Utility as a max of experts

A candidate is as good as its best style thinks it is. Two more obvious designs cannot represent a cross-island comparison at all.

The form

A candidate $x$ has the feature vector $\varphi(x)$, which the standardizer maps to $z(x)$ with per-feature means $\mu$ and scales $s$. The taste model holds $K$ styles, and style $k$ is a weight vector $\theta_k$: a linear functional on $z$. Utility is the maximum over styles, not a weighted mixture:

$$u(x) = \max_{k \in 1..K} ; \theta_k^\top z(x), \qquad z(x) = \frac{\varphi(x) - \mu}{s}$$

At $K = 1$ this reduces exactly to Bayesian linear regression on $z$, which is a useful property: the mixture is a strict generalization with no special-casing at the boundary.

One posterior draw’s utility (TasteSample::utility_mix):

pub fn utility_mix(&self, phi: &[f64]) -> f64 {
    self.theta.iter()
        .map(|t| dot(t, phi))
        .fold(f64::NEG_INFINITY, f64::max)
}

Why a maximum

Taste is multi-modal. One person can love dark drones and bright plucks (“ambient-me” and “acid-me”), and those are not points on one axis. A single linear utility would average them into a preference for neither, and would then be confidently wrong about both.

The max form gives each island its own style, and every judgment, including a duel across two islands, compares candidates on the shared scale $u = \max_k u_k$. A dark drone and a bright pluck are both scored, each by whichever style rates it highest, and the comparison is well-formed.

Two rejected designs, and why

A per-session style latent $z_s$

“One mood per session: sample which style is active, then use it.”

Fails because it cannot represent several islands inside a session. A user who auditions a pad, then a bass, then a pad in one sitting is not switching moods; they have two tastes at once. Whenever the session’s latent is wrong for the current candidate, every observation in that session is scored by the wrong style.

A per-observation marginalized style

“Marginalize over which style judges each observation.”

Fails on a sharper point: it forces both duel items through the same style, so a cross-island comparison is unrepresentable. There is no style under which “the drone beats the pluck” is a sensible statement if the drone lives in style 1 and the pluck in style 2, and a duel between them is exactly the question the acquisition rule will ask.

This is not a theoretical objection. A synthetic bimodal user exposed it: the marginalized mixture failed to beat $K = 1$. Adding capacity made the model no better, which is the signature of capacity the likelihood cannot use.

What max-utility buys structurally

There are no discrete latent sites at all. No style assignment to sample, no categorical variables, no label-switching during inference to fight. Every site in the model is an f64, which means fugue’s generic adaptive single-site MH applies unchanged, with no custom kernel and no Rao-Blackwellization.

Label permutation is resolved post hoc instead, by TastePosterior::aligned.

$K$ is an upper bound, not a claim

$K = 5$ by default (SessionConfig::k_styles, in engine.rs), and the fitted number of live styles grows with evidence.

Nothing enforces that; it falls out. A style with no evidence to explain stays near its prior, and style_share reports what fraction of the pool each style actually claims as its best. A style claiming ≈0% is idle: the player’s taste has fewer islands than $K$, and the app dims it rather than inventing a name for it.

So $K$ is capacity, and the data decides how much gets used.

The prior, and the correction $K$ forces

$$\theta_{k,j} \sim \mathcal{N}(0, \sigma_\theta^2), \qquad \sigma_\theta = \frac{1}{\sqrt{d}; s_K}$$

The $1/\sqrt{d}$ factor is standard: with $\lVert z \rVert^2 \approx d$ for a standardized vector, it makes the prior utility of a candidate roughly unit-variance, so likelihood scales stay sane at any feature count.

The $s_K$ factor is the correction the max form forces, and it is easy to miss.

Under the prior each $u_k$ is marginally $\mathcal{N}(0,1)$, so $u = \max_k u_k$ is the maximum of $K$ iid standard normals, whose standard deviation falls with $K$ (MAX_NORMAL_SD, in auracle-taste’s model.rs):

The mean shift cancels in duels (both sides shift equally) and is absorbed by $\tau$ and the cutpoints elsewhere. The variance shrinkage does not cancel. Left uncorrected, $\mathrm{Var}(u_a - u_b)$ drops from 2.0 at $K=1$ to 0.90 at $K=5$, so growing $K$ mid-session would quietly make the model less able to express a strong preference.

That is the opposite of what adding capacity should do, and a player would meet it as a model that grows vaguer the longer they play.

Dividing by $s_K$ restores invariance: $\mathrm{Var}(u_a - u_b)$ is the same at every $K$.

What the interface reads off this

responsibilities is a posterior distribution over which style applies, which is strictly more informative than an argmax and is what lets a candidate sit visibly between two islands.