatic analysis of data. Recently, a “machine learning” algorithm has been implemented for cluster analysis (mean shift) to treat difficult case where several local maxima of the likelihood function are present [3]. It has been implemented for the analysis of spectra of different nature : X-ray emission spectra from heavy highly charged ions and pionic atoms [1,4–6], photoemission spectra from nanoparticles [7,8] and nuclear decays [9].

Figure 1
Left : Profile curves corresponding to the likelihood maxima of the models with different number of peaks. Right : Probability distribution of the main peak position from the single probabilities of the models (figures from Ref. [1]).


[1] M. Trassinelli, Bayesian data analysis tools for atomic physics, Nucl. Instrum. Methods B 408, 301-312 (2017)

[2] M. Trassinelli, The Nested_fit Data Analysis Program, Proceedings 33, 14 (2019)

[3] M. Trassinelli and P. Ciccodicola, Mean Shift Cluster Recognition Method Implementation in the Nested Sampling Algorithm, Entropy 22, 185 (2020)

[4] M. Trassinelli, D.F. Anagnostopoulos, G. Borchert, A. Dax, J.P. Egger, D. Gotta, M. Hennebach, P. Indelicato, Y.W. Liu, B. Manil, N. Nelms, L.M. Simons, and A. Wells, Measurement of the charged pion mass using X-ray spectroscopy of exotic atoms, Phys. Lett. B 759, 583-588 (2016)

[5] M. Trassinelli, D.F. Anagnostopoulos, G. Borchert, A. Dax, J.-P. Egger, D. Gotta, M. Hennebach, P. Indelicato, Y.-W. Liu, B. Manil, N. Nelms, L.M. Simons, and A. Wells, Measurement of the charged pion mass using a low-density target of light atoms, EPJ web conf. 130, 01022 (2016)

[6] J. Machado, G. Bian, N. Paul, M. Trassinelli, P. Amaro, M. Guerra, C.I. Szabo, A. Gumberidze, J.M. Isac, J.P. Santos, J.P. Desclaux and P. Indelicato, Reference-free measurements of the 1s 2s 2p 2PO1/2,3/2 → 1s2 2s 2S1/2 and 1s 2s 2p 4P5/2 → 1s2 2s 2S1/2 transition energies and widths in lithiumlike sulfur and argon ions, accepted for publication in Phys. Rev. A (2020)

[7] I. Papagiannouli, M. Patanen, V. Blanchet, J.D. Bozek, M. de Anda Villa, M. Huttula, E. Kokkonen, E. Lamour, E. Mevel, E. Pelimanni, A. Scalabre, M. Trassinelli, D.M. Bassani, A. Lévy, and J. Gaudin, Depth Profiling of the Chemical Composition of Free-Standing Carbon Dots Using X-ray Photoelectron Spectroscopy, The Journal of Physical Chemistry C 122, 14889-14897 (2018)

[8] M. De Anda Villa, J. Gaudin, D. Amans, F. Boudjada, J. Bozek, R. Evaristo Grisenti, E. Lamour, G. Laurens, S. Macé, C. Nicolas, I. Papagiannouli, M. Patanen, C. Prigent, E. Robert, S. Steydli, M. Trassinelli, D. Vernhet, and A. Lévy, Assessing the Surface Oxidation State of Free-Standing Gold Nanoparticles Produced by Laser Ablation, Langmuir 35, 11859-11871 (2019)

[9] F.C. Ozturk, B. Akkus, D. Atanasov et al., New test of modulated electron capture decay of hydrogen-like 142Pm ions : Precision measurement of purely exponential decay, Phys. Lett. B 797, 134800 (2019)

Born’s rule (and Quantum Mechanics formalism) from two postulates

Relational Quantum Mechanics (RQM) is an approach for the foundation of Quantum Mechanics with only three postulates. Initially formulated by Rovelli in 1996, RQM is based on the limited amount of information that can be extracted from interaction of different systems, with a third postulate to define the properties of the probability function. We demonstrate that from a rigorous definition of the conditional probability for the possible outcomes of different measurements, the third postulate is unnecessary and the Born’s rule naturally emerges from the first two postulates by applying the Gleason’s theorem. We demonstrate in addition that the probability function is uniquely defined for classical and quantum phenomena. The presence or not of interference terms is, in fact, related to the correct formulation of the conditional probability where distributive property on its arguments cannot be taken for granted.



[1] M. Trassinelli, Relational Quantum Mechanics and Probability, Found. Phys. 48, 1092-1111 (2018), DOI:10.1007/s10701-018-0207-7, arXiv:1803.02644 [quant-ph]