Basic Concepts
(P3-P8) Sample Spaces and Events Ω sample space
ω outcome (point or
element)
A event (subset of Ω)
Aᶜ complement of A
(not A)
A ∪ B union (A or
B)
A ∩ B or AB intersection (A and
B)
A − B set difference
(ω in A but not in B)
A ⊂ B set
inclusion
∅ null event (always false)
Ω true event (always true)
Indicator Function :
- switch set concept to digital concept, like One-Hot and Mask concept in ML.
- the connection between probability and expectation.
Two Interpretations of Probability (P6)
- The Frequentist View
- Degree of beliefs(The Bayesian View)
Three Axioms (P5)
- P(A) ≥ 0 for every A
- P(Ω) = 1
- If A1, A2, …
are disjoint then
Properties （P6-7） One can derive many properties of P from axioms
- P(⌀) = 0
- A ⊂ B ⇒ P(A) ≤ P(B)
- 0 ≤ P(A) ≤ 1
- P(Ac) = 1 − P(A)
- A ∩ B = ⌀ ⇒ P(A ∪ B) = P(A) + P(B)
Lemma(容斥原理): For any events A and B , P(A ∪ B) = P(A) + P(B) − P(AB)
Theorem (Continuity of Probabilities): If An → A , then P(An) → P(A) as n → ∞ . （Use Axiom3 to prove）
uniform probability distribution:
Bayes
Independent Events (P8) Two events A and B are independent if P(AB) = P(A)P(B) A set of events {Ai : i ∈ I} is independent if P (⋂i ∈ JAi) = ∏i ∈ JP(Ai) for every finite subset J of I.
Conditional Probability (P10) If P(B) > 0 then the
conditional probability of A
given B is
| D | Dc | ||
|---|---|---|---|
| + | .009 (真阳性TP) | .099（假阳性 FP） | |
| - | .001 (假阴性FN) | .891 (真阴性TN) |
如果前提有病/没病，能正确检测的概率是 90%（Recall）
Lemma(P11) : For any events A and B with P(B) > 0 , P(AB) = P(A ∣ B)P(B) = P(B ∣ A)P(A)
The Law of Total Probability(P12) : Let {A1, …, Ak}
be a partition of Ω . Then
Bayes' Theorem : (P12)
