Math/Statistics
Probability vs. Likelihood
- Probability: During the testing phase, given a learned model, we determine the probability of observing the outcome
- Likelihood: During training, given some outcome we determine the likelihood of observing theta that maximizes the probability of that outcome (MLE).
Bayes Theorem
- Revise prediction using new evidence
- Naive Bayes Classifier: Generative Classification model
Linear Transformation
- Rotation (produced by shearing) and scaling
\[
f(\alpha x + \beta y)=\alpha f(x)+\beta f(y)
\]
Affine Transformation
AKA a linear transformation plus translation
\[
f(\alpha x + (1-\alpha)y)=\alpha f(x)+(1-\alpha)f(y)
\]
\[
f(x)=\vec{a}^Tx+\vec{b}
\]
Properties Perserved:
- Collinearity between points: points on same line (collinear points) remain on same line after transformation
- Parallelism: Parallel lines remain parallel
- Convexity of sets: Furthermore, extreme points of original set map to extreme points of transformed set
- Length Ratios of Parallel lines
- Barycenters of weighted collections of points. Aka center of mass. ?