The Jensen-Shannon Csiszar-function in log-space

A Csiszar-function is a member of F = { f:R_+ to R : f convex }.

vi_jensen_shannon(logu, self_normalized = FALSE, name = NULL)

Arguments

logu	float-like Tensor representing log(u) from above.
self_normalized	logical 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	name prefixed to Ops created by this function.

Value

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

Details

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].

Warning: this function makes non-log-space calculations and may therefore be numerically unstable for |logu| >> 0.

References

• Lin, J. "Divergence measures based on the Shannon entropy." IEEE Trans. Inf. Th., 37, 145-151, 1991.

See also

Other vi-functions: vi_amari_alpha(), vi_arithmetic_geometric(), vi_chi_square(), vi_csiszar_vimco(), vi_dual_csiszar_function(), vi_fit_surrogate_posterior(), vi_jeffreys(), vi_kl_forward(), vi_kl_reverse(), vi_log1p_abs(), vi_modified_gan(), vi_monte_carlo_variational_loss(), vi_pearson(), vi_squared_hellinger(), vi_symmetrized_csiszar_function()