Benutzer:Alexander Brock

${\sqrt {140^{2}-80^{2}}}\cong 115$

$e^{\ln {x}}=x,\forall x\in \mathbb {R} ^{+}$

$\ln({a\cdot b})=\ln a+\ln b$

$\ln {a \over b}=\ln a-\ln b$

$P(w_{i}|C)={N_{C,i} \over N_{C}}$

$P(A|B)={{P(B|A)\cdot P(A)} \over {P(B)}}$

$P(w_{1},w_{2},...,w_{n}|C)=P(w_{1}|C)\cdot P(w_{2}|C)\cdot ...\cdot P(w_{n}|C)={\prod _{k=1}^{n}{N_{C,k} \over N_{C}}}$

$P(w_{1},w_{2},...,w_{n}|C)=P(w_{1}|C)\cdot P(w_{2}|C)\cdot ...\cdot P(w_{n}|C)={\prod _{k=1}^{n}{N_{C,k} \over N_{C}}}$

$P(Spam|w_{1},w_{2},...,w_{n})={\frac {{\prod _{k=1}^{n}{{\color {Red}N_{{\text{Spam}},k}} \over {\color {Brown}N_{\text{Spam}}}}}\cdot P(Spam)}{P(w_{1},w_{2},...,w_{n})}}$

$P(Ham|w_{1},w_{2},...,w_{n})={\frac {{\prod _{k=1}^{n}{{\color {OliveGreen}N_{{\text{Ham}},k}} \over {\color {Blue}N_{\text{Ham}}}}}\cdot P(Ham)}{P(w_{1},w_{2},...,w_{n})}}$

$V={\frac {{\prod _{k=1}^{n}{{\color {Red}N_{{\text{Spam}},k}} \over {\color {Brown}N_{\text{Spam}}}}}\cdot P(Spam)}{P(w_{1},w_{2},...,w_{n})}}\cdot {\frac {{\prod _{k=1}^{n}{{\color {Blue}N_{\text{Ham}}} \over {\color {OliveGreen}N_{{\text{Ham}},k}}}}\cdot P(w_{1},w_{2},...,w_{n})}{P(Ham)}}$

$V={\frac {P(Spam)}{P(Ham)}}\cdot {\prod _{k=1}^{n}{{\color {Red}N_{{\text{Spam}},k}} \over {\color {Brown}N_{\text{Spam}}}}}\cdot {\prod _{k=1}^{n}{{\color {Blue}N_{\text{Ham}}} \over {\color {OliveGreen}N_{{\text{Ham}},k}}}}$

$V={{\color {Brown}N_{\text{Spam}}}\prod _{k=1}^{n}{{\color {Red}N_{{\text{Spam}},k}} \over {\color {Brown}N_{\text{Spam}}}} \over {{\color {Blue}N_{\text{Ham}}}\prod _{k=1}^{n}{{\color {OliveGreen}N_{{\text{Ham}},k}} \over {\color {Blue}N_{\text{Ham}}}}}}$

${\prod _{k=1}^{n}{{\color {Red}N_{{\text{Spam}},k}} \over {\color {Brown}N_{\text{Spam}}}} \over {\prod _{k=1}^{n}{{\color {OliveGreen}N_{{\text{Ham}},k}} \over {\color {Blue}N_{\text{Ham}}}}}}$

$V={{{\color {Brown}N_{\text{Spam}}} \over {\color {Blue}N_{\text{Ham}}}}{\prod _{k=1}^{n}{{\color {Red}N_{{\text{Spam}},k}} \over {\color {OliveGreen}N_{{\text{Ham}},k}}}}{\prod _{k=1}^{n}{{\color {Blue}N_{\text{Ham}}} \over {\color {Brown}N_{\text{Spam}}}}}}\quad |\ln(...)$

$\ln(V)={\ln({\color {Brown}N_{\text{Spam}}})-\ln({\color {Blue}N_{\text{Ham}}})+\sum _{k=1}^{n}{\ln \left({{\color {Red}N_{{\text{Spam}},k}} \over {\color {OliveGreen}N_{{\text{Ham}},k}}}\right)}+\sum _{k=1}^{n}{\ln \left({{\color {Blue}N_{\text{Ham}}} \over {\color {Brown}N_{\text{Spam}}}}\right)}}$

$\ln(V)={{\Bigl (}n-1{\Bigr )}{{\Bigl (}\ln({\color {Blue}N_{\text{Ham}}})-\ln({\color {Brown}N_{\text{Spam}}}){\Bigr )}}+\sum _{k=1}^{n}{{\Bigl (}\ln({\color {Red}N_{{\text{Spam}},k}})-\ln({\color {OliveGreen}N_{{\text{Ham}},k}}){\Bigr )}}}$

$V=\exp {\Biggl (}{{\Bigl (}n-1{\Bigr )}{{\Bigl (}\ln({\color {Blue}N_{\text{Ham}}})-\ln({\color {Brown}N_{\text{Spam}}}){\Bigr )}}+\sum _{k=1}^{n}{{\Bigl (}\ln({\color {Red}N_{{\text{Spam}},k}})-\ln({\color {OliveGreen}N_{{\text{Ham}},k}}){\Bigr )}}{\Biggr )}}$

$P(Spam)=\exp {\bigl (}{sum(ln)+(n-1)({\color {Blue}ham}-{\color {Brown}spam}){\bigr )}}$

$(n-1)(\ln({\color {Blue}N_{\text{Ham}}})-\ln({\color {Brown}N_{\text{Spam}}}))$

$10^{-1500}$

$\ln {N_{Spam,i} \over N_{Ham,i}}$

$spam=spam+\Delta spam$

$\ln(spam+\Delta spam)-\ln {e^{(}\ln(spam)-ln)},spam=spam+n$