One if by Land, Two if by Sea, Three if by Four Seas, and More to Come -- Values of Perception, Prediction, Communication, and Common Sense in Decision Making

Technical Summary: Values of Perception, Prediction, Communication, and Common Sense in Decision Making

Problem Formulation
This work addresses the challenge of rigorously defining and quantifying the value of specific cognitive operations—perception, prediction, communication, and common sense—within the context of autonomous decision-making systems. The authors frame a generic decision-making problem where an agent selects an action $u$ from a space $\mathcal{U}$ to minimize the expected loss $\mathbb{E}[\ell(X, Y, u)]$. Here, $X$ represents a random observable state, $Y$ represents a random unobservable state, and $\ell$ is a loss function dependent on both states and the action. The core problem involves determining the quantitative benefit of acquiring and utilizing different types of knowledge (marginal distributions, joint distributions, observed states, or unobserved states) to reduce the minimum achievable expected loss (risk).

Methodology
The paper adopts a decision-theoretic framework to define the "value" of an operation as the reduction in risk achieved by incorporating that operation compared to a baseline where the operation is absent. The methodology proceeds as follows:

1. Definitions of Operational Values:

   • Common Sense: Defined as the value of knowing the joint distribution $P_{X,Y}$ (statistical dependence) compared to assuming independence ($P_X P_Y$). It quantifies the benefit of reasoning about correlations without direct observation.
   • Perception (Narrow): Defined as the value of observing $X$ and acting based on it, without updating beliefs about $Y$ based on $X$ (i.e., treating $Y$ as independent of $X$).
   • Prediction: Defined as the value of updating the belief about $Y$ based on the observation of $X$ (utilizing the posterior $P_{Y|X}$) compared to acting on $X$ while ignoring the dependence on $Y$.
   • Perception + Prediction: The combined value of observing $X$ and predicting $Y$ based on $X$.
   • Communication: Defined as the value of accessing the unobservable state $Y$ directly (in addition to $X$) to inform the decision.

2. Information-Theoretic Realization:
   The authors map these decision-theoretic values to information-theoretic measures by considering a specific case where the loss function is the negative log-likelihood ($\ell = -\log u(x,y)$). In this setting, the defined values correspond to:

   • Common Sense $\leftrightarrow$ Mutual Information $I(X; Y)$.
   • Perception $\leftrightarrow$ Conditional Entropy $H(X|Y)$.
   • Prediction $\leftrightarrow$ Mutual Information $I(X; Y)$.
   • Perception + Prediction $\leftrightarrow$ Entropy $H(X)$.
   • Communication $\leftrightarrow$ Conditional Entropy $H(Y|X)$.

3. Functional Analysis:
   The paper analyzes the mathematical properties of these values, specifically their concavity and convexity with respect to the underlying probability distributions. It utilizes decision-theoretic divergences (generalizations of Kullback-Leibler divergence) to prove these properties.

4. Multi-Agent Extension:
   The framework is extended to scenarios involving $n$ agents. The authors define the importance of a specific agent based on the risk reduction gained by observing and predicting that agent, either in isolation or sequentially after other agents have been accounted for.

Key Results and Findings

• Non-Negativity of Combined Values: The value of prediction alone, the value of communication, and the combined value of perception and prediction are proven to be always non-negative.
• Negative Value of Perception: A critical finding is that the value of perception without prediction can be negative. The paper demonstrates through a numerical example (and a driving scenario involving a truck and an occluded vehicle) that observing a variable $X$ and acting on it without updating the belief about the dependent variable $Y$ can lead to worse decisions than acting solely on prior knowledge (common sense). This suggests that raw sensory data, if not properly contextualized through prediction, can be detrimental.
• Order Dependence in Multi-Agent Systems: In multi-agent settings, the importance of an agent is not absolute but depends on the order of observation. The paper provides a greedy algorithm to determine the optimal order of perception and prediction to maximize risk reduction, relying solely on the known joint distribution and loss function.
• Mathematical Properties:
  • The value of perception + prediction is concave in the marginal distribution $P_X$ (for fixed $P_{Y|X}$).
  • The value of communication is concave in $P_{Y|X}$ (for fixed $P_X$) and linear in $P_X$.
  • The values of common sense and prediction are convex in the conditional distribution $P_{Y|X}$ (for fixed marginals).

Significance and Claims
The authors claim that this work provides a rigorous, formal path to answer practical questions in the design of autonomous systems, such as:

• Whether observing and predicting a specific agent is necessary.
• The relative importance of different agents.
• The optimal sequence for processing information from multiple agents.

The paper posits that these defined quantities offer insights into cognitive and neural sciences, potentially explaining how natural decision-makers utilize information from different sources. Furthermore, the results suggest that the "value of information" found in earlier literature (e.g., Blackwell, Lindley, DeGroot) can be viewed as a special case of the combined value of perception and prediction defined here. The work also proposes that these theoretical values could serve as "ground truth" labels for training machine learning models that prioritize agents in resource-constrained environments, addressing a gap where current methods often rely on human labels or heuristic rules.

The title and framing are inspired by a talk by Prof. Robert Gallager, drawing a parallel to the historical signal system ("One if by Land, Two if by Sea") to illustrate the hierarchy of information value in decision-making.