Title: Theory / Important / Math / Probability / Jaynes
Tags: fromMath mathToDef fromImplication mathToSee récupéré toBinaryRelation

Name: Their logical product
Definition: Both \(A\) and \(B\) are true
Page: 10
Chapter: 1
Section: 5
Variables: \(A,B\) propositions
Notation: \(AB\)
Name2: Their conjunctions

Name: Their logical sum
Definition: At least one of \(A\) or \(B\) is true
Page: 10
Chapter: 1
Section: 5
Variables: \(A,B\) propositions
Notation: \(A+B\)
Name2: Their disjunctions

Page: 10
Definition: Two propositions which are equivalent.
Chapter: 1
Section: 5
Notation: \(=\)
Name: Having the same truth value
Context: Boolean Algebra

Name: The denial of \(A\)
Page: 10
Chapter: 1
Section: 5
Variables: \(A\) a proposition
Notation: \(\overline A\)

Question: \(A=\overline A\) is
Answer: false.
Page: 10
Chapter: 1
Section: 5
Variables: \(A\) a proposition

Q: \([...] =[...]\).
A: \overline{AB}; \overline A+\overline B
Page: 11
Chapter: 1
Section: 5
Variables: \(A, B\) propositions
Name: Duality

Q: \([...] =[...]\).
A: \overline{A+B}; \left(\overline A\right)\left(\overline B\right)
Page: 11
Chapter: 1
Section: 5
Variables: \(A, B\) propositions
Name: Duality

Q: Logical sum is:
-[...],
-[...],
-[...].
A: Idempotent; commutative; associative
Page: 11
Chapter: 1
Section: 5
Variables: \(A, B\) propositions

Logical product is:
-Idempotent,
-commutative,
-associative,
-distributive over logical sum.
Page: 11
Chapter: 1
Section: 5
Variables: \(A, B\) propositions

Name: \(A\) implies \(B\)
Page: 11
Chapter: 1
Section: 5
Variables: \(A, B\) propositions
Notation: \(A\implies B\)

Name: \(A\) nand \(B\)
Definition: \(\overline{AB}\)
Page: 16
Chapter: 1
Section: 6
Variables: \(A, B\) propositions
Notation: \(A\uparrow B\)

Name: \(A\) nor \(B\)
Definition: \(\overline{A+B}\)
Page: 16
Chapter: 1
Section: 6
Variables: \(A, B\) propositions
Notation: \(A\downarrow B\)

Name: A logical function
Definition: A function which assigns True or False to each \(n\)-tuple of Booleans, for \(n\) fixed.
Page: 13
Chapter: 1
Section: 5

Question: The number of \(n\)-ary logical function is
Answer: \(2^n\).
Page: 13
Chapter: 1
Section: 5
Variables: \(n\in\mathbb N\)

Name: A formula in disjunctive normal form
Definition: A disjonction of conjunctions, or False.
Page: 15
Chapter: 1
Section: 6

Name: An adequate set
Definition: A set of logical functions which generates every logical functions.
Page: 16
Chapter: 1
Section: 6
Variables: In Boolean algebra
Example: \(\{\uparrow\}\) and \(\{\downarrow\}\)

Question: Degrees of plausibility are represented by
Answer: real numbers.
Page: 17
Chapter: 1
Section: 7

Question: Greater plausibility corresponds to
Answer: greater real number.
Page: 17
Chapter: 1
Section: 7

Name: The plausibility that \(A\) is true, assuming \(B\) is true
Page: 17
Chapter: 1
Section: 7
Variables: \(A,B\) propositions
Notation: \(A|B\)
Name2: \(A\) given \(B\)
Assuming: \(B\) is possible

Page: 21
Chapter: 1
Section: 8.1
Name: Logically equivalent, commonly uniquivalent
Q: [...].
[...]
A: «He believe what he sees», and «He doesn't see what he doesn't believe»; For \(L\) a straight line in a plane, \(S\) an infinite set of points in that plane, each of which is projected onto \(L\):
-«The projection of the limite is the limit of the projections» and
-«The limit of the projections is the projections of the limit».

