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p(x|y) =
p(y|x)p(x)
p(y)
p(x, y) = p(y|x)p(x)
p(x) =
Z
p(x, y)dy

6
p(✓|D) =
p(D|✓)p(✓)
p(D)
p(x|D) =
X
✓
p(x|✓, D)p(✓|D)

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log p(x) = + L
= KL(q(z|x)||p(z))L + Eq(z|x)[log p(x|z)]
KL(q(z|x)||p(z|x))
log p(x) = log p(x)
Z
q(z|x)dz
= log
p(x, z)
p(z|x)
Z
q(z|x)dz
=
Z
q(z|x) log
q(z|x)
p(z|x)
p(x, z)
q(z|x)
dz
=
Z
q(z|x) log
p(z|x)
q(z|x)
dz +
Z
q(z|x)log
p(x, z)
q(z|x)
dz
= KL(q(z|x)||p(z|x)) + L

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p✓(x|z)
p✓(x|z)
q (z|x)
q (z|x)

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q (z|x) p✓(x|z)
L(x; ✓, ) = KL(q (z|x)||p(z)) + Eq (z|x)[log p✓(x|z)]

14
L(x; ✓, ) = KL(q (z|x)||p(z)) + Eq (z|x)[log p✓(x|z)]
L(x; ✓, ) = 1
2
dX
(1 + log ( 2
d) µ2
d d) + 1
L
lX
log p✓(x|zl)

15
z ⇠ N(µ, )
z ⇠ N(µ, )

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✏ ⇠ N(0, I) z = µ + ✏

17
q (z|x) p✓(x|z)

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