Title: Theory / Unimportant / CS / Fundamental / Information Theory, Inference and Learning Algorithm / Proba
Tags: examplesSwitchExample1

Name: An ensemble
Definition: \(\mathcal A_X\) a finite set of elements \(a_i\)
Page: 22
Chapter: 2
Section: 1
Context: Probability
Definition2: \(\mathcal P_X\) a set of probability \(p_i\) for each \(a_i\), such that \(\sum_{a_i\in\mathcal A_X}p_i=1\)
Conjdef: A triple \((x,\mathcal A_X,\mathcal P_X)\)
Definition3: \(x\) a random variable taking value in \(\mathcal A_X\)
Denoted by: \(X\)

Definition: \(\sum_{a_i\in T}P(x=a_i) \)
Page: 22
Chapter: 2
Section: 1
Variables: \((x,\mathcal A_X,\mathcal P_X)\) an ensemble,
\(T\subseteq \mathcal A_X\)
Notation: \(p(T)\)

Name: The joint ensemble of \(X_1\) and \(X_2\)
Definition: The ensemble whose variable is the pair \(x_1,x_2\), \(\mathcal A_{X_1X_2}\) is the cartesian products and \(\mathcal P_{X_1X_2}\) is the numberical product
Page: 22,23
Chapter: 2
Section: 1
Variables: \((x_i,\mathcal A_{X_i},\mathcal P_{X_i})\) ensembles.
Notation: \(X_1X_2\)
Index: ,1
Kind: , Ex
Example: Pair of successive letters in an English document

Name: The marginal probability over the first variable
Definition: The probability \(P(x_1=a_i)\) defined by \(\sum_{y\in X_{2}}P(x=a_i,y)\)
Page: 22
Chapter: 2
Section: 1
Variables: \((x_i,\mathcal A_{X_i},\mathcal P_{X_i})\) ensembles.
Notation: \(P(x)\)

Name: The conditional probability
Definition: \(\frac{P(x=a_i,y=b_j)}{P(y=b_j)}\) if \(P(y=b_j)\ne0\)
Page: 23
Chapter: 2
Section: 1
Variables: \((x_i,\mathcal A_{X_i},\mathcal P_{X_i})\) ensembles.
Notation: \(P(x=a_i\mid y=b_j)\)
Name2: the probability that \(x\) equals \(a_i\), given \(y\) equals \(b_ j\).

Smaller0: \(P(x,y\mid\mathcal H)\)
Greater0: \(P(x\mid y,\mathcal H)P(y\mid \mathcal H)\)
Page: 24
Chapter: 2
Section: 1
Variables: \((x_i,\mathcal A_{X_i},\mathcal P_{X_i})\) ensembles.
\(\mathcal H\) assumption
Smaller to greater: =
Name: The chain rule
Name2: The product rule