Subjective Function Overview

Updated 19 December 2025

Subjective function is an agent-specific mapping from internal states—such as beliefs and cognitive parameters—to numerical values.
It underpins decision-making in fields like reinforcement learning, psychology, and economics by reflecting intrinsic valuation and learning error.
Mathematical formulations include nonlinear models and information measures that enable dynamic goal adaptation and policy evaluation.

A subjective function is a mapping whose specification and evaluation are endogenous to an agent’s own internal features, beliefs, or psychological states, rather than being imposed externally. Subjective functions have emerged across artificial intelligence, decision theory, psychology, economics, information theory, policy evaluation, and the cognitive sciences as crucial constructs for modeling autonomous agency, preference formation, learning progress, belief updating, and subjective evaluation. Their mathematical formalization, empirical estimation, and practical applications span a wide variety of domains.

1. Formal Definition and Endogeneity

A subjective function is defined as an agent-specific, higher-order utility or objective whose value can be computed only with reference to the agent’s current internal state (feature vector, weights, beliefs, cognitive parameters, etc.), and not merely as a function of the external environment or an exogenous target. Formally, given an agent with internal feature space $\mathcal{F}$ , a subjective function is a mapping

$S: \mathcal{F} \longrightarrow \mathbb{R}, \qquad \phi \mapsto S(\phi)$

where $\phi$ encodes the agent's internal parameters. When nested in reinforcement learning or planning with self-generated goals, one typically considers $U_g^\pi:\mathcal S \times \mathcal{F} \to \mathbb{R}$ , where $U_g^\pi(s, \phi)$ measures, for a given goal $g$ , policy $\pi$ , and agent state $s$ , a subjective utility computed via $\phi$ . The essential property is endogeneity: the function’s value changes with the agent’s own features, beliefs, and evolving models (Gershman, 17 Dec 2025).

Subjective functions differentiate from exogenous objective functions, which are fixed externally (e.g., task reward, penalty, loss) and are independent of the agent’s feature space. Classical paradigms posit an external reward structure, whereas subjective functions are internal and agent-relative.

2. Illustrative Examples in Learning and Decision Theory

One canonical construction is the expected prediction error (EPE)—the discrepancy between true and estimated value functions or outcomes, as assessed by the agent itself. In reinforcement learning, let $\hat V^\pi_\phi(s)$ denote an agent’s internal estimator under policy $\pi$ . The temporal-difference error: $\delta_t(\phi) = R_g(s_t) + \gamma \hat V^\pi_\phi(s_{t+1}) - \hat V^\pi_\phi(s_t)$ leads to a subjective function

$U_g^\pi(s, \phi) = \mathbb{E} \left[\sum_{t=0}^\infty \gamma^t \delta_t(\phi) \mid s_0 = s \right]$

where the value depends on the agent’s $\phi$ and thus is strictly subjective. Telescoping yields $U_g^\pi(s, \phi) = V_g^\pi(s) - \hat V^\pi_\phi(s)$ , showing how subjective valuation tracks the gap between reality and the agent’s model.

In the context of subjective expected utility (SEU), the "subjective" probability measure $\mu$ on events $E \subset S$ is not externally determined, but derived from the agent’s own discounting of time or risk, and calibrated via time-equivalent trades (Bastianello et al., 2024). For acts $f: S \times T \to X$ , the SEU functional,

$U(f) = \int_0^\infty e^{-\lambda t}\left( \int_S u(f(s,t))\,d\mu(s) \right) dt,$

embeds subjective beliefs within the agent’s temporal and probabilistic preferences.

Subjective functions are foundational in psychological wellbeing modeling. The nonlinear response-function model for subjective well-being (RFSWB) integrates multiple psychological dimensions through partial response functions: $IISW_{\rm mod} = IISW_{\max} \cdot \prod_{j=1}^{9} f_j(x_j)$ where $x_j$ are scores on psychometric scales (inner-directedness, sociotropy, neuroticism, etc.) and $f_j$ are empirical, often nonlinear, dose-response curves calibrated to the individual (Malkina-Pykh, 2015). The resultant composite, multiplicative function forms a subject-specific predictive surface for well-being, sensitive to intervention (e.g., therapy) and to variation across agents.

In subjective logic, belief/opinion update functions such as

$\omega^{A[t+1]}_X = \omega^{A[t]}_X \oplus (\omega^A_B \otimes \omega^{B[t]}_X)$

govern the evolution of subjective opinions under trust and consensus dynamics. These update “subjective functions” lack universal rationality properties—idempotence, monotonicity, and convergence—but can model phenomena such as radicalization and consensus beyond purely Bayesian or logical fusion (Alvim et al., 2024).

4. Subjective Functions in Policy, Economic, and Information Domains

Subjective functions operationalize resident well-being and value alignment in policy evaluation. For instance, a subjective target function is derived by regressing self-reported well-being on individual, social, ecological, and economic features: $U_s(\mathbf{x}) = \beta_0 + \sum_{i=1}^8 \beta_i x_i$ This is combined with objective multi-agent simulation indices to form a coupled target function used for optimization over policy alternatives (Owa et al., 2023). These composite subjective-objective functions capture pluralistic values in multi-stakeholder environments.

