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La vraisemblablité sont modélisées par des probabilités. + +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}. + +\begin{center} + \includegraphics[scale=1]{./fig/test_baysien} +\end{center} + +\subsection*{Test médical} + +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 +\[ + A = \left\{ \mbox{Malade} \right\} \qquad \qquad B = \left\{ \mbox{Test positif} \right\} +\] +\paragraph{Paramètres du test:} +\begin{itemize} + \item \textbf{Sensibilité}: la probabilité que le test soit positif sachant que l'on est malade + \[ + P_A(B) = 0.9 + \] + \item \textbf{Spécificité}: la probabilité que le test soit négatif sachant que l'on est pas malade + \[ + P_{\overline{A}}(\overline{B}) = 0.99 + \] +\end{itemize} +\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 +\[ + P(A) = 1\% = 0.01 +\] +J'ai donc une chance sur 100 d'avoir cette maladie. +\paragraph{Mise en situation:} On fait un test qui est positif. Comment réévaluer la probabilité d'être malade? C'est à dire connaître +\[ + P_B(A) = ? +\] + +Imaginons une population de 1000 individus. En respectant les proportions, on peut construire le tableau des effectifs: + +\begin{center} + \begin{tabular}{|*{4}{p{3cm}|}} + \hline + & Test positif ($B$) & Test négatif ($\overline{B}$) & Total \\ + \hline + Malade ($A$) & & & \\ + \hline + Pas malade ($\overline{A}$) & & & \\ + \hline + Total & & & 1000 \\ + \hline + \end{tabular} +\end{center} + +\afaire{Compléter le tableau et calculer la probabilité cherchée} + + + + +\end{document} diff --git a/Complementaire/02_Inference_Baysienne/fig/test_baysien.png b/Complementaire/02_Inference_Baysienne/fig/test_baysien.png new file mode 100644 index 0000000000000000000000000000000000000000..7bca6116dd0e74bf9f2e913b7eeef1bc35866834 GIT binary patch literal 6737 zcmY+p2T&8;8}^+-kuE*-4nex~CRMr!f=CmQfP~INSBeB_fzTA`B_V zoEKc8p|rkG&K^of7i{Jf3IH(o|2u#~LDim=L7uSdj$w8|55gkvhj;-ZBO~Q~0{uc^ z_k+FUgF?KEf9mj3VwTK}4eX;zwo4xtaD4uZf4n8!4NXJeXLnsmNIlISa*0u z9bd3)FsLEIKl%=tgB(9_aSGT3woqRHt-x&n!PM;*nq9_S;a$#M(FV;16?-x(xhVK| zBd+I50Ua(Y-VcFAIN{#o^xZjY_at{C^0vpu)bxk0x_*Cy%lGQYG^gbQxa5D)QULEf z|Dt6ck$xV_!1i)GyWt(HOt;`VP={J4Gi;R;t-I^#$OWEA{>lwh2S)<12zgw9z8=}$ zox_aKL&FHD62B6mp(voCUdIB=1O)1Hk#^#15e_&;_vYQhVzWz9pn1?9LI@Q7{Rmh_ z8#rP+UE-Rsv9NF^=xCF?SrDdJ^!w5rcHPT4Pz`k+3qL9;BzXf)1Wtf5h_&HMTD(pH zdkE$Jp7WgiD5k4vKSAFBLm~#!s=$4wZvhZ7VW5Wy|!RcwUu{Qd>lh_dGti^zi80i)`RukMZ&)g~VCmA}Qv6eq|ak(bJZmP}WGpxTVUB+<6sKIvx zR%{6ZEx16|Z(yDOIg0 zbCS|fDUg{g!Q0cjsm_cXU%q6d=k4G!$L}K zgrB>bFD7-JK8A%PQ`gJLPxlAt=1aVzj(_p|UA!FPCk_ey87|=KU&46Cnp8DzWDu7f z8R_Q8l@jV-tg&-@c(m4NF68oL^~2XWStrcFG;Gxq{RZiXyYZY~aHSuP+TLISd1G@b z1PeOIRX9!0E}4?#lhRmb*SS=o0mw7AYokQw$^Y{_hUsZUXa>7RIh*-ij=BUeHUYR zGrDQK-aA?ke=g_xr;HV+1${k1X!t|DWT-(c^Z_Q}DwlTziawMRXq}^|-Sk6Der8tcU8SQOEyu4VIxNYtm1A|#LH?R~j#yraW zZEm(}l2>bZ<2_fAQtMa^G}>Rj3X?Y;b862-@xZrOLSpXam4;Ctl#kNClyWrxL=L(e zDC1io|Fr%8u9-EXOQ{y(wP<)@R=(7)#eyM- zDs;|~?xy5G^5dr5dq}EnN%z}4n_gC`R|PsMK{JbWpqVoTpES&N#x3B@5!-9Xk6RP? z0$%YO32KliW>t{qS0^&lm9u0;ynUuhXNg`AI)-O=F~t6b2@kWsJuYqnu%rIV*$>_D zzDcrdhzRjvQ?YaEu|8+zDYbPC6>PZLcD}4~mk^iXT>E%(V6T9;Kj8uWT2+HFKUE13 zMY~jEmVq*-xL^5g6BmbjguAtxU~RnUE83<%z0?IK4$;CZ<|E_9Y##T?<38N~>w{zW zs_r(|^zr3T{5%2A1$qULIde70L)Fi`2h?yu*`r!KQ4DmXHm}z;J~<>q<;n`~5CbWj z-g0s+E(v!IPn}b@5vti^8)~O7k4%oeMiV!K+E;kW{-(qZ`{b}LFXw#MioeUP}F-!u)X-2eLc_9 z=g2jY=Ek8B9d!g?9H>TB?{nc>l!u}o%S~&yy#x~=4}uQL+4D~0_iPtr*;+W^1;t7k zeM*${l3-#pglbG`oSa?5zTzPN56cVYm8ZCv+FL^JGr7kK}6w{qRh7hf#>W(k|hRiHE`H zO};=}+@Pv*N2EJiND|6d1%B(r?<7@B-xWsgkqH9rVaMLS$`Rc~1I; 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Explications po Conclusion sur l'utilité d'un test comme outils pour affiner un a priori. -Bilan: Reprise d'un exemple traité et vocabulaire associé aux tests +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. + +.. image:: ./2B_vocabulaire.pdf + :height: 200px + :alt: Vocabulaire autour des tests Étape 3: Application aux tests ADN ==================================