Toward Machine Learning with the Unit as a Primitive: Learning from Unit-Linked Events

Heyang Gong

cs.LGAug 25, 2026

Toward Machine Learning with the Unit as a Primitive: Learning from Unit-Linked Events

Authors: Heyang Gong

Abstract

Machine learning is usually formalized through samples, while the persistent individual to which multiple observed or possible events refer often remains implicit. We propose the unit as an explicit primitive at the level of task semantics. A learning task first declares a population of persistent referents and a sameness criterion; the realized value $u$ denotes the selected referent. Supervised learning is the main formal specialization. Its semantic object is a family of unit-conditioned response laws. Homogeneity is the special case in which those laws coincide; a sample-only conditional is silent as to whether the world is homogeneous or the observed law is only the marginal of a heterogeneous family. What is learned from data is a pair $(T_φ,R_θ)$ : a tokenizer that produces a contextual unit token and one shared response-law form that reads it. The structured class takes that form to be a simple relation in the token; a linear predictor is the running instance. The token is the learner-side representation through which the task-side unit affects prediction, while a learner specification that omits unit information is unit-insensitive; homogeneity remains a property of the world-side response family. When identity is unresolved, the world-side law mixes unit-conditioned targets, while the learner composes its shared form with a token. A trusted resolver may fix the unit and supply a lookup token; otherwise unit abduction forms a token of the same type from factual evidence. Unlinked single-row observations can fail to distinguish a heterogeneous unit world from a homogeneous pooled world; trusted same-unit pairs separate a restricted witness. The formal results concern this supervised specialization.

Figures & tables

Panel of Figure 1 — Figure 1: From the unit declaration to linked events and response laws. The task specifies the unit boundary and sameness criterion; $U\sim\Pi$ is the population statement and $U=u$ its realized unit. Several event records may be attributed to the same $u$ , and that same referent indexes a world-side response law. Holding $u$ fixed preserves the referent while allowing context and event variation.

Panel of Figure 2 — Figure 2: The computational assumption for the supervised specialization. (a) A unit-insensitive learner: every unit receives a common token $z_{0}$ , or the shared form ignores the token. (b) Structured heterogeneity: a tokenizer forms contextual unit tokens, and all units share one response-law form $R_{\theta}$ , a simple relation in the token (linear running instance). (c) A saturated class assigns an unrelated law to each unit.

Panel of Figure 3 — Figure 3: Direct unit access and unit abduction return tokens of the same type. A trusted resolver uses a key $k$ to fix the persistent referent and its stable address; lookup then supplies $Z_{u(k)}^{c}$ , typically the learned row $z_{\theta}(k)$ , without forming a which-unit belief. When no resolver identifies the unit, a tokenizer forms a contextual token $Z^{c}$ from factual evidence. A which-unit belief $Q_{\phi}$ may approximate the world-side identity conditional ( Appendix B ). Both access modes use the same learned response model, with the token mode using the shared form $R_{\theta}$ .

Access to $U$	unit-homogeneous response	unit-heterogeneous response
resolved or informative	The referent may be resolved directly, or the formed token may be informative; this response target is invariant across units.	Personalization is substantive; a resolved identity or an informative token changes the prediction.
unresolved; uninformative	Unit uncertainty remains at the population level; the response is unit-invariant.	Oracle predictive value of knowing $U$ may exist, but the current evidence cannot access it; mix, abstain, or collect new evidence.

Table 1: For a unit boundary fixed by the task, learner access to the unit and unit-response dependence are independent axes. “Current evidence” means the evidence declared by the protocol.

Panel of Figure 4 — Figure 4: Single-row marginals do not certify unit structure. Distinct fixed-unit Bernoulli laws and a unit-independent Bernoulli law can both yield $P(Y=1)=1/2$ after marginalizing $U$ . In the restricted witness of Proposition 4 , one response from each fresh unit remains indistinguishable for every sample size, whereas a conditionally independent pair with trusted same-unit linkage reveals within-unit covariance.

Panel of Figure 5 — Figure 5: Repeated units change both the estimand and the split question. Row averaging and unit averaging assign different weights when units contribute unequal numbers of records. A record-wise split can measure a new event for a known unit, whereas a whole-unit split targets generalization to a new unit. The declared deployment question determines the appropriate choice.

Object in the unit formulation	Recommendation interpretation
Population
population unit $U$	possible user individuals
Individual referent
realized unit $u$	one persistent user; not a uniquely identified embedding coordinate
Direct-access model
trusted key $k$	stable referential address that associates interaction records and parameter updates with the same user slot

Table 2: For this worked task, users instantiate the declared unit boundary.

