![Baye's Theorem
Bayes' Theorem is named after Thomas Bayes.There are
two types of probabilities −
Posterior Probability [P(H/X)]
Prior Probability [P(H)]
where X is data tuple and H is some hypothesis.
According to Bayes' Theorem,
P(H/X)= P(X/H)P(H) / P(X)](https://image.slidesharecdn.com/module4bayesclassification-210423165205/85/Module-4-bayes-classification-2-320.jpg)
Module 4 bayes classification
- 2. Baye's Theorem Bayes' Theorem is named after Thomas Bayes.There are two types of probabilities − Posterior Probability [P(H/X)] Prior Probability [P(H)] where X is data tuple and H is some hypothesis. According to Bayes' Theorem, P(H/X)= P(X/H)P(H) / P(X)
- 3. Naïve Bayes It is a classification technique based on Bayes’ Theorem with an assumption of independence among predictors. In simple terms, a Naive Bayes classifier assumes that the presence of a particular feature in a class is unrelated to the presence of any other feature. For example, a fruit may be considered to be an apple if it is red, round, and about 3 inches in diameter. Even if these features depend on each other or upon the existence of the other features, all of these properties independently contribute to the probability that this fruit is an apple and that is why it is known as ‘Naive’. Naive Bayes model is easy to build and particularly useful for very large data sets. Along with simplicity, Naive Bayes is known to outperform even highly sophisticated classification methods.
- 4. Bayes theorem provides a way of calculating posterior probability P(c|x) from P(c), P(x) and P(x|c). Look at the equation below: Formula
- 5. Formula P(c|x) is the posterior probability of class (c, target) given predictor (x, attributes). P(c) is the prior probability of class. P(x|c) is the likelihood which is the probability of predictor given class. P(x) is the prior probability of predictor.
- 6. Dataset Day Outlook Temperature Humidity Wind Class: Play ball D1 Sunny Hot High False No D2 Sunny Hot High True No D3 Overcast Hot High False Yes D4 Rain Mild High False Yes D5 Rain Cool Normal False Yes D6 Rain Cool Normal True No D7 Overcast Cool Normal True Yes D8 Sunny Mild High False No D9 Sunny Cool Normal False Yes D10 Rain Mild Normal False Yes D11 Sunny Mild Normal True Yes D12 Overcast Mild High True Yes D13 Overcast Hot Normal False Yes D14 Rain Mild High True No
- 7. Problem The weather data, with counts and probabilities outlook temperature humidity windy play yes no yes no yes no yes no yes no sunny 2 3 hot 2 2 high 3 4 false 6 2 9 5 overcast 4 0 mild 4 2 normal 6 1 true 3 3 rainy 3 2 cool 3 1 sunny 2/9 3/5 hot 2/9 2/5 high 3/9 4/5 false 6/9 2/5 9/14 5/14 overcast 4/9 0/5 mild 4/9 2/5 normal 6/9 1/5 true 3/9 3/5 rainy 3/9 2/5 cool 3/9 1/5 A new day outlook temperature humidity windy play sunny cool high true ? Outlook Temp Humidity Wind Overcast Mild Normal True
- 8. Problem P(outlook=Sunny|Yes) = 2/9 P(temp=cool|yes) = 3/9 P(humidity=high|yes)=3/9 P(Windy=true|yes)=3/9 P(outlook=Sunny|temp=cool|humidity=high|Windy=true|Yes)=2/9*3/9*3/9*3/9*9/14 = 0.00529 P(outlook=Sunny|No) = 3/5 P(temp=cool|No) = 1/5 P(humidity=high|No)= 4/5 P(Windy=true|No)= 3/5 P(outlook=Sunny|temp=cool|humidity=high|Windy=true|No) = 3/5*1/5*4/5*3/5*5/14 = 0.0206 P(Yes)<P(No) Prediction = No
- 9. Likelihood of yes Likelihood of no Therefore, the prediction is No 0053 . 0 14 9 9 3 9 3 9 3 9 2 0206 . 0 14 5 5 3 5 4 5 1 5 3
- 10. Predict stolen for Color=red Type=suv Origin=domestic
- 11. Color Type Origin Yes No Yes No Yes No Red 3 2 Sports 4 2 Dom 2 3 Yellow 2 3 SUV 1 3 Imp 3 2 Red 3/5 2/5 Sports 4/6 2/6 Dom 2/5 3/5 Yellow 2/5 3/5 SUV 1/4 3/4 Imp 3/5 2/5 Total Rows = 10 P(Yes) = 5/10 P(No) = 5/10 Predict stolen for Likelihood forYes Color=red = 3/5 Type=suv = 1/4 Origin=domestic = 2/5 P(X|Yes) = 3/5*1/4*2/5*5/10 = 0.003 Likelihood for No Color=red = 2/5 Type=suv = 3/4 Origin=domestic = 3/5 P(X|No) = 2/5*3/4*3/5*5/10 = 0.033 Prediction = Stolen = No