# tfp.vi.jensen_shannon

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

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)`.