https://arxiv.org/abs/1709.02082

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. 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