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p(x|y) = p(y|x)p(x)p(y)
p(x, y) = p(y|x)p(x)
p(x) =
Zp(x, y)dy
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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 + E
q(z|x)[log p(x|z)]KL(q(z|x)||p(z|x))
log p(x) = log p(x)
Zq(z|x)dz
= log
p(x, z)
p(z|x)
Zq(z|x)dz
=
Zq(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 +
Zq(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)]
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L(x; ✓,�) = �KL(q�(z|x)||p(z)) + Eq�(z|x)[log p✓(x|z)]
L(x; ✓,�) = �1
2
dX(1 + log (�2
d)� µ2d � �d)
+ 1
L
lXlog p✓(x|zl)
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z ⇠ N (µ,�)
z ⇠ N (µ,�)
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✏ ⇠ N (0, I) z = µ+ ✏�
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q�(z|x) p✓(x|z)
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