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==Epistemology==

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| align="right" width="250" |<small>''The black swan symbolizes one of the historical problems of epistemology: if all the swans we have seen so far are white, can we decide that all swans are white?''</small>
| align="center" |[[File:Black_Swan_(Cygnus_atratus)_RWD.jpg|175px|center]]
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| align="center" |[[File:Duck-Rabbit illusion.jpg|203px|center]]
| width="250" |<small>''Kuhn used optical illusions to demonstrate how a paradigm shift can make a person perceive the same information in a completely different way.''</small>
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'''Epistemology''' (from the Greek ἐπιστήμη, epistēmē, “certain knowledge” or “science”, and λόγος, logos, “discourse”) is the branch of philosophy that studies the necessary conditions for acquiring scientific knowledge and the methods through which it is achieved.{{Tooltip|<sup>[5]</sup>|<ref>The term was coined by the Scottish philosopher [[Wpen:James Frederick Ferrier|James Frederick Ferrier]], in his ''Institutes of Metaphysic'' (1854); see Internet Encyclopedia of Philosophy, ''[https://www.iep.utm.edu/ferrier/ James Frederick Ferrier (1808—1864)]''</ref>|<small>The term was coined by the Scottish philosopher [[Wpen:James Frederick Ferrier|James Frederick Ferrier]], in his ''Institutes of Metaphysic'' (1854); see Internet Encyclopedia of Philosophy, ''[https://www.iep.utm.edu/ferrier/ James Frederick Ferrier (1808—1864)]</small>|}}

In particular, epistemology analyzes the foundations, validity, and limits of scientific knowledge. In English-speaking countries, the term "epistemology" is often used as a synonym for the theory of knowledge or gnoseology.

The central problem of epistemology, today as in the time of Hume,{{Tooltip|<sup>[6]</sup>|<ref>[[wikipedia:David_Hume|David Hume]] (1711–1776) was a Scottish philosopher.</ref>|<small>📌 David Hume, son of lawyer Joseph Home of Chirnside and Katherine Falconer, daughter of the president of the College of Justice, was born the third child in a mansion on the north side of the Lawnmarket in Edinburgh. Although of noble origins, his family was not very wealthy, and he was entrusted with a small portion of their estate. He changed his surname from Home to Hume in 1734 to better maintain the Scottish pronunciation even in England.</small>}}{{Tooltip|<sup>[7]</sup>|<ref>{{cita libro
| autore = Srivastava S
| titolo = Verifiability is a core principle of science
| url = https://www.cambridge.org/core/journals/behavioral-and-brain-sciences/article/verifiability-is-a-core-principle-of-science/D46462A598492AFDB7AFB4975A313446#
| opera = Behav Brain Sci
| anno = 2018
| editore = Cambridge University Press
| DOI = 10.1017/S0140525X18000869
}}</ref>|<small>📌 Scientific knowledge should be verifiable. Replications promote verifiability in several ways. Most directly, replications can confirm empirical claims. Replication research also promotes the dissemination of information necessary for other aspects of verification; creates meta-scientific knowledge about which results to consider credible even in the absence of replications; and reinforces a broader norm that scientists should check each other's work.</small>}} is the issue of verifiability.

According to Hempel's paradox, every example that does not contradict a theory confirms it. This is expressed in propositional logic as:

<math>A \Rightarrow B = \lnot A \lor B</math> {{Tooltip||2=Consider the following statement: ✅ “If a person has TMD, then they experience orofacial pain.” We can represent this in logic as <math>A \Rightarrow B = \lnot A \lor B</math>, where: 🎯<math>A</math> represents "The person has TMD." 🎯 <math>B</math> represents "The person experiences orofacial pain." In this case, "If a person has TMD, then they experience orofacial pain" is equivalent to saying “either the person does not have TMD (<math>\lnot A</math>), or they experience orofacial pain (<math>B</math>).” 🧠 The formula is true in the following cases: If the person does not have TMD (<math>\lnot A</math>), the statement is true regardless of orofacial pain. If the person has TMD (<math>A</math>) and experiences orofacial pain (<math>B</math>), the statement is true. The statement is false only if the person has TMD (<math>A</math>) but does not experience orofacial pain (<math>\lnot B</math>), contradicting the implication condition.}}

