Estimate Conditional Probabilities for Continuous Attributes

Estimate Conditional Probabilities for
Continuous Attributes (Additional Notes)
ο‚— Notes:
1
𝑃 𝑋𝑖 π‘Œ = 𝑦𝑗 =
2πœ‹πœŽπ‘–π‘–2
𝑒
𝑋𝑖 βˆ’πœ‡π‘–π‘–
βˆ’
2
2πœŽπ‘–π‘–
2
Probability
density function
ο‚— The probability density function is continuous, the
P(y|x)
probability that the random variable Xi takes a
particular value xi is zero.
1
xk
x
Estimate Conditional Probabilities for
Continuous Attributes (Additional Notes)
ο‚— Instead, we should compute
π‘₯π‘˜ +πœ€
𝑃 π‘₯π‘˜ ≀ 𝑋𝑖 ≀ π‘₯π‘˜ + πœ€ π‘Œ = 𝑦𝑗 = οΏ½
P(y|x)
π‘₯π‘˜
xk
xk+Ξ΅
x
β‰ˆ
1
2πœ‹πœŽπ‘–π‘–2
1
2πœ‹πœŽπ‘–π‘–2
𝑒
𝑒
𝑋𝑖 βˆ’πœ‡π‘–π‘–
βˆ’
2
2πœŽπ‘–π‘–
𝑋𝑖 βˆ’πœ‡π‘–π‘–
βˆ’
2
2πœŽπ‘–π‘–
2
2
𝑑𝑋𝑖
×πœ€
ο‚— Since Ι› appears as a constant multiplicative factor
for each class, it cancels out when normalizing the
posterior probability for P(Y|X).
2
Estimate Conditional Probabilities for
Continuous Attributes (Additional Notes)
ο‚— Therefore, we can still apply the following
equation to approximate the class-conditional
probability, 𝑃 𝑋𝑖 = π‘₯π‘˜ |π‘Œ = 𝑦𝑗
𝑃 𝑋𝑖 = π‘₯π‘˜ π‘Œ = 𝑦𝑗 =
3
1
2πœ‹πœŽπ‘–π‘–2
𝑒
π‘₯π‘˜ βˆ’πœ‡π‘–π‘–
βˆ’
2
2πœŽπ‘–π‘–
2