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Variational AutoEncoder

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    https://twitter.com/one_meets_seven

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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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