But no theory can be definitively confirmed: an infinite number of future experiments could always refute it.{{Tooltip|<sup>[8]</sup>|<ref>{{cita libro
| autore = Evans M
| titolo = Measuring statistical evidence using relative belief
| url = https://www.ncbi.nlm.nih.gov/pubmed/26925207
| opera = Comput Struct Biotechnol J
| anno = 2016
| DOI = 10.1016/j.csbj.2015.12.001
}}</ref>|<small>📌 A fundamental issue in the theory of statistical inference concerns how one should measure statistical evidence. Certainly, terms like “statistical evidence” or simply “evidence” are widely used in statistical contexts. However, it is fair to say that a precise characterization of this concept remains somewhat elusive. Our goal here is to provide a definition of how to measure statistical evidence in relation to a specific statistical problem. Since evidence is what causes belief change, it is proposed to measure evidence based on the extent of belief change, from the prior to the posterior moment. 🧠 Consequently, our definition implies the existence of pre-existing beliefs, which raises questions about subjectivity and objectivity in statistical analyses. This aspect is addressed through a principle that requires the falsifiability of every element involved in the statistical analysis. These considerations lead to the need to verify any conflicts between prior beliefs and observed data, and to measure the prior bias present in an initial distribution</small>|}} 

{{qnq|But it's not all that obvious...}}

=== '''P-value''' ===
In medicine, we often rely on statistical inference to validate experimental results. One of the most well-known tools is the 'P-value', or probability value, an indicator used in significance testing.{{Tooltip||2=The P-value represents the probability that the observed results are due to chance, assuming the null hypothesis <math> H_0 </math> is true. It should not be used as a binary criterion (e.g., <math> p < 0.05 </math>) for scientific decisions, as values close to the threshold require additional verification, such as cross-validation. ''P-hacking'' (repeating tests to achieve significance) increases false positives. Rigorous experimental designs and transparency about all tests conducted can mitigate this risk. Type I error increases with multiple tests: for <math> N </math> independent tests at threshold <math> \alpha </math>, the Family-Wise Error Rate (FWER) is <math> FWER = 1 - (1 - \alpha)^N </math>. The Bonferroni correction divides the threshold by <math>N</math>, <math>p < \frac{\alpha}{N}</math>, but can increase false negatives. The Benjamini-Hochberg False Discovery Rate (FDR) allows more discoveries with an acceptable proportion of false positives. The Bayesian approach uses prior knowledge to balance prior and data with a posterior distribution, offering a valid alternative to the P-value. To combine P-values from multiple studies, meta-analysis uses methods like Fisher's: <math> \chi^2 = -2 \sum \ln(p_i) </math>. 🧠 In summary, the P-value remains useful if contextualized and integrated with other measures, such as confidence intervals and Bayesian approaches.}}

However, even the P-value, for years a fundamental criterion in evidence-based medicine, is now undergoing profound revision. In 2019, a campaign published in "Nature", signed by over 800 scientists, questioned the rigid use of statistical significance.{{Tooltip|<sup>[9]</sup>|<ref>{{cita libro
| autore = Amrhein V
| autore2 = Greenland S
| autore3 = McShane B
| titolo = Scientists rise up against statistical significance
| url = https://www.ncbi.nlm.nih.gov/pubmed/30894741
| opera = Nature
| anno = 2019
| DOI = 10.1038/d41586-019-00857-9
}}</ref>|<small>📌 In the March edition of Nature, over 800 scientists signed a commentary calling for the retirement of the term “statistical significance” [1]. The main arguments of the authors concern the fact that the scientific literature is full of erroneous and potentially harmful interpretations of associations based on an arbitrary and binary classification, founded on a p-value of 0.05. The authors illustrate the critical issues of this approach, providing concrete examples where it has led to erroneous conclusions within and between different studies. 🧠 Additionally, analyzing 791 articles published in five academic journals, they found that 51% of them misinterpreted a statistically non-significant result as an indication of the absence of an effect.</small>}} This "silent revolution" in the field of statistical inference promotes a more reflective, contextual, and scientifically honest approach. Among the most authoritative voices in this debate are:

* Rodgers JL – who speaks of a “silent methodological revolution”{{Tooltip|<sup>[10]</sup>|<ref>{{cita libro|autore=Rodgers JL|titolo=The epistemology of mathematical and statistical modeling: a quiet methodological revolution|url=https://www.ncbi.nlm.nih.gov/pubmed/20063905|opera=Am Psychol|anno=2010|DOI=10.1037/a0018326}}</ref>|<small>📌 In recent decades, a silent methodological revolution has occurred almost without discussion: a revolution in modeling. In contrast, the 20th century ended with lively debates about the utility of null hypothesis significance testing (NHST). However, this controversy may have been at least partly irrelevant, as the modeling revolution has rendered the NHST debate superfluous in various ways. I begin by presenting a history of NHST and modeling, and the relationships between the two. Next, I define and illustrate the principles guiding the development and evaluation of mathematical models. This is followed by a discussion on the difference between using statistical procedures in a rule-based framework and constructing mathematical models within a scientific epistemology. 🧠 In postgraduate psychology education, almost exclusive attention is given to the first, rule-based approach. The pedagogical implications of this imbalance and the need for revised teaching to account for the modeling revolution are then described. Finally, the discussion turns to how focusing on modeling leads to an evolution of statistical practice in more progressive directions. The epistemological basis of statistics has shifted: from a set of mechanically applied procedures to the construction and evaluation of statistical and scientific models.</small>|}}

