Base Model And Reference Notation

This note records the notation and prose conventions for the base model, frozen reference model, and z-free base-model surprisal.

Core Convention

Use $p_\theta(y|x)$ as the primary notation for the base model: the conditional language model whose response distribution we seek to factorize.

Use base model as the prose term for $p_\theta$.

Use frozen reference copy of $p_\theta$ when discussing scoring, weighting, or normalization quantities computed from a nonmoving copy of the base model.

Use z-free base-model surprisal for losses computed under $p_\theta(y|x)$ without conditioning on a latent variable.

Reference phrasing:

Let $p_\theta(y|x)$ denote the base conditional language model whose distribution we seek to factorize. We learn a router $r_\phi(z|x)$ and strategy-conditioned generator $g_\phi(y|x,z)$ such that marginalizing over $z$ preserves $p_\theta(y|x)$ while exposing strategy structure.

Why Use $p_\theta$ Rather Than $p_{\mathrm{base}}$

The notation $p_\theta$ emphasizes that the target conditional distribution is defined by a model with parameters $\theta$, not by an external data distribution.

This matters because the paper studies factorization of an already-capable model. The base distribution is not merely an empirical data law; it is the distribution induced by the pretrained or task-trained language model at parameter point $\theta$.

The notation also makes the adaptation story visible. The learned router-generator parameterization uses parameters $\phi$, initialized from or built on top of the base parameters $\theta$. This supports parameter-space language such as:

$$\phi \approx \theta + \delta_\phi,$$

where $\delta_\phi$ denotes the adaptation parameters schematically.

Use this relation as intuition, not as a literal universal equality. In LoRA settings, the more literal form is that adapted weights satisfy

$$W_\phi = W_\theta + \Delta W_\phi,$$

with $\Delta W_\phi$ constrained to be low rank. Other added components such as heads, latent projections, or special embeddings may not be expressible as a simple additive perturbation to every base parameter.

Reference phrasing:

In pretrained-model experiments, the router and generator are initialized from the base parameters $\theta$ and adapted by a small set of additional parameters $\delta_\phi$; schematically, $\phi$ denotes the base model plus this adaptation.

Parameter Roles

Use the following parameter convention:

• $\theta$: fixed base model parameters defining $p_\theta(y|x)$.
• $\phi$: learned router-generator parameters defining $r_\phi(z|x)$ and $g_\phi(y|x,z)$.
• $\xi$: training-time posterior parameters defining $q_\xi(z|x,y)$.

The compact notation should keep this separation visible:

$$p_\theta(y|x) \quad\leadsto\quad \int r_\phi(z|x) g_\phi(y|x,z)\,d\mu(z).$$

Reference phrasing:

We write $\theta$ for the fixed base model, $\phi$ for the adapted router-generator factorization, and $\xi$ for the training-time posterior.

Factorized-Model Distribution Notation

Use $p_\phi$ as compact notation for the distribution family induced by the router-generator factorization. The arguments indicate which component or marginal is being used:

$$p_\phi(z|x) := r_\phi(z|x), \qquad p_\phi(y|x,z) := g_\phi(y|x,z).$$

The joint and latent-marginal conditionals are then:

$$p_\phi(y,z|x) := p_\phi(z|x)p_\phi(y|x,z), \qquad p_\phi(y|x) := \int p_\phi(z|x)p_\phi(y|x,z)\,d\mu(z).$$

This notation is useful for distributional-fidelity statements:

$$p_\phi(y|x) \approx p_\theta(y|x).$$

It also emphasizes that the router and generator have different probabilistic roles but are implemented as interfaces of the same adapted model parameterization $\phi$.

Important convention:

Here $p_\phi(y|x)$ denotes the latent-marginal distribution of the factorized model, not a z-free pass through the adapted generator.

If the learned generator is evaluated without latent conditioning, call that the z-free generator path or generator without latent conditioning.

Frozen Reference Copy

When the objective computes base-model scores, use frozen reference copy of $p_\theta$. This makes clear that the surprisal or weighting signal is not moving with the learned generator.

Reference phrasing:

We compute base-model surprisal using a frozen reference copy of $p_\theta(y|x)$.

The frozen reference model provides the z-free surprisal scale used to weight or normalize reconstruction.

Avoid using base model ambiguously for both the fixed target distribution and a z-free path through the learned adapted model. If the learned generator is evaluated without $z$, call that the z-free generator path or generator without latent conditioning, not the base model.

z-Free Base-Model Surprisal

Use z-free base-model surprisal for the negative log probability assigned by the base model to $y$ given $x$:

$$b_\theta(x,y) = -\log p_\theta(y|x).$$

For token-level quantities, use:

$$b_{\theta,t}(x,y) = -\log p_\theta(y_t|x,y_{<t}).$$

The phrase z-free means that the score is computed without conditioning on a latent variable. In objective sections, z-free is shorter and clearer than latent-free once $z$ has been introduced.

Reference phrasing:

We use a frozen reference copy of the base model to compute z-free surprisal, which provides the scale and weighting signal for the reconstruction objective.

Entangled Base Model

Use entangled base model when making the scientific point that the base model mixes strategies implicitly.

Reference phrasing:

The entangled base model already represents multiple strategies implicitly, but does not expose a separable strategy variable in its generative procedure.

Do not use entangled model as the general technical name everywhere. The neutral technical term is base model; entangled base model is interpretive.

Adaptation Language

Use adapted-from-base for experiment conditions initialized from the base model.

Use random-reinit or randomly reinitialized for the contrast condition.

Use factorizing a base model or adapting the base model into a router-generator factorization for the methodology.

Avoid post-training as the main term because it is overloaded with RLHF and SFT. Avoid distillation unless explicitly contrasting the present objective against mere imitation of $p_\theta(y|x)$.

Reference phrasing:

The learned generator is initialized from the base model, so at initialization it can predict $y$ from $x$ without using $z$. This makes the z-ignoring factorization immediately available.

Terms That Need Care

• $p_{\mathrm{base}}$: readable, but not the main notation; it hides the parameter-space relationship between $\theta$ and $\phi$.
• reference model: useful for frozen scoring copies, but should not replace base model as the target distribution.
• source model: implementation-flavored and can sound like distillation or transfer learning.
• latent-free: acceptable synonym for z-free, but less compact after $z$ is introduced.
• post-training: overloaded with RLHF/SFT.
• distillation: overlaps with sampling from or matching the base model, but misnames the factorization goal.