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. La-dernière-Égalité-est-appelée-formule-de-bayes, De cette définition, il découle les premiers moments conditionnels, que sont l'espérance et la variance conditionnelles définies par

E. Pour-toute-sous-tribu-a-?-f,-l-'espérance-conditionnelle, suit les propriétés suivantes : i) si X est intégrable, E[X|A] l'est aussi De même, si X ? 0, alors E[X|A] ? 0 ii) E[·|A] est linéaire, c'est à dire que ?(a, b) ? R, X et Y deux variables aléatoires réelles : E[aX + bY |A] = aE[X|A] + bE[Y |A] iii) orthogonalité : Y ? E[X|Y ] est orthogonal à X iv)

. La-variance-conditionnelle,-quant-À-elle, suit les propriétés suivantes : vi) Var, X|Y) 2 |Y ] vii) Var[X|Y ] = E[X 2 |Y ] ? (E[X|Y ]) 2

. Dans-cette-thèse, ces relations sont toutes utilisées dans les développements des algorithmes utilisant le filtre de Kalman. Il est possible d'en trouver des démonstrations dans des ouvrages de tels que [Dalang and Conus, 2008.