# Quantum-Like Sampling

## Abstract

**:**

## 1. Introduction

## 2. Kolmogorovs Probabilities

- Humans can believe in a subjective viewpoint, which can be determined by some empirical psychological experiments. This approach is a very subjective way to determine the numerical degree of belief.
- For a finite sample we can estimate the true fraction. We count the frequency of an event in a sample. We do not know the true value because we cannot access the whole population of events. This approach is called frequentist.
- It appears that the true values can be determined from the true nature of the universe, for example, for a fair coin, the probability of heads is $0.5$. This approach is related to the Platonic world of ideas. However, we can never verify whether a fair coin exists.

#### Frequentist Approach and ${l}_{1}$ Sampling

## 3. Quantum Probabilities

#### Conversion

## 4. Quantum-Like Sampling and the Sigmoid Function

#### Combination

## 5. Naïve Bayes Classifier

#### Titanic Dataset

## 6. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**$f\left(\omega \right)=\frac{\omega}{\Omega}$ compared to $g\left(\omega \right)=\frac{{\omega}^{2}}{{\omega}^{2}+{(\Omega -\omega )}^{2}}$ for $\Omega $, (

**a**) $\Omega =4$, (

**b**) $\Omega =10$, (

**c**) $\Omega =100$ and (

**d**) $\Omega =1000$. With growing size of $\Omega $ the function $g\left(x\right)$ converge to a sigmoid function.

**Figure 2.**Sigmoid functions: $g\left(x\right)=\frac{{x}^{2}}{{x}^{2}+{(10-x)}^{2}}$ versus logistic function $\sigma \left(x\right)=\frac{1}{1+{e}^{-0.9523\xb7(x-5)}}$.

**Figure 3.**(

**a**) $g\left(x\right)=\frac{{x}^{2}}{{x}^{2}+{(10-x)}^{2}}$ and ${g}^{\prime}\left(x\right)=-\frac{5(-10+x)x}{{(50-10x+{x}^{2})}^{2}}$; (

**b**) ${g}^{\prime}\left(x\right)=-\frac{5(-10+x)x}{{(50-10x+{x}^{2})}^{2}}$ versus $\mathcal{N}\left(x\right|5,1.{9}^{2})=\frac{1}{1.9\ast \sqrt{2\ast \pi}}\xb7{e}^{-\frac{1}{2}\xb7{\left(\frac{x-5}{1.9}\right)}^{2}}$.

**Figure 4.**Shannon’s entropy H with $\Omega =100$, blue discrete plot for ${l}_{1}$ sampling and yellow discrete plot ${l}_{2}$ sampling.

**Figure 5.**Multiplying two independent sampled events x and y with $\Omega =10$ leads to different results: (

**a**) ${p}_{l1}(x,y)$, (

**b**) ${p}_{l2}(x,y)$, (

**c**) counterplot of ${p}_{l1}(x,y)$, (

**d**) counterplot of ${p}_{l2}(x,y)$.

**Figure 6.**Different ${H}_{1}$ and ${H}_{2}$ values for sampled events x and y with $\Omega =10$: (

**a**) ${H}_{1}(x,y)$, (

**b**) ${H}_{2}(x,y)$, (

**c**) counterplot of ${H}_{1}(x,y)$, (

**d**) counterplot of ${H}_{2}(x,y)$.

**Figure 7.**The Titanic data set is represented in an Excel table that contains data for 891 of the real Titanic passengers, some entries are not defined.

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Wichert, A. Quantum-Like Sampling. *Mathematics* **2021**, *9*, 2036.
https://doi.org/10.3390/math9172036

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Wichert A. Quantum-Like Sampling. *Mathematics*. 2021; 9(17):2036.
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Wichert, Andreas. 2021. "Quantum-Like Sampling" *Mathematics* 9, no. 17: 2036.
https://doi.org/10.3390/math9172036