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conditional_probability [2016/12/27 14:28] nikolaj |
conditional_probability [2016/12/27 14:29] nikolaj |
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| Consider a pair of functions $S_L, S_R : (A\to {\mathbb R})\to A\to {\mathbb R}$, then | Consider a pair of functions $S_L, S_R : (A\to {\mathbb R})\to A\to {\mathbb R}$, then | ||
| - | $\dfrac{f}{S_Lf} = \dfrac{f}{S_Rf}\dfrac{S_Rf}{S_Lf} $ | + | $\dfrac{f}{S_Lf} = \dfrac{f}{S_Rf}\dfrac{S_Rf}{S_Lf} = \dfrac{f}{S_Rf}\dfrac{\frac{1}{S_LS_Rf}S_Rf}{\frac{1}{S_LS_Rf}S_Lf}$ |
| - | If the pair of functions commute, | + | If the pair of functions commute, we can write |
| - | $\dfrac{f}{S_Lf} = \dfrac{f}{S_Rf}\dfrac{\frac{1}{S_LS_Rf}S_Rf}{\frac{1}{S_LS_Rf}S_Lf} = \dfrac{f}{S_Rf}\dfrac{\frac{1}{S_L(S_Rf)}S_Rf}{\frac{1}{S_R(S_Lf)}S_Lf}$ | + | $\dfrac{f}{S_Lf} = \dfrac{f}{S_Rf}\dfrac{\frac{S_Rf}{S_L(S_Rf)}S_Rf}{\frac{S_Lf}{S_R(S_Lf)}}$ |
| ** Bayes rule: ** | ** Bayes rule: ** | ||
