Title: Theory / Important / Math / Probability / Statistics / Data analysis, A Bayesian Tutorial/
Tags: relationHasDefinition

Name: The negation of \(X\)
Page: 5
Chapter: 1
Section: 2
Variables: \(X\) a propositional random variable
Notation: \(\bar X\)

Name: The basic assumptions of probability theory
Definition: The sum rule: \(\mathrm{prob}(X\mid I)+\mathrm{prob}(\neg X\mid I)=1\)
Page: 5
Chapter: 1
Section: 2
Variables: \(X,Y,I\) propositional random variables
Definition2: The product rule: \(\mathrm{prob}(X,Y\mid I)=\mathrm{prob}(Y\mid I)\mathrm{prob}(X\mid Y,I)\)
Conjdef: all of

Name: The probability of \(X\) given \(I\)
Page: 5
Chapter: 1
Section: 2
Variables: \(X,Y,I\) propositional random variables
Notation: \(\mathrm{prob}(X\mid I)\)

Name: The probability of both \(X\)and \(Y\)
Page: 5
Chapter: 1
Section: 2
Variables: \(X,Y,I\) propositional random variables
Notation: \(\mathrm{prob}(X, Y)\)

Smaller0: \(\mathrm{prob}(X, Y\mid I)\)
Greater0: \(\frac{\mathrm{prob}(Y\mid X,I)\mathrm{prob}(X\mid I)}{\mathrm{prob}(Y\mid I)}\)
Page: 6
Chapter: 1
Section: 3
Variables: \(X,Y,I\) propositional random variables
Smaller to greater: =
Name: Bayes Theorem

Smaller0: \(\mathrm{prob}(X\mid I)\)
Greater0: [$]\int_{-\infty}^\infty\mathrm{prob}(X,Y\mid I)\dd{Y}[/$]
Page: 6
Chapter: 1
Section: 3
Variables: \(X,Y,I\) propositional random variables
Index: 4
Kind: eq
Smaller to greater: =
Name: Bayes' Marginalization

Name: Background knowledge
Page: 6
Chapter: 1
Section: 3
Variables: \(prior, hypothesis,I\) propositional random variables
Kind: 1.4
Denoted by: \(I\)

Name: The prior probability
Definition: \(\mathrm{prob}(hypothesis\mid I)\)
Page: 6
Chapter: 1
Section: 3
Variables: \(hypothesis,I\) propositional random variables
Kind: 1.4

Name: The likelihood
Definition: \(\mathrm{prob}(data\mid hypothesis, I)\)
Page: 6
Chapter: 1
Section: 3
Variables: \(data, hypothesis,I\) propositional random variables
Kind: 1.4

Name: The posterior probability
Definition: \(\mathrm{prob}(hypothesis\mid data, I)\)
Page: 6
Chapter: 1
Section: 3
Variables: \(data, hypothesis,I\) propositional random variables
Kind: 1.4

Name: The evidence
Definition: \(\mathrm{prob}(data\mid I)\)
Page: 6
Chapter: 1
Section: 3
Variables: \(data,I\) propositional random variables
Kind: 1.4

Name: Probability density function
Definition: \(\lim_{\delta y\to 0}\frac{\mathrm{prob}(X,y\le Y<y+\delta y\mid I)}{\delta y}\)
Page: 8
Chapter: 1
Section: 3
Variables: \(X,I\) propositional random variables,
\(Y\) continuous
Kind: 1.4
Abbreviation: \(\mathrm{pdf}(X,Y=y\mid I)\)

Name: Distinct historical definition of probability
Definition: Degree-of belief
Page: 9
Chapter: 1
Section: 3
Kind: 1.4
Definition2: Long-run relative frequency with which an even occured

Name: Uniform pdf
Definition: [$]\defFun{\mathbb R}{\mathbb R}{H}{1\text{ if $H\in[0,1]$}, 0\text{ otherwise}}[/$]
Page: 15
Chapter: 2
Section: 1
Name2: flat pdf

Smaller0: \(X_0\) is the best estimate for \(X\)
Greater0: [$]\eval{\derivative{P}{X}}_{X_0}=0[/$]
Page: 21
Chapter: 2
Variables: \(X\) a real variable,
\(P\) a pdf on \(X\)
\(X_0\) a possible value for \(X\)
Smaller to greater: iff
Greater1: [$]\eval{\derivative[2]{P}{X}}_{X_0}<0[/$]