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A Note on ZINB-VAE

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https://arxiv.org/abs/1709.02082

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A Note on ZINB-VAE

  1. 1. A Note on ZINB-VAE[1] Tomonari MASADA @ Nagasaki University September 8, 2017 p(xn,g, hn,g, yn,g, wn,g, zn; W h, W w) = p(xn,g|hn,g, yn,g)p(hn,g|zn; W h)p(yn,g|wn,g)p(wn,g|zn; W w)p(zn) = δ0(xn,g)hn,g δyn,g (xn,g)1−hn,g × fh(zn)hn,g (1 − fh(zn))1−hn,g × w yn,g n,g e−wn,g yn,g! × fw,2(zn) fw,1(zn) Γ(fw,1(zn)) wfw,1(zn)−1 n,g e−fw,2(zn)wn,g × K k=1 1 √ 2π exp z2 n,k 2 (1) p(xn,g, yn,g, wn,g, zn; W h, W w) = p(xn,g, hn,g = 0, yn,g, wn,g, zn; W h, W w) + p(xn,g, hn,g = 1, yn,g, wn,g, zn; W h, W w) = {δyn,g (xn,g)(1 − fh(zn)) + δ0(xn,g)fh(zn)} × w yn,g n,g e−wn,g yn,g! × fw,2(zn) fw,1(zn) Γ(fw,1(zn)) wfw,1(zn)−1 n,g e−fw,2(zn)wn,g × K k=1 1 √ 2π exp − z2 n,k 2 (2) p(xn,g, wn,g, zn; W h, W w) = p(xn,g|wn,g, zn; W h)p(wn,g|zn; W w)p(zn) = p(xn,g, yn,g, wn,g, zn; W h, W w)dyn,g = (1 − fh(zn)) × w xn,g n,g e−wn,g xn,g! + δ0(xn,g)fh(zn) × fw,2(zn) fw,1(zn) Γ(fw,1(zn)) wfw,1(zn)−1 n,g e−fw,2(zn)wn,g × K k=1 1 √ 2π exp − z2 n,k 2 = (1 − fh(zn)) × w xn,g n,g e−wn,g xn,g! + δ0(xn,g)fh(zn) × fw,2(zn) fw,1(zn) Γ(fw,1(zn)) wfw,1(zn)−1 n,g e−fw,2(zn)wn,g × K k=1 1 √ 2π exp − z2 n,k 2 (3) p(xn,g|zn; W h, W w) = p(xn,g|wn,g, zn; W h)p(wn,g|zn; W w)dwn,g = (1 − fh(zn)) × w xn,g n,g e−wn,g xn,g! + δ0(xn,g)fh(zn) fw,2(zn) fw,1(zn) Γ(fw,1(zn)) wfw,1(zn)−1 n,g e−fw,2(zn)wn,g dwn,g = (1 − fh(zn)) 1 xn,g! fw,2(zn) fw,1(zn) Γ(fw,1(zn)) w{xn,g+fw,1(zn)}−1 n,g e−{1+fw,2(zn)}wn,g dwn,g + δ0(xn,g)fh(zn) = (1 − fh(zn)) 1 xn,g! fw,2(zn) fw,1(zn) Γ(fw,1(zn)) Γ(xn,g + fw,1(zn)) {1 + fw,2(zn)}xn,g+fw,1(zn) + δ0(xn,g)fh(zn) (4) 1
  2. 2. The ELBO is obtained as follows. log p(xn,g; W h, W w) = log p(xn,g|zn; W h, W w)p(zn)dzn ≥ q(zn|xn) log p(xn,g|zn; W h, W w)p(zn) q(zn|xn) dzn (5) We can perform a Monte Carlo approximation as below. q(zn|xn) log p(xn,g|zn; W h, W w)p(zn) q(zn|xn) dzn ≈ 1 S S s=1 log p(xn,g|z(s) n ; W h, W w) + q(zn|xn) log p(zn)dzn − q(zn|xn) log q(zn|xn)dzn (6) where z (s) n ≡ (s) σn(xn) + µn(xn), and (s) ∼ N(0, I). References [1] R. Lopez, J. Regier, M. Jordan, and N. Yosef. A deep generative model for gene expression profiles from single-cell RNA sequencing. ArXiv e-prints, September 2017. 2

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