Family	Established object or assumption	Relation to the unit formulation
random effects	A supplied subject or group index and a population model for subject-specific parameters [ 25 , 13 ] .	Direct lookup of a subject token with a shared $R_{\theta}$ ; the random effect is a distributional token, not a private response mechanism.
latent-variable models and mixtures of experts	Incomplete-data estimation, input-dependent gating over experts, and amortized inference over latent codes [ 9 , 20 , 23 ] .	The encoder or gating map is a tokenizer; the decoder or expert family is a shared form, provided the task assigns the latent the role of a persistent unit.
multitask, grouped-latent, Neural-Process, and meta-learning methods	Shared parameters or representations across related tasks, and dataset-, group-, or task-level latents or deterministic context representations derived from an already associated context set [ 6 , 1 , 10 , 5 , 12 , 11 , 16 ] .	An episode or context encoder is a tokenizer and the decoder is a shared form, provided the outer task declares a persistent unit. In the cited methods the context-to-group association is supplied before prediction, which differs from forming a token from otherwise unattributed evidence.
record linkage, entity resolution, and probabilistic identity representations	Posterior linkage structures, learned entity matching, and distribution-valued instance representations [ 41 , 42 , 22 , 26 , 30 , 39 ] .	These methods infer which records share a task-side unit. The present formulation additionally asks for a learner-side token that instantiates one shared $R_{\theta}$ .
recommenders and personalization	Known user indices, latent user factors, history encoders, and user-conditioned response scores [ 28 , 24 , 37 , 18 , 21 , 27 ] .	A known user index provides a lookup token $z_{\theta}(k)$ while the scoring model is a shared $R_{\theta}$ . Unresolved user attribution forms a token of the same type rather than looking it up.
potential outcomes, structural causal models, and Abductive Learning	units under treatment comparisons, counterfactual abduction–intervention–prediction, and logical abduction coupled to learning [ 35 , 32 , 31 , 8 ] .	Here $u$ remains a task-side referent across treatments or worlds. Their abduction objects (noise, logical facts, identity posteriors) are not the unit token of assumption 1 , and each tradition keeps its own estimands and identification conditions.

Table 3: Relations to six established model families.

Appendix figures & tables3 assets

Supplementary material from the paper’s appendix.

Appendix

Symbol	Intended type	Boundary
World and population
$U:\Omega\to\mathcal{U}$	population unit selection variable	which individual in $\mathcal{U}$
$\Pi(du)$	population unit law	which-individual randomness under $P$
Fixed individual and event
$u\in\mathcal{U}$	realized persistent unit	fixed value of $U$ ; indexes a unit-conditioned law but is not synonymous with a learned numerical representation
$K_{u}(dx,dy)$	fixed-individual law	retains exogenous/event variation after $U=u$

Appendix

Table 4: Notation by semantic layer. Learner quantities are listed with the computational pair first;

Q_{\phi}

belongs to the identity-mixture learner specialization.

Case or class	Restriction	Meaning
sample-only	no persistent unit is declared	Learns a row-level law such as $P(dy\mid x)$ ; it makes no direct statement about persistent individuals.
homogeneous unit extension	$K_{u}^{w}=K^{w}$ for $\Pi$ -almost every $u$	After persistent units are declared, unit identity adds no response information at the queries in $\mathfrak{Q}$ .
pure-unit, input-invariant	$K_{u}^{w}=K_{u}$ while $K_{u}$ varies with $u$	Outcomes vary across units while remaining invariant to the supplied query.
structured unit-responsive class	$K_{u}^{w}=K^{w}(\eta,\zeta_{u})$ ; simple relation in a token, linear running instance ( Definition 2 )	Both query and unit may matter. The running instance is a linear predictor $\alpha_{\theta}+\langle\psi_{\theta},Z\rangle$ in the unit token $Z$ ; other declared simple maps of $Z$ instantiate the same class. $\psi_{\theta}$ may be nonlinear in the query $w$ .
saturated unit-responsive class	$(u,w)\mapsto K_{u}^{w}$ is otherwise unrestricted across units	The class permits one arbitrary response family per unit, including homogeneous families; unseen-unit learning requires additional structure or information.

Appendix

Table 5: Response-law cases and model classes inside a unit-explicit task. Standard sample-only learning is listed separately because its basic object does not require a declared unit population.

Unit formalism	DiscoSCM object	Additional causal content
population law $U\sim\Pi$	unit selection variable $U$ with law $P(u)$	Population over individuals participating in a structural causal system.
realized unit $u$	individual retained across factual and counterfactual worlds	The same referent appears in every structural assignment $f_{j}(\cdot,\cdot;u)$ .
fixed-unit event variation	exogenous variables $\mathbf{E}$	A realized event depends on both $(u,\mathbf{e})$ ; fixing $u$ need not make the outcome deterministic.
response law $K_{u}^{w}$	law induced by $V_{j}\leftarrow f_{j}(\operatorname{pa}_{j},E_{j};u)$	The response is generated by a directed mechanism rather than specified only as a conditional kernel.
response query $x^{Q}$	intervention value $x$ in $do(X=x)$	Reading a query as an intervention requires the causal graph, structural assignments, and an intervention semantics.
factual evidence $\mathcal{O}$	an observed factual trace	Evidence updates unit information about the already-realized unit; DiscoSCM represents that update as an identity posterior, which is one specialization of a formed token. The trace may also contain variables with separately declared direct response roles.

Appendix

Table 6: Object-level correspondence between the unit formalism and DiscoSCM. The final column records structure contributed by the causal specialization.

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