1 Computing a partition function
p(x; w) = 1 exp (x⊤w) (1) Z(w)
p(x;w)dx=1 (x∈R) or p(x;w)=1 (x∈Q) (2) x
p(x;w) = 1 exp(x⊤w) (3) x x Z(w)
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1= 1 exp(x⊤w) (4) Z(w) x
log1 = log 1 exp(x⊤w) (5) Z(w) x
0 = − log Z(w) + log exp (x⊤w) (6) x
log Z(w) = log exp (x⊤w) x
2 Log-Sum-Exp Trick
Given x = {x1,x2,…,xn}, where n ∈ N
y = ln exn ⇔ ey = exn (8)
n=1 n=1 NN
e−aey =e−aexn ⇔ ey−a =exn−a (9) n=1 n=1
y − a = ln exn −a ⇔ y = a + ln exn −a (10)
ln exn = a + ln exn −a , where a = max (xi ), i = 1..n
3 Compute a log of Gaussian PDF
Given D = (x(1), x(2), …x(N)), D ∈ R(N,d)
lnp(D;(m,S)) = lnp(x(i);(m,S)) = lnp(x(i);(m,S)) = i=1 i=1
d ln 2π + ln |S| + (x(i) − m)⊤S−1(x(i) − m) =
−2 Ndln2π+Nln|S|+
i=1 D = D − m[np.newaxis],
x(i) = trx(i)
(14) tr ABC =tr BCA (15)
(x(i) − m)⊤S−1(x(i) − m) = tr(x(i) − m)⊤S−1(x(i) − m) =
trS−1 (x(i) − m)(x(i) − m)⊤ = trS−1D⊤D = trDS−1D⊤ (16)
(x(i)−m)⊤S−1(x(i)−m) (12) D ∈ R(N,d) (13)
SA = D⊤ ⇔ A = S−1D⊤ (17) A = np.linalg.solve(S, D⊤) (18) trDS−1D⊤ = trDA (19)
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