tfp.substrates.numpy.vi.jensen_shannon

The Jensen-Shannon Csiszar-function in log-space.

tfp.substrates.numpy.vi.jensen_shannon(
    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 Jensen-Shannon Csiszar-function is:

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

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

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

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

In a sense, this divergence is the "reverse" of the Arithmetic-Geometric f-Divergence.

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

For more information, see: Lin, J. "Divergence measures based on the Shannon entropy." IEEE Trans. Inf. Th., 37, 145-151, 1991.

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.

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