Adversarial

Implicit generative models as factors — models you can sample from but whose density you cannot evaluate — and the discriminator read as what it is: a density-ratio estimator.

Three factors

src = NoiseSource(:z, 8; nsamples = 512)              # q(z), emits particles
gen = GeneratorFactor(my_decoder, (z = 8, x = 2))     # x = G_θ(z)
rat = RatioFactor(my_critic, 2)                        # log p(x) - log q(x)

All three are unidirectional. A generator is a function with no residual behind it, so there is nothing to run backwards — inverting $G_\theta$ is the GAN-inversion problem, and factor_message on the latent channel says so rather than pretending.

This package is the only one that produces a SampleBelief: a weighted particle set rather than a point or a parametric distribution.

The ratio identity

Train a classifier to separate $p$ from $q$ with equal class priors, and at the optimum

\[\operatorname{logit} D^\ast(x) = \log p(x) - \log q(x)\]

so the discriminator's logit is the log density ratio. logratio and discriminator are the two readings of that number.

Reweighting

The reason a ratio is useful here has little to do with generating images: importance reweighting needs a ratio, not a density, and that is what a classifier gives you.

pb, st = reweight(rat, particle_belief, ps, st)   # w_i ∝ w_i · r(x_i)
effective_sample_size(pb)                          # ...and check it meant something
weighted_mean(pb), weighted_cov(pb)

Always look at effective_sample_size. Reweighting concentrates, and the particle set can collapse onto a handful of samples with no error raised.

What it cannot do

Train adversarially. The generator descends the same quantity the discriminator ascends, and the graph's free energy has one sign. You can express a GAN's wiring here — and it is a DAG, since every factor is unidirectional — but not its objective. local_free_energy on a RatioFactor computes the number both players care about; nothing acts on it.

Also: no training of either network, and reweight is not idempotent (applying the same ratio twice squares the weights), so it is not wired into Mycelium.combine.

Adversarial.Adversarial — Module

Adversarial

Implicit generative models as Lenticulum factors — Mohamed & Lakshminarayanan's Learning in Implicit Generative Models (arXiv:1610.03483), and the GAN diagram read as a factor graph.

The three factors

factor	is	polarities
`NoiseSource`	the latent prior $q(z)$, as an emitting factor	1
`GeneratorFactor`	$x = G_\theta(z)$ — samples out, no density	1
`RatioFactor`	$\log r(x) = \log p(x) - \log q(x)$ — the discriminator	1

Every one of them is unidirectional, and that is the finding rather than a shortcoming: a generator is a function with no residual behind it, so there is nothing to run backwards. See [[Three Senses of Implicit]] for why a model can be "implicit" and still have exactly one direction.

Two things this package is for

1. It is the first thing in the project that produces a SampleBelief. LenticulumCore has declared that type since the beginning and nothing has ever constructed one. A generator is what it was for — and the moment one exists, Mycelium.combine throws, which is messages.md §1's recorded main gap made concrete.

2. A discriminator is a way around that gap. The central identity of the paper is

\[D^\ast(x) = \frac{p(x)}{p(x)+q(x)} \qquad\Longrightarrow\qquad \operatorname{logit} D^\ast(x) = \log p(x) - \log q(x)\]

so a classifier estimates a log-density ratio without either density — and a ratio is precisely what importance reweighting needs. reweight is the operation; effective_sample_size is the diagnostic that says whether it meant anything.

What it cannot do

Train adversarially. The generator descends the same quantity the discriminator ascends, and a Bethe free energy has one sign. The graph expresses the GAN's wiring exactly and its objective not at all — see [[GANs as Two Factors]] §4, which argues the missing structure is Ghani–Hedges–Winschel–Zahn's open game, a third lens-shaped object alongside the parametric lens and the statistical game.

Dependencies are LuxCore, Random, LinearAlgebra — no Lux, no AD, no training loop. This package scores and reweights; fitting the two networks is somebody else's job.

Concept notes: [[Implicit Generative Models]], [[GANs as Two Factors]], [[Three Senses of Implicit]]. Per-file notes: [[generator]], [[ratio]].

in	out	why
`DiracBelief`	`DiracBelief`	a function of a point is a point
`SampleBelief`	`SampleBelief`	map each particle; weights are carried through unchanged
`TrivialBelief`	`TrivialBelief`	no latent, no samples