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tfp.vi.jensen_shannon

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

tfp.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.

Args:

  • 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.

Returns:

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