vi_symmetrized_csiszar_function | R Documentation |
A Csiszar-function is a member of F = { f:R_+ to R : f convex }
.
vi_symmetrized_csiszar_function(logu, csiszar_function, name = NULL)
logu |
|
csiszar_function |
function representing a Csiszar-function over log-domain. |
name |
name prefixed to Ops created by this function. |
The symmetrized Csiszar-function is defined as:
f_g(u) = 0.5 g(u) + 0.5 u g (1 / u)
where g
is some other Csiszar-function.
We say the function is "symmetrized" because:
D_{f_g}[p, q] = D_{f_g}[q, p]
for all p << >> q
(i.e., support(p) = support(q)
).
There exists alternatives for symmetrizing a Csiszar-function. For example,
f_g(u) = max(f(u), f^*(u)),
where f^*
is the dual Csiszar-function, also implies a symmetric
f-Divergence.
Example: When either of the following functions are symmetrized, we obtain the Jensen-Shannon Csiszar-function, i.e.,
g(u) = -log(u) - (1 + u) log((1 + u) / 2) + u - 1 h(u) = log(4) + 2 u log(u / (1 + u))
implies,
f_g(u) = f_h(u) = u log(u) - (1 + u) log((1 + u) / 2) = jensen_shannon(log(u)).
Warning: this function makes non-log-space calculations and may therefore be
numerically unstable for |logu| >> 0
.
symmetrized_g_of_u: float
-like Tensor
of the result of applying the
symmetrization of g
evaluated at u = exp(logu)
.
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_jensen_shannon()
,
vi_kl_forward()
,
vi_kl_reverse()
,
vi_log1p_abs()
,
vi_modified_gan()
,
vi_monte_carlo_variational_loss()
,
vi_pearson()
,
vi_squared_hellinger()
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