tfp.substrates.numpy.vi.symmetrized_csiszar_function

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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)).