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Maximum Likelihood Estimator of Quantum Probabilities

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Abstract

Classical probability theory is based on assumptions which are often violated in practice. Therefore quantum probability is a proposed alternative not only in quantum physics, but also in other sciences. However, so far it mostly criticizes the classical approach, but does not suggest a working alternative. Maximum likelihood estimators were given very low attention in this context. We show that they can be correctly defined and their computation in closed form is feasible at least in some cases.

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Notes

  1. Only if there is a risk of confusion, we speak of edges rather than operations.

  2. The notion of a cycle in a hypergraph is quite natural, though mathematically complicated. We refer to [3, 4] for details.

  3. We really achieve maximum likelihood for \(p(x)=0\). This does not appear in (6); more exactly, it gives rise to a factor \(p(x)^{n(x)}=0^0=1\) which does not affect the product. The sum of other probabilities is upper bounded by \(1-p(x)\), which becomes the maximal bound, 1, and thus allows their maximal values.

  4. There is one significant difference: without such outcomes, condition (G1) may be violated. However, this is only a technical assumption needed for the Greechie diagram to represent an orthomodular poset. In order to have a hypergraph which properly represents the probabilities, this is not important and our computations work without this condition, too.

  5. Here \(p|A_i\) denotes the restriction of p to \(A_i\) like a conditional probability; it may be understood this way, too.

  6. Here p(.|C) is really an ordinary conditional probability, not a restriction as in Section 5.

  7. Notice that we cannot sum over all \(B_i\) if it has more edges (operations).

References

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Acknowledgements

The first author was supported by the European Regional Development Fund, project “Center for Advanced Applied Science” (No. CZ.02.1.01/0.0/0.0/16_019/0000778). The second author received support from the Czech Science Foundation grant 20-09869L.

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  1. Department of Cybernetics, Faculty of Electrical Engineering, Czech Technical University in Prague, Technická 2, Praha, CZ-166 27, Czech Republic

    Mirko Navara & Jan Ševic

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  1. Mirko Navara
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  2. Jan Ševic
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Both authors contributed equally to this work.

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Correspondence to Mirko Navara.

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Navara, M., Ševic, J. Maximum Likelihood Estimator of Quantum Probabilities. Int J Theor Phys 62, 214 (2023). https://doi.org/10.1007/s10773-023-05469-8

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