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# tfp.vi.arithmetic_geometric

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

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)`.

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