Name: Mind projection fallacy
Definition: One's own private thought and sensations are realities existing externally in Nature.
Page: 21
Chapter: 1
Section: 8.1

Smaller0: \(B|C'>B|C\)
Greater0: \(AB|C'\ge AB|C\)
Page: 26
Chapter: 2
Section: 1
Variables: \(A,B,C,C'\) propositions
Smaller1: \(A|BC'=A|BC\)

Smaller0: \(B|C'=B|C\)
Greater0: \(AB|C'> AB|C\)
Page: 26
Chapter: 2
Section: 1
Variables: \(A,B,C,C'\) propositions
Smaller1: \(A|BC'>A|BC\)

Smaller0: \(B|C=0\)
Greater0: \(AB|C'=AB|C\)
Page: 26
Chapter: 2
Section: 1
Variables: \(A,B,C,C'\) propositions
Smaller1: \(A|BC'>A|BC\)

Smaller0: \(B|C'>B|C\)
Greater0: \(AB|C'=AB|C\)
Page: 26
Chapter: 2
Section: 1
Variables: \(A,B,C,C'\) propositions
Smaller1: \(A|BC=0\)

Smaller0: \(C\) implies \(A\)
Greater0: \(B\mid C\)
Page: 26
Chapter: 2
Section: 1
Variables: \(A,B,C\) propositions
Greater1: \(AB\mid C\)
Connect Greater0: =

Question: \(f\) can be canonically written as
Answer: a disjunctive normal form.
Page: 15
Chapter: 1
Section: 6
Variables: \(f\) a propositional formula

Smaller0: \(p(AB\mid C)\)
Greater0: \(p(A\mid BC)p(B\mid C)\)
Page: 33
Chapter: 2
Variables: \(A,B,C\) propositional varaibles
Smaller to greater: =
Name: The product rule

Smaller0: \(p(A+B\mid C)\)
Greater0: \(p(A\mid C)+p(B\mid C)-p(AB\mid C)\)
Page: 34
Chapter: 2
Variables: \(A,B,C\) propositional varaibles
Smaller to greater: equals
Name: The sum rule

Smaller0: the \(A_i\)'s are mutually exclusive and exhaustive
Greater0: \(p(A_i\mid B)=\frac 1n\)
Page: 40
Chapter: 2
Section: 4
Variables: \(A_1,\dots, A_n,B\) propositional varaibles
Smaller1: each \(A_i\) is equiprobable,given \(B\)
Name: Principle of indifference

Name: Original definition of probability of \(A\)
Definition: The ratio of the number of cases favorable to \(A\), to the number of all cases possible when nothing leads us to expect that any one of these cases should occur more than any other, which renders them, for us, equally possible.
Page: 42-43
Chapter: 2
Section: 4
Variables: \(A\) an event

Name: Probability of propositions
Page: 43
Chapter: 2
Section: 5
Context: Probability (Jaynes)
Denoted by: Upper case

Name: Probabilities whose arguments are numerical values
Page: 43
Chapter: 2
Section: 5
Context: Probability (Jaynes)
Extra: CHAPTER DONE; goto 3
Denoted by: lower case

Question: The probability to get \(r\) red in \(n\) draw is
Answer: \({ {M\choose r} {{N-M}\choose {n-r} } }/{N\choose n}\)
Page: 65
Chapter: 3
Section: 4
Variables: \(M\) red balls
\(N\) blue balls

Name: The cumulative probability distribution
Definition: [$]x\mapsto\int_{-\infty}^xh(r)\dd r[/$]
Page: 56
Chapter: 3
Section: 1
Variables: \(h:\mathbb R\to[0,1]\) a probability distribution
Denoted by: \(H\)

Name: The \(k\)-th decile of \(h\)
Definition: A real \(m\) such that [$]\abs{\frac{1-k}{10}H(r<m)-\frac k{10}H(r>m)}[/$] is as small as possible
Page: 57
Chapter: 3
Section: 1
Variables: \(h:\mathbb R\to[0,1]\) a probability distribution
\(0\le k\le 10\)

Name: The median of \(h\)
Definition: A real \(m\) such that [$]\abs{H(r<m)-H(r>m)}[/$] is as small as possible
Page: 57
Chapter: 3
Section: 1
Variables: \(h:\mathbb R\to[0,1]\) a probability distribution

Name: The first quartile of \(h\)
Definition: A real \(m\) such that [$]\abs{3H(r<m)-H(r>m)}[/$] is as small as possible
Page: 57
Chapter: 3
Section: 1
Variables: \(h:\mathbb R\to[0,1]\) a probability distribution

Name: The first quartile of \(h\)
Definition: A real \(m\) such that [$]\abs{H(r<m)-3H(r>m)}[/$] is as small as possible
Page: 57
Chapter: 3
Section: 1
Variables: \(h:\mathbb R\to[0,1]\) a probability distribution

Name: The expectation of \(X\)
Definition: [$]\int X(x)p(x)\dd x[/$]
Page: 66
Chapter: 3
Section: 4
Variables: \(S\) a measurable space,
\(X:S\to[0,1]\) a probability distribution
Notation: \(\langle X\rangle\)
Intuition: Weighted average of the possible values

