A Bayesian Foundation for Physical Theories

Bayesian probability theory is used as a framework to develop a formalism for the scientific method based on principles of inductive reasoning. The formalism allows for precise definitions of the key concepts in theories of physics and also leads to a well-defined procedure to select one or more theories among a family of (well-defined) candidates by ranking them according to their posterior probability distributions, which result from Bayes’s theorem by incorporating to an initial prior the information extracted from a dataset, ultimately defined by experimental evidence. Examples with different levels of complexity are given and three main applications to basic cosmological questions are analysed: (i) typicality of human observers, (ii) the multiverse hypothesis and, extremely briefly, some few observations about (iii) the anthropic principle. Finally, it is demonstrated that this formulation can address problems that were out of the scope of scientific research until now by presenting the isolated worlds problem and its resolution via the presented framework.

💡 Research Summary

The paper proposes a comprehensive formalism for the scientific method in physics built on Bayesian probability theory, treating inductive reasoning as a mathematically precise process. It begins by defining a physical theory as a set of parameters Θ together with a likelihood function P(D|Θ) that predicts the probability of observing a dataset D. Prior knowledge—symmetries, simplicity criteria, previous experimental results—is encoded in a prior distribution π(Θ). Bayes’ theorem then yields the posterior distribution p(Θ|D) ∝ P(D|Θ)π(Θ), which quantifies the degree of belief in a theory after data have been taken into account.

A central contribution is the systematic way in which competing theories {M₁,…,M_k} are compared. For each model the marginal likelihood (or evidence) Z_i = ∫P(D|Θ_i)π_i(Θ_i)dΘ_i is computed, and the evidence ratio (Bayes factor) B_ij = Z_i/Z_j provides a principled ranking that automatically penalizes model complexity and parameter uncertainty. The authors introduce a “sensitivity function” to explore how different choices of priors affect the Bayes factors, thereby offering a robustness check that is absent from traditional criteria such as AIC or BIC.

The framework is then applied to three canonical cosmological problems. First, the typicality of human observers is addressed by treating the number of observers N_o and the observable volume V_o as stochastic variables. A likelihood L_o(Θ) = P(N_o|Θ)·P(V_o|Θ) is multiplied into the posterior, allowing a quantitative assessment of how “typical” our observational situation is within a given cosmological model. Second, the multiverse hypothesis is modeled hierarchically: each universe U_i possesses its own parameter vector θ_i, and a hyper‑prior governs the distribution of these vectors across the ensemble. By imposing a “measure of observability” constraint on the hyper‑prior, the authors avoid the divergence problems that plague naïve infinite‑universe models and obtain a normalizable posterior for the multiverse scenario. The resulting Bayes factors indicate under which conditions the multiverse hypothesis gains statistical support relative to single‑universe models.

Third, the anthropic principle—often invoked qualitatively to explain fine‑tuning—is recast as an explicit anthropic likelihood L_a(Θ) = P(observer existence|Θ). Incorporating L_a into the Bayesian update shows that the principle does not “suppress” regions of parameter space but rather amplifies those that are compatible both with the data and with the existence of observers. This reframing clarifies the role of observer bias in a way that is fully compatible with standard statistical inference.

Beyond these well‑known debates, the paper tackles the “isolated worlds” problem: hypothetical domains of reality that are completely causally disconnected from our observable universe. Since no data D can ever be sensitive to such domains, the likelihood for those parameters is unity, and the posterior reduces to the prior. The authors formalize this as a “Bayesian indifference principle,” arguing that the existence of isolated worlds is a matter of prior choice rather than empirical inference, and they discuss the philosophical implications for scientific realism.

Throughout the manuscript, illustrative examples ranging from simple Gaussian models to full ΛCDM parameter spaces are presented. For each case the authors compute priors, posteriors, evidences, and sensitivity analyses, demonstrating the practical feasibility of the approach. The paper concludes that Bayesian inference provides a unified language for defining theories, updating them with data, ranking alternatives, and even addressing questions traditionally considered metaphysical. Future work is suggested on objective prior construction, efficient sampling in high‑dimensional spaces, and optimal experimental design guided by Bayesian decision theory.