Feat: Cours sur les tests bayésiens
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\documentclass[a4paper,10pt]{article}
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\usepackage{myXsim}
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\author{Benjamin Bertrand}
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\title{Probabilités conditionnelles - Cours}
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\date{Mars 2021}
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\pagestyle{empty}
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\begin{document}
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\maketitle
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\setcounter{section}{1}
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\section{Tests Bayésiens}
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Un test bayésien permet d'affiner la vraisemblablité d'hypothèses. La vraisemblablité sont modélisées par des probabilités.
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On part d'un \textbf{a priori} (notre évaluation de la vraisemblablité des hypothèses avant le test). Puis nous faisons le test ce qui permet d'ajuster la vraisemblablité des hypothèses. Nous obtenons un \textbf{a posteriori}.
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\begin{center}
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\includegraphics[scale=1]{./fig/test_baysien}
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\end{center}
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\subsection*{Test médical}
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Étudions l'intérêt d'un test médical. Pour faire simple, on considèrera que l'on est soit \textbf{malade} soit pas malade et que le test donne deux résultats possibles \textbf{positif} ou négatif. On notera alors
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\[
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A = \left\{ \mbox{Malade} \right\} \qquad \qquad B = \left\{ \mbox{Test positif} \right\}
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\]
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\paragraph{Paramètres du test:}
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\begin{itemize}
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\item \textbf{Sensibilité}: la probabilité que le test soit positif sachant que l'on est malade
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\[
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P_A(B) = 0.9
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\]
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\item \textbf{Spécificité}: la probabilité que le test soit négatif sachant que l'on est pas malade
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\[
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P_{\overline{A}}(\overline{B}) = 0.99
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\]
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\end{itemize}
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\paragraph{A priori:} on estime que 1\% de la population est malade. On appelle cela la \textbf{la prévalence} d'un maladie. On peut noter
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\[
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P(A) = 1\% = 0.01
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\]
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J'ai donc une chance sur 100 d'avoir cette maladie.
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\paragraph{Mise en situation:} On fait un test qui est positif. Comment réévaluer la probabilité d'être malade? C'est à dire connaître
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\[
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P_B(A) = ?
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\]
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Imaginons une population de 1000 individus. En respectant les proportions, on peut construire le tableau des effectifs:
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\begin{center}
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\begin{tabular}{|*{4}{p{3cm}|}}
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\hline
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& Test positif ($B$) & Test négatif ($\overline{B}$) & Total \\
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\hline
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Malade ($A$) & & & \\
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\hline
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Pas malade ($\overline{A}$) & & & \\
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\hline
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Total & & & 1000 \\
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\hline
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\end{tabular}
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\end{center}
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\afaire{Compléter le tableau et calculer la probabilité cherchée}
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\end{document}
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@ -2,7 +2,7 @@ Inférence Bayésienne
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####################
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:date: 2021-03-15
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:modified: 2021-03-15
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:modified: 2021-03-16
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:authors: Benjamin Bertrand
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:tags: Probabilité, Bayes
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:category: Complementaire
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@ -36,7 +36,11 @@ En groupe, justification des valeurs avancées dans le document. Explications po
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Conclusion sur l'utilité d'un test comme outils pour affiner un a priori.
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Bilan: Reprise d'un exemple traité et vocabulaire associé aux tests
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Bilan: Reprise d'un exemple traité et vocabulaire associé aux tests. On reprendra le schéma des tests pour représenter la mise à jour de la vraisemblance.
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.. image:: ./2B_vocabulaire.pdf
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:height: 200px
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:alt: Vocabulaire autour des tests
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Étape 3: Application aux tests ADN
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==================================
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Reference in New Issue