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Symmetrizes a Csiszar-function in log-space.
tfp.substrates.numpy.vi.symmetrized_csiszar_function(
logu, csiszar_function, name=None
)
A Csiszar-function is a member of,
F = { f:R_+ to R : f convex }.
The symmetrized Csiszar-function is defined as:
f_g(u) = 0.5 g(u) + 0.5 u g (1 / u)
where g
is some other Csiszar-function.
We say the function is "symmetrized" because:
D_{f_g}[p, q] = D_{f_g}[q, p]
for all p << >> q
(i.e., support(p) = support(q)
).
There exists alternatives for symmetrizing a Csiszar-function. For example,
f_g(u) = max(f(u), f^*(u)),
where f^*
is the dual Csiszar-function, also implies a symmetric
f-Divergence.
Example:
When either of the following functions are symmetrized, we obtain the Jensen-Shannon Csiszar-function, i.e.,
g(u) = -log(u) - (1 + u) log((1 + u) / 2) + u - 1
h(u) = log(4) + 2 u log(u / (1 + u))
implies,
f_g(u) = f_h(u) = u log(u) - (1 + u) log((1 + u) / 2)
= jensen_shannon(log(u)).
Returns | |
---|---|
symmetrized_g_of_u
|
float -like Tensor of the result of applying the
symmetrization of g evaluated at u = exp(logu) .
|