# Probability Theory They sometimes say that [[self.stats/Statistics]] is just inverse probability. What does that really mean? A key concept in the field of [[Artificial Intelligence]], and life in general, is that of uncertainty. It arises both through noise on measurements, as well as through finite size datasets. ## Definition of Probability *Probability theory provides a consistent framework for the quantification and manipulation of uncertainty and forms one of the central foundations of pattern recognition.* ## Axioms of Probability $P(A) \geq 0$ $P(\Omega) = 1$ $P\big(\bigcup_n^\infty A_n\big) = \sum_{n=1}^{\infty}P(A_n) (A_i \bigcap A_j = \emptyset$ $ \begin{aligned} P(X = x_i) &= \sum_{j=1}^{L}P(X=x_i, Y=y_j) \\ \\ P(X) &= \sum_Y P(X,Y) \rightarrow \text{sum rule}\\ \\ P(X,Y) &= P(Y|X)P(X) \rightarrow \text{product rule}\\ \end{aligned} $ ## Probability Densities $ \begin{aligned} p(x \in (a,b)) &= \int_a^b p(x)dx \\ \\ p(x) &\geq 0 \\ \\ \int_{-\infty}^{\infty}p(x)dx &= 1 \end{aligned} $ Under a nonlinear change of variable a probability density transforms differently form a simple function due to the Jacobian Factor. ![[probability_mindmap_mml.png]]