# 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; ,) = 12

dX(1 + log (2d) 2d d)+

1

L

lXlog p(x|zl)

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z N (,)

z N (,)

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N (0, I) z = +

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q(z|x) p(x|z)

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