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--- data_mining:naive_bayes [2013/09/15 15:33] – [Bayes rule] phreazer
+++ data_mining:naive_bayes [2015/08/04 15:02] (current) – [Bayes rule] phreazer
@@ Line 16: / Line 16: @@
 ====== Bayes rule ======
-Datensatz:
+**Datensatz:**
 ^ R ^ B |
@@ Line 23: / Line 23: @@
 | y | n |
-n: Einträge
+  * n: Einträge
-R=y für r Fälle
+  * R=y für r Fälle
-B=y für k Fälle
+  * B=y für k Fälle
-R=y und B=y für i Fälle
+  * R=y und B=y für i Fälle
-p(B|R) = i/r
+  * $p(B|R) = i/r$
-p(R) = r/n
+  * $p(R) = r/n$
-p(R und B) = i/n = (i/r) * (r/n)
+  * $p(R und B) = i/n = (i/r) * (r/n)$
-p(B,R) = p(B|R) p(R)
+  * $p(B,R) = p(B|R) p(R)$
 Bayes Rule:
 $P(B,R) = P(B|R) P(R) = P(R|B) P(B)$
+Bayes Theorem:
+$P(B|R) = \frac{P(B,R)}{P(R)} = \frac{P(R|B)P(B)}{P(R)}$
+A-Priori-Wkeit: $P(B)$
+Likelihood: $P(R|B)$
+A-Posterior-Wkeit: $P(B|R)$
+===== Beispiel =====
+Question 1:
+If a person has malaria (mp), there is 90% chance that the blood test for malarial parasite comes up positive (tp); however, 1% of the time the test gives a false positive (tp and mn). Also, there is a 1% chance of getting malaria in general (mp).
+Unfortunately, you happen to test positive. What is the chance of your having malaria?
+Geg.:
+  * $P(tp | mp) = 0,9$
+  * $P(tp | mn) = 0,01$
+  * $P(mp) = 0,01$
+Gesucht:
+$P(mp|tp) = \frac{P(tp|mp) * P(mp)}{P(tp)} = \frac{0,9 * 0,01}{0,9*0,01+0,01*(1-0,01)} = 0,476$
+Question 2:
+Now suppose your doctor had employed a far superior, more expensive test, one with only a .1% chance of a false positive. (Other parameters are the same - 90% chance of a true positive, 1% chance of malaria in general.)
+What is the chance that you have malaria if you test positive with this improved procedure?
+  * $P(tp | mp) = 0,9$
+  * $P(tp | mn) = 0,001$
+  * $P(mp) = 0,01$
+Gesucht:
+$P(mp|tp) = \frac{P(tp|mp) * P(mp)}{P(tp)} = \frac{0,9 * 0,01}{0,9*0,01+0,001*(1-0,01)} = 0,901$
+====== Unabhängigkeit ======
+Angenommen B und R sind voneinander unabhängig.
+P(R) = r/n, P(C)= c/n
+P(R|C) = i/c, P(C|R) = i/r
+R und B sind voneinander unabhängig wenn und nur wenn
+i/c = r/n; i/r = c/n
+P(R|B) = P(R); P(B|R) = P(B)
+====== Naive Bayes ======
+Naiv wegen Annahme: R und C sind unabhängig gegeben B
+$$
+P(B|R,C) * P(R,C) = P(R,C|B) * P(B)\\
+= P(R|C,B) * P(C|B) * P(B)    (Bayes Rule)\\
+= P(R|C) * P(C|B) * P(B)      (Unabhängigkeit)
+$$
+Verhältnis berechnen:
+$$
+\frac{p(r|B=y) * p(c|B=y) * p(B=y)}{p(r|B=n) * p(c|B=n) * p(B=n)}
+$$
+B=y, wenn $>\alpha$ (z.B. 1), sonst B=n
+====== Naive Bayes für N features ======
+$$
+L = \prod_{i=1}^N \frac{p(x_i|B=y)}{p(x_i|B=n)} * \frac{p(B=y)}{p(B=n)}
+$$
+B=y, wenn $>\alpha$ (z.B. 1), sonst B=n
+log-likelihood => Anstelle von Multiplikation zu Addition um Rundungsfehler zu vermeiden
+====== Beispielhafte Sentiment Analysis ======
+p(+),p(-),
+p(like|+), p(enjoy|+), p(hate|+), ...
+P(hate|-), p(enjoy|-), p(lot|-), ...
+Im Text: like, simple, lot
+$$
+L = \frac{p(like|+)p(lot|+)[1-p(hate|+)][1-p(waste|+)]p(simple|+)}{p(like|-)p(lot|-)[1-p(hate|-)][1-p(waste|-)]p(simple|-)} * \frac{p(+)}{p(-)}
+$$