tfp.substrates.numpy.vi.arithmetic_geometric

The Arithmetic-Geometric Csiszar-function in log-space.

tfp.substrates.numpy.vi.arithmetic_geometric(
    logu, self_normalized=False, name=None
)

A Csiszar-function is a member of,

F = { f:R_+ to R : f convex }.

When self_normalized = True the Arithmetic-Geometric Csiszar-function is:

f(u) = (1 + u) log( (1 + u) / sqrt(u) ) - (1 + u) log(2)

When self_normalized = False the (1 + u) log(2) term is omitted.

Observe that as an f-Divergence, this Csiszar-function implies:

D_f[p, q] = KL[m, p] + KL[m, q]
m(x) = 0.5 p(x) + 0.5 q(x)

In a sense, this divergence is the "reverse" of the Jensen-Shannon f-Divergence.

This Csiszar-function induces a symmetric f-Divergence, i.e., D_f[p, q] = D_f[q, p].

logu float-like Tensor representing log(u) from above.
self_normalized Python bool indicating whether f'(u=1)=0. When f'(u=1)=0 the implied Csiszar f-Divergence remains non-negative even when p, q are unnormalized measures.
name Python str name prefixed to Ops created by this function.

arithmetic_geometric_of_u float-like Tensor of the Csiszar-function evaluated at u = exp(logu).