ABSTRACT
In this appendix, we list a few basic definitions that are used throughout the book.
A.1 Probabilities and Bayes’ Rule To begin, the probability of an event A occurring is often denoted p(A). The probability of an event A occurring, given that another event B has happened, is written as p(A|B). This is known as a conditional probability. Two events occurring simultaneously is given by:
p(A∩B) = p(A|B) p(B) p(B)
(A.1)
If the conditional probability of event A occuring, given that event B has already occurred, and given the probabilities of A and B happening in isolation, we can compute the conditional probability of B occurring, given event A. This is known as Bayes’ rule:
p(B|A) = p(A|B) p(B) p(A)
(A.2)
For a set of N events Bi, if we have:
Bi =Ω (A.3)
∀i 6= j : Bi ⋃
B j = /0 (A.4)
then we can compute the following probability:
p(A) = N
p(A∩Bi) (A.5)
A.2 Gaussian Distribution A univariate Gaussian probability distribution (also known as the univariate normal distribution) is given by:
N(x|µ,σ) = 1 σ √
2pi exp ( − (x−µ)
) (A.6)
This can be extended to multiple dimensions d, leading to the multivariate Gaussian distribution:
N(x|µ,Σ) = 1 (2pi)d/2
√|Σ| exp ( −1
2 (x−µ)TΣ−1(x−µ)
) (A.7)
Here, µ is a d-dimensional vector of means and Σ is the covariance matrix.
