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Variational AutoEncoder

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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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