* Meehl P – who suggests replacing significance tests with 'confidence intervals' and 'verifiable numerical predictions'{{Tooltip|<sup>[11]</sup>|<ref>{{cita libro
| autore = Meehl P
| titolo = The problem is epistemology, not statistics: replace significance tests by confidence intervals and quantify accuracy of risky numerical predictions
| anno = 1997}}</ref>|<small>📌 Significance tests have a role in social science research, but their widespread use in theory evaluation is often harmful. The cause of this does not lie in the mathematics itself, but in the poor understanding, by social scientists, of the logical relationship between theory and facts, i.e., a lack of methodological or epistemological clarity.🧭 Theories imply observations, but the reverse is not true. Although a theory's success in deriving a fact tends to corroborate it, this confirmation is weak unless the fact has a very low a priori probability and there are few plausible alternative theories. 🧭 Detecting a non-zero difference or correlation — as occurs when rejecting the null hypothesis — generally does not have a very low a priori probability, because in social sciences practically everything is correlated with everything else, regardless of the theory. 🎯 In the "strong" use of significance tests, the theory predicts a precise numerical value, or a very narrow range, so the test poses a serious risk of falsification if the theory were objectively incorrect. In general, it is preferable to construct a confidence interval, which provides richer information and still implies the null hypothesis's refutation if a difference falls outside the interval. 🧠 Significance tests are usually more justifiable in technological contexts (e.g., evaluating an intervention) rather than in theory evaluation. It would be useful to have a quantitative index measuring how accurately a theory predicts a risky fact, and an example of such an index is proposed. Unlike current widespread practices, textbooks and statistics courses should clarify and emphasize the significant semantic (logical) gap separating a substantive (causal, compositional) theory from a statistical hypothesis.</small>|}}

* Sprenger & Hartmann – proponents of the 'Bayesian philosophy of science'{{Tooltip|<sup>[12]</sup>|<ref>{{cita libro
| autore = Sprenger J
| autore2 = Hartmann S
| titolo = Bayesian Philosophy of Science. Variations on a Theme by the Reverend Thomas Bayes
| anno = 2019
| editore = Oxford University Press
}}</ref>|<small>📌 How should we reason in science? Jan Sprenger and Stephan Hartmann offer an innovative view on classic themes in the philosophy of science, using a single key concept to explain and clarify numerous aspects of scientific reasoning. 🧭 They propose that good arguments and good inferences are characterized by their effect on our rational degrees of belief. 🧠 Contrary to the view that there is no room for subjective attitudes in "objective science," Sprenger and Hartmann explain the value of compelling evidence through a cycle of variations on the theme of representing rational degrees of belief through subjective probabilities (and their modification through Bayesian conditioning). In this way, they integrate Bayesian inference — the main theory of rationality in the social sciences — with the scientific practice of the 21st century. Bayesian Philosophy of Science thus shows how modeling such attitudes improves our understanding of causes, explanations, confirmatory evidence, and scientific models in general. Their approach combines a scientifically oriented and mathematically refined perspective with conceptual analysis and a particular focus on the methodological problems of modern science, especially in statistical inference, making it a valuable resource for both philosophers and practitioners of science.</small>}}

The 'American Statistical Association' has supported this change by publishing a special issue of the journal 'The American Statistician', titled “Statistical Inference in the 21st Century: A World Beyond p < 0.05”.{{Tooltip|<sup>[13]</sup>|<ref name="wasser">{{cita libro
| autore = Wasserstein RL
| autore2 = Schirm AL
| autore3 = Lazar NA
| titolo = Moving to a World Beyond ''p'' < 0.05
| url = https://www.tandfonline.com/doi/full/10.1080/00031305.2019.1583913
| opera = Am Stat
| anno = 2019
| DOI = 10.1080/00031305.2019.1583913
}}</ref>|<small>🧠 Some of you, exploring this special issue of The American Statistician, might wonder if it is a lecture from pedantic statisticians intent on moralizing about what not to do with p-values, without offering real solutions to the difficult problem of separating signal from noise in data and making decisions under uncertainty. Fear not. In this issue, thanks to 43 innovative and stimulating articles written by forward-thinking statisticians, the help we need arrives.</small>|}} The volume proposes new ways of representing uncertainty and invites us to move beyond the dependence on the P-value as the sole metric of scientific truth.