This document discusses several mathematical concepts related to probability and statistics: - It defines the Kullback-Leibler divergence KL(P||Q) as a measure of how one probability distribution P differs from a second distribution Q. - It presents an equation for Qθ(y|x) as a Gaussian distribution with mean fθ(x) and variance 1/2, commonly used in probabilistic modeling. - It defines the Wasserstein distance Wp(μ,ν) as a way to measure the distance between two probability distributions based on the minimum cost of transporting mass between them. - It provides equations for the probability density function and cumulative distribution function of a Dirac delta function and a

4. ☓
☓
KL(P ||Q) = P(x)log
P(x)
Q(x)
dx∫

5. KL(P ||Q)
= P(x)logP(x)dx − P(x)logQ(x)dx∫∫

6. Qθ (y | x) = Aexp −
1
2
y − fθ (x)( )2⎛
⎝⎜
⎞
⎠⎟
− P(x)logQθ (x)dx∫
= EP −logQθ x( )⎡⎣ ⎤⎦
= EP −log A +
1
2
y − fθ (x)( )2⎡
⎣⎢
⎤
⎦⎥

7. Wp (µ,ν)p
= inf
π∈Γ(µ,ν )
c(x,y)p
dπ(x,y)∫

17. δ x − a( )−∞
∞
∫ dx = 1
δ x( )= 0 x ≠ 0( )
ˆP(x) = 1
m δXi
i=1
m
∑

20. lp
p
P,Q( )= | FP (x)− FQ (x)|p
dx
−∞
∞
∫

21. W1(P,Q) = | FP
−1
(
0
1
∫ µ)− FQ
−1
(µ)| dµ
l1 P,Q( )= | FP (x)− FQ (x)| dx
−∞
∞
∫

22. c(x1,y1)+ c(x2,y2 ) < c(x1,y2 )+ c(x2,y1)