Name: The hypergeometric function
Definition: \(\frac{ {M\choose r} {N-M\choose n-r} }{ {N \choose n} }\)
Page: 55, 65, 76
Chapter: 3,
Section: 1, 4, 15
Notation: \(h(r\mid N,M,n)\)
Definition2: \(\frac{ {n\choose r} {N-n\choose M-r} }{ {N \choose M} }\)
Intuition: \(\frac{ {R\choose r} {W\choose w} }{ {R+W \choose r+w} }\) with \(N=R+W\) the number of ball being the sum of number of red and white, and \(n=r+w\), the number of draw beeing the sum of the number of red and white draw

Name: The generalized hypergeometric function
Definition: \(\frac{\prod_{i=1}^k {N_i\choose r_i}}{ {\sum N_i \choose\sum r_i} }\)
Page: 76
Chapter: 3,
Section: 1, 4, 15
Context: Probability to draw \(r\) red in \(M\) draw in an urn with \(n\) red balls and \(N\) balls
Notation: \(h(r_1,\dots,r_k\mid N_1\dots N_k)\)
Definition2: \(\frac{ {n\choose r} {N-n\choose M-r} }{ {N \choose M} }\)
Intuition: \(\frac{ {R\choose r} {W\choose w} }{ {R+W \choose r+w} }\) with \(N=R+W\) the number of ball being the sum of number of red and white, and \(n=r+w\), the number of draw beeing the sum of the number of red and white draw

Definition: \(\frac{ \prod { N_i\choose r_i } }{ {\sum N_i \choose \sum r_i} }\)
Page: 55, 68
Chapter: 3
Section: 1, 5
Variables: \(N_1,\dots,N_k\ge1\),
\(0\le r_i\le N_i\)
Context: In an urn with \(\sum N_i\) balls, \(N_i\) balls of color \(c_i\), with \(\sum r_i\) draw, probability to draw \(r_i\) balls of color \(c_i\)
Notation: \(h(r_1,\dots,r_k\mid N_1,\dots,N_k)\)

Name: The multinomial distribution
Definition: \(\frac{(\sum r_i)!\prod f_i^{r_i}}{\prod r_i!}\)
Page: 71
Chapter: 3
Section: 7
Variables: \(r_1,\dots,r_k,n,f_1,\dots,f_k\) integers
Context: Given \(n\) independant events where \(P_i\) occurs with probability \(f_i\) each time, it is the probability to have \(r_i\) times \(P_i\) for each \(i\)
Notation: \(b((r_i)\mid n,(f_i))\)

Name: The binomial distribution
Definition: \({n\choose r}f^r(1-f)^{n-r}\)
Page: 68
Chapter: 3
Section: 7
Variables: \(r,n,f\) integers
Context: Given \(n\) independant events where \(P\) occurs with probability \(f\) each time, it is the probability to have \(r\) times \(P\)
Notation: \(b(r\mid n,f)\)

Question: What is the difference between physic's causality and logical's causality
Answer: Take a urn with two ball, white and red.
Take a ball. You know later you'll take a red ball.
It implies that the current ball is white, but does not physically cause it
Page: 69
Chapter: 3
Section: 8

Name: An exchangeable probability distribution
Definition: A probability distribution over many variables, such that \(p(x_1,\dots,x_n\mid B)=p(x_{\sigma1},\dots,x_{\sigma n}\mid B)\) for any permutation \(B\) and any value \(x_i\)
Page: 71
Chapter: 3
Section: 10

Smaller0: \(P(X_1=Red\mid X_N=Red)\)
Greater0: \(P(X_1=Red\mid \text{Some other ball are red})\)
Page: 73
Chapter: 3
Section: 10
Variables: \(N\) a number of ball
\(M\) a number of white ball
\(X_1,\dots,X_N\) \(N\) draws of the ball
Smaller to greater: \(\le\)
Greater to smaller: \(\ge\)
Context: taking distinct balls each time
Construction: Each universe where \(X_N=Red\) corresponds to two* universes where some \(X_i=Red\), in one of those universe, it's easier for \(X_1\) to be red, in the other, it's equally diffucult.
*With the exception of the case where each \(X_i\), \(1{\lt}i{\lt}n\) is white, and the case where \(M=N-1\)

Name: The expectation of \(X\)
Definition: \(\sum_{i=1}^np_ix_i\)
Page: 75
Chapter: 3
Section: 14
Variables: \(X\) a variable quantity
\(x_1,\dots,x_n\) its potential values
\(p_i\) the probability of \(X=x_i\)
Notation: \(\langle X\rangle\)
Notation2: \(E(X)\)