Abstract
Probability theory is built around Kolmogorov’s axioms. To each event, a numerical degree of belief between 0 and 1 is assigned, which provides a way of summarizing the uncertainty. Kolmogorov’s probabilities of events are added, the sum of all possible events is one. The numerical degrees of belief can be estimated from a sample by its true fraction. The frequency of an event in a sample is counted and normalized resulting in a linear relation. We introduce quantum-like sampling. The resulting Kolmogorov’s probabilities are in a sigmoid relation. The sigmoid relation offers a better importability since it induces the bell-shaped distribution, it leads also to less uncertainty when computing the Shannon’s entropy. Additionally, we conducted 100 empirical experiments by quantum-like sampling 100 times a random training sets and validation sets out of the Titanic data set using the Naïve Bayes classifier. In the mean the accuracy increased from 78.84% to 79.46%.
Highlights
The l2 sampling leads to an interpretation of the amplitudes as a normalized frequency of occurrence of an event multiplied by the phase
When developing empirical experiments that are explained by quantum cognition models l2 sampling should be used rather than l1 sampling
The l2 sampling leads to an interpretation of the amplitudes as a normalized frequency of occurrence of an event multiplied by the phase card( x )
Summary
The incapability between Kolmogorov’s probabilities and quantum probabilities results from the different norms that are used. In quantum probabilities the length of the vector in l2 norm representing the amplitudes of all events is one. Kolmogorov’s probabilities are converted in Quantum-like probabilities by the squared root operation, it is difficult to attach any meaning to the squared root operation. Motivated by the lack of interpretability we define quantum-like sampling, the l2 sampling. The resulting Kolmogorov’s probabilities are not any more linear but related to the sigmoid function. We will indicate the relation between the l2 sampling, the sigmoid function and the normal distribution. The quantum-like sampling leads to less uncertainty. We fortify this hypothesis by empirical experiments with a simple Naïve Bayes classifier on the Titanic dataset
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