Evidence lower bound
lower bound on the log-likelihood of some observed data
In variational Bayesian methods, the evidence lower bound (often abbreviated ELBO, also sometimes called the variational lower bound or negative variational free energy) is a useful lower bound on the log-likelihood of some observed data.
The ELBO is useful because it provides a guarantee on the worst-case for the log-likelihood of some distribution (e.g.
p
(
X
)
{\displaystyle p(X)}
) which models a set of data. The actual log-likelihood may be higher (indicating an even better fit to the distribution) because the ELBO includes a Kullback-Leibler divergence (KL divergence) term which decreases the ELBO due to an internal part of the model being inaccurate despite good fit of the model overall. Thus improving the ELBO score indicates either improving the likelihood of the model
p
(
X
)
{\displaystyle p(X)}
or the fit of a component internal to the model, or both, and the ELBO score makes a good loss function, e.g., for training a deep neural network to improve both the model overall and the internal component. (The internal component is
q
ϕ
(
⋅
|
x
)
{\displaystyle q_{\phi }(\cdot |x)}
, defined in detail later in this article.)
Definition
Let
X
{\displaystyle X}
and
Z
{\displaystyle Z}
be random variables, jointly distributed with distribution
p
θ
{\displaystyle p_{\theta }}
. For example,
p
θ
(
X
)
{\displaystyle p_{\theta }(X)}
is the marginal distribution of
X
{\displaystyle X}
, and
p
θ
(
Z
∣
X
)
{\displaystyle p_{\theta }(Z\mid X)}
is the conditional distribution of
Z
{\displaystyle Z}
given
X
{\displaystyle X}
. Then, for a sample
x
∼
p
data
{\displaystyle x\sim p_{\text{data}}}
, and any distribution
q
ϕ
{\displaystyle q_{\phi }}
, the ELBO is defined as
L
(
ϕ
,
θ
;
x
)
:=
E
z
∼
q
ϕ
(
⋅
|
x
)
[
ln
p
θ
(
x
,
z
)
q
ϕ
(
z
|
x
)
]
.
{\displaystyle L(\phi ,\theta ;x):=\mathbb {E} _{z\sim q_{\phi }(\cdot |x)}\left[\ln {\frac {p_{\theta }(x,z)}{q_{\phi }(z|x)}}\right].}
The ELBO can equivalently be written as
L
(
ϕ
,
θ
;
x
)
=
E
z
∼
q
ϕ
(
⋅
|
x
)
[
ln
p
θ
(
x
,
z
)
]
+
H
[
q
ϕ
(
z
|
x
)
]
=
ln
p
θ
(
x
)
−
D
K
L
(
q
ϕ
(
z
|
x
)
|
|
p
θ
(
z
|
x
)
)
.
{\displaystyle {\begin{aligned}L(\phi ,\theta ;x)=&\mathbb {E} _{z\sim q_{\phi }(\cdot |x)}\left[\ln {}p_{\theta }(x,z)\right]+H[q_{\phi }(z|x)]\\=&\mathbb {\ln } {}\,p_{\theta }(x)-D_{KL}(q_{\phi }(z|x)||p_{\theta }(z|x)).\\\end{aligned}}}
In the first line,
H
[
q
ϕ
(
z
|
x
)
]
{\displaystyle H[q_{\phi }(z|x)]}
is the entropy of
q
ϕ
{\displaystyle q_{\phi }}
, which relates the ELBO to the Helmholtz free energy.
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