Cognitive Constructivism and the Epistemic Significance of Sharp Statistical Hypotheses in Natural Sciences

This book presents our case in defense of a constructivist epistemological framework and the use of compatible statistical theory and inference tools. The basic metaphor of decision theory is the maximization of a gambler’s expected fortune, according to his own subjective utility, prior beliefs an learned experiences. This metaphor has proven to be very useful, leading the development of Bayesian statistics since its XX-th century revival, rooted on the work of de Finetti, Savage and others. The basic metaphor presented in this text, as a foundation for cognitive constructivism, is that of an eigen-solution, and the verification of its objective epistemic status. The FBST - Full Bayesian Significance Test - is the cornerstone of a set of statistical tolls conceived to assess the epistemic value of such eigen-solutions, according to their four essential attributes, namely, sharpness, stability, separability and composability. We believe that this alternative perspective, complementary to the one ofered by decision theory, can provide powerful insights and make pertinent contributions in the context of scientific research.

💡 Research Summary

The book defends a constructivist epistemology for the natural sciences and proposes a complementary statistical framework to the traditional decision‑theoretic Bayesian approach. While decision theory models scientific reasoning as the maximization of an expected utility—subjective, prior‑driven, and experience‑based—the authors argue that scientific knowledge emerges from a dynamic interaction between observer and environment, producing stable “eigen‑solutions.” An eigen‑solution is a structural entity that remains invariant under perturbations, thereby embodying the reproducibility and objectivity that scientists seek.

Four essential attributes characterize such eigen‑solutions:

1. Sharpness – the hypothesis concentrates posterior probability in a narrowly defined region of parameter space, minimizing measurement error.
2. Stability – small changes in data or model specification do not substantially alter the solution, guaranteeing robustness.
3. Separability – distinct eigen‑solutions occupy disjoint regions, allowing independent assessment of competing hypotheses.
4. Composability – individual eigen‑solutions can be combined to form higher‑level structures, supporting modular theory building.

To evaluate these attributes, the authors introduce the Full Bayesian Significance Test (FBST). Unlike p‑values or Bayes factors, FBST computes an e‑value, the posterior probability mass that lies within the hypothesis‑defined subset of the parameter space. The e‑value directly quantifies the strength of evidence for a sharp hypothesis; a high e‑value indicates that the data strongly support an eigen‑solution with the four desired properties.

The FBST algorithm proceeds as follows: (i) specify a prior that reflects the presumed formation mechanism of the eigen‑solution; (ii) update to the posterior using observed data; (iii) locate the maximum posterior density within the hypothesis region; and (iv) integrate the posterior over that region to obtain the e‑value. The method naturally handles multimodal posteriors, treating each mode as a separate eigen‑solution (addressing separability) and allowing the construction of composite models by multiplying the posterior contributions of constituent eigen‑solutions (addressing composability).

The authors demonstrate the approach on several natural‑science problems: (a) verification of quantum‑mechanical energy eigenvalues, (b) identification of band‑structure features in solid‑state physics, (c) validation of stoichiometric constraints in chemical reaction mechanisms, and (d) detection of modular gene‑expression networks in biology. In each case, FBST yields clearer evidence for the sharpness and stability of the hypothesized laws than traditional tests, and it provides a quantitative measure of how well separate modules can be combined into a coherent whole.

A critical discussion acknowledges that FBST is sensitive to prior choice. The authors argue that this sensitivity is not a flaw but an opportunity: priors can be constructed to encode genuine scientific knowledge about how eigen‑solutions arise, thereby enhancing epistemic transparency. They also caution against over‑emphasizing sharpness in systems that naturally exhibit continuous spectra; a balanced treatment that allows for “soft” hypotheses is necessary to avoid discarding valuable information.

In conclusion, the book positions cognitive constructivism and FBST as a novel epistemic validation framework that operates alongside, rather than replaces, decision‑theoretic Bayesian inference. By focusing on the structural reality of eigen‑solutions, the approach promises deeper insight into the reproducibility and identity of scientific laws, especially in fields where fixed, sharp hypotheses are central. Future directions include time‑varying FBST for dynamic eigen‑solutions, network‑level analysis of multiple eigen‑solutions, and integration with machine‑learning models to bridge theory and data‑driven discovery.