In production economics, subjective expectations data from managers are employed in estimation procedures that supplant exogenous optimization assumptions. Structural models utilize subjectively expected future output and inputs: $y_{it} = f(k_{it}, l_{it}; \beta) + \omega_{it} + \epsilon_{it}$ with $\omega_{it}$ inferred from

$g(\omega_{it}) = E_{it}[y_{it+1}|\Omega_{it}] - \int f(k_{it+1}, l; \beta)dF_{it}(l)$

leading to robust estimation even under non-optimal input selection (Keiller et al., 2024).

In information theory, subjective information measures generalize the classical formulation. For messages $y_j$ inducing fuzzy sets $A_j$ , the subjective information of $x_i$ given $y_j$ is

$I(x_i;y_j) = \log \frac{Q(A_j|x_i)}{Q(A_j)}$

and subjective mutual information is

$I_s(X;Y) = \sum_{i,j} P(x_i,y_j)\;\log\frac{Q(A_j|x_i)}{Q(A_j)}$

which evaluates information relative to the agent’s graded set-membership model $Q$ , not just statistical probability. Subjective rate-fidelity functions thus optimize coding relative to an agent’s perception or semantic model (0705.3644).

5. Mechanisms for On-the-Fly Construction and Adaptation

Agents equipped with subjective functions can synthesize new objectives and adapt goals dynamically. The process is bi-level: the agent first selects an internal policy $\pi^*_g = \arg\max_\pi U_g^\pi$ , then chooses a goal $g^* = \arg\max_g U_g^{*,}(s_0, \phi)$ . Such mechanisms underpin intrinsic motivation, skill acquisition, lifelong learning, and curiosity-driven exploration in artificial agents (Gershman, 17 Dec 2025). The definition of the subjective function itself can change as the agent learns, enabling generalization beyond fixed-task scenarios.

In cognition and neurobiology, subjective functions relate to observed phenomena such as hedonic adaptation, preference for increasing reward sequences, dopamine-mediated prediction error signaling, and exploratory behavior—all of which entail domain- and agent-specific valuation of stimuli or experiences.

6. Mathematical, Geometric, and Entropic Structures

Subjective functions can take diverse mathematical forms depending on context. In quantum-inspired evaluation, an individual’s preference state is modeled as a Bloch sphere vector $\lvert\psi\rangle$ with the subjective function

$f(\lvert\psi\rangle) = \cos^2(\frac{\theta}{2})$

mapping cognitive state to a scalar propensity-to-like score. Cohesion, polarization, and uncertainty are then analyzed using Shannon and von Neumann entropy over individual and group-level state mixtures, generalizing subjective evaluation to a geometric- and entropic-framework (Soodchomshom, 2 Jun 2025).

Multiplicative models (e.g., the RFSWB model), nonlinear composite functions, and partial response surfaces commonly appear in psychological and multi-variable contexts. In production and econometric estimation, subjective functions may involve nonparametric monotonic transforms (e.g., using shape-constrained P-splines for unknown functions $\Psi$ constrained by survey expectations (Keiller et al., 2024)).

7. Implications, Limitations, and Research Directions

The subjective function paradigm enables open-ended agency, enhanced adaptability, and more realistic models of human-like behavior and value formation. In machine learning, it supports scalable meta-learning architectures where reward, goal, or utility functions are updated as a function of task mastery and ongoing internal modeling. In policy and economics, subjective function estimation better accommodates diversity of stakeholder values, dynamic reallocation, and adjustment to incomplete optimization.

Key limitations include the challenge of intersubjective calibration (how to relate different agents’ subjective functions), difficulties in ensuring convergence and rationality of update dynamics, and the increased complexity of constructing functions that are well-aligned with externally desirable outcomes.

Ongoing research is systematizing subjective function construction in reinforcement learning, formalizing subjective Bayesianism in decision theory, integrating subjective rating spaces with geometric and entropic quantifications, and leveraging large-scale subjective data in empirical economics, social science, and organizational policy (Gershman, 17 Dec 2025, Bastianello et al., 2024, Keiller et al., 2024, 0705.3644, Malkina-Pykh, 2015, Soodchomshom, 2 Jun 2025, Alvim et al., 2024, Owa et al., 2023).

Key References Table

Domain	Paper (arXiv)	Subjective Function Instantiation
Learning/Agency Theory	(Gershman, 17 Dec 2025)	Endogenous mapping $S(\phi)$ , EPE
Psychometric Well-being	(Malkina-Pykh, 2015)	Multiplicative nonlinear RFSWB model
Decision Theory (SEU)	(Bastianello et al., 2024)	Time-calibrated subjective probability
Quantum-inspired Evaluation	(Soodchomshom, 2 Jun 2025)	Bloch sphere map $f((\theta,\phi))$
Production/Econometrics	(Keiller et al., 2024)	Expectation-based estimator via subjective CDFs
Information Theory	(0705.3644)	Generalized/fuzzy-set information functions
Opinion Dynamics/Social Netw.	(Alvim et al., 2024)	SL belief update via cumulative fusion
Policy/Values Integration	(Owa et al., 2023)	Subjective–objective target coupling