What is the role of probability theory in data analysis?

La théorie moderne des probabilités a été créée notamment par Kolmogorov. Elle généralise la notion de probabilité discrètes (ex: quelle probabilité de tirer 5 fois de suite pile dans une série de 100 pile ou face) à des variables plus complexe (continues, listes de valeurs, objectifs mathématiques composites) en introduisant la notion de "mesure" sur des espaces de données. Ces mesures permettent d'identifier les points de l'espace des données ou il y a le plus de chance de rencontrer des occurrences des données qu'on modélise.

As an element of applied mathematics, probability theory provides a coherent framework for analysts to decompose a problem space across various applications. Probability theory is an essential tool for any quantitative analyst as the foundation of statistical inference (i.e. deductive reasoning), particularly when designing more comprehensive analyses and exploring complex datasets. It is crucial to understand the spectrum of techniques and approaches, the advantages and limitations of these, as well as the implications and meaning of outputs. An experienced analyst will have a strong command of probability theory to design and implement a range of analyses with well-structured design choices to communicate clearly a range of audiences.

Probability theory helps data scientists to quantify uncertainty, model complex phenomena, and design effective algorithms. Some examples of how probability theory is used in data science are: 1.Probability distributions, such as binomial, normal, Poisson, etc., are used to describe the behavior of random variables and the likelihood of different outcomes, 2.Conditional probability and Bayes’ theorem are used to update beliefs based on new evidence and to perform Bayesian inference. 3.Hypothesis testing and confidence intervals are used to make statistical decisions and to measure the reliability of estimates. 4.Markov chains and Monte Carlomethods are used to model stochastic processes and to simulate complex systems.

I would just place more emphasis on 'Handling Uncertainty.' The truth is that in the real world, data often comes with inherent uncertainty. Probability theory provides the mathematical framework to quantify and manage this uncertainty

La théorie des probabilité est utilisée pour valider scientifiquement la plupart des méthodes d'analyse et de modélisation des données. Le Machine Learning repose largement dessus. Les notions de probabilités conjointes ou de probabilités conditionnelles permettent de modéliser les systèmes dynamiques en tenant compte des erreurs de modèles physiques et des erreurs de mesures lors de l'observation du système.

Data is not noisy as much recently found out that population workers of a hospital had null values assigned according to data cleaning methods one did not account for things legitimate such as people hired at a certain date and not fired yet had null values for end date. I think the datasets were correct this whole time.

I had an equation to predict the trigonometric outcome of a logistic regression equation, probability is used for data analysis when it comes odds ratio in logistics regression.

Probability theory supports descriptive statistics, probability distributions, hypothesis testing, confidence intervals, Bayesian inference, machine learning algorithms, risk evaluation, simulation, and Monte Carlo techniques. They provide light on central tendency and dispersion, as well as help construct confidence intervals to predict the range of values that a population parameter is likely to take. Bayesian inference updates assumptions about parameters or hypotheses based on prior knowledge and observed data, which makes Bayesian approaches useful for decision making and forecasting. Overall, probability theory provides a powerful framework for data analysis, forecasting, and informed decision-making in a variety of disciplines.

Monte Carlo statistical methods are widely used in data analysis and they rely massively on probability theory. Monte Carlo Markov chains, genetic algorithms are typical algorithms whose outcome is probabilistic. Probability theory is also used to demonstrate their convergence and the accuracy of the results they produce.

Il y a de nombreuses limitations, dont les deux plus importantes, à mon avis : 1/ les aspects computationnels qui peuvent être très intensifs et nécessiter de très fortes puissances de calcul, avec des algorithmes pas toujours parallélisables 2/ L'interprétation par l'homme : les concepts et résultats probabilistes ne sont souvent contre-intuitifs, il faut une solide formation pour bien les interpréter.