Baye’s rule

Baye’s rule

Theorem:

Let $\mu$ and $\nu$ be two probability measures on a measurable space $(\mathrm{\Omega },𝒢)$ such that

$$d\nu (\omega )=f(\omega )d\mu (\omega )$$

for some $f\in L^1(\mu )$.Let $X$ be a random variable on $(\mathrm{\Omega },𝒢)$ such that

($\nu$-integrable)

Let $\mathscr{H}$ be a $\sigma$-algebra, $\mathscr{H}\subset 𝒢$. Then,

$$E_{\nu }[X|\mathscr{H}].E_{\mu }[f|\mathscr{H}]=E_{\mu }[fX|\mathscr{H}]\mathit{\hspace{1em}}a.s.$$

or

$E_{\nu }[X|\mathscr{H}]={\displaystyle \frac{E_{\mu }[fX|\mathscr{H}]}{E_{\mu }[f|\mathscr{H}]}}\mathit{\hspace{1em}}a.s.$