Bayesian Meta-Theory: Scientific Inference Framework

• Bayesian Meta-Theory is a rigorous framework for formalizing inductive reasoning and the scientific method using Bayesian probability to rank and test competing theories.
• It operationalizes key epistemological concepts by precisely defining theory, falsifiability, and complexity within a probabilistic model.
• The framework iteratively updates priors with new data, enabling systematic theory refinement and objective comparison of scientific hypotheses.

Bayesian Meta-Theory is the formal paper of inductive logic, model selection, and theory construction in science under the Bayesian paradigm. It provides precise mathematical foundations for the scientific method, articulates how theories are ranked and falsified, and operationalizes key epistemological concepts within a probabilistic framework based on Bayes’s theorem.

1. Bayesian Inductive Reasoning and Scientific Method

Bayesian meta-theory asserts that Bayesian probability is the unique, mathematically consistent extension of deductive logic for reasoning and learning under uncertainty. The scientific method is formalized as an iterative algorithmic process consisting of three steps:

1. Information Acquisition: Gathering data via measurement or reasoning.
2. Modeling: Forming candidate scientific theories to explain the data.
3. Testing and Selection: Ranking theories by updating priors with data and selecting according to their posterior probabilities.

This process is rendered explicit by Bayes’s theorem: $$P(T_i | D_t) = \frac{P(D_t|T_i)\,P(T_i)}{\sum_j P(D_t|T_j) P(T_j)}$$ where $T_i$ are candidate theories, $D_t$ is observed data, $P(T_i)$ is the prior (encoding preferences), and $P(D_t|T_i)$ the likelihood. Posterior probabilities dictate theory preference and relegate the role of observation in scientific advance to probabilistic inference (Alamino, 2010).

2. Mathematical Precision in Theory Definition

Bayesian meta-theory provides precise definitions for previously ambiguous concepts in theoretical science:

• Theory: Formally, $T \equiv (\alpha, \pi)$, with $\alpha$ an algorithmic structure (axioms) and $\pi$ a set of fundamental constants. The theory must generate $P(D_t|T)$.
• Falsifiability: Explicitly, $T$ is falsifiable if $\exists D_t$ such that $P(D_t|T) = 0$.
• Fundamental Status: A theory addressing an identical question set is more "fundamental" if characterized by fewer constants.
• Complexity: Quantified by the Kolmogorov complexity of $\alpha$, independently from the number of constants in $\pi$.
• Power of Theory: Measured by the cardinality of the question set the theory satisfactorily answers.

These definitions transcend subjective or philosophical argument, enabling unambiguous operationalization in scientific discourse.

3. Bayesian Theory Selection and Ranking

Bayesian theory selection is viewed as probabilistic updating over a family of candidate theories. The selection process, generalized, is:

• Prior: $P(T_i)$ encodes meta-theoretical criteria such as simplicity (via penalizing algorithmic complexity) and fundamental status (preferring theories with fewer constants):

$P(T) = P(\alpha, \pi) = P(\pi | \alpha)\,P(\alpha)$

• Likelihood: $P(D_t|T_i)$ quantifies how likely observed data are under $T_i$.
• Posterior: $P(T_i|D_t)$ is used for rational decision and selection.

Parameter uncertainty is marginalized: $P(\alpha|D_t) = \int P(\alpha, \pi|D_t)\;d\pi$ Bayesian meta-theory thus operationalizes theory selection as a quantitative ranking and supports continual theory refinement and replacement.

Concept	Formal Definition/Fomula	Role in Meta-Theory
Theory ranking	$$P(T_i|D_t) = \frac{P(D_t|T_i)P(T_i)}{\sum_j P(D_t|T_j)P(T_j)}$$	Algorithmic selection
Falsifiability	$\exists D_t\ \mathrm{s.t.\ } P(D_t|T) = 0$	Criterion for scientific status
Marginalization	$P(\alpha|D_t) = \int P(\alpha, \pi|D_t)\,d\pi$	Uncertainty treatment

4. Application to Cosmological and Epistemological Problems

Bayesian meta-theory has been used to formalize prominent scientific problems previously inaccessible to strict empiricist or positivist science:

• Typicality of Observers: Bayesian ranking does not favor "typical" observers a priori; instead, it ranks theories by the probability assigned to the observed data, given explicit question sets and priors.
• Multiverse Hypothesis: Claims that multiverse models are always preferred are rejected; Bayesian comparison considers only the probability assigned to the actual observed data and the prior.
• Anthropic Principle: Recast as a deterministic observation step—any theory inconsistent with observer existence is assigned zero likelihood.
• Isolated Worlds Problem: Bayesian inference permits scientific belief in entities or regions never observed if the top-ranked theory implies their existence with high posterior probability.

5. Extension Beyond Positivism: Expanding Scientific Scope

The Bayesian meta-theoretical framework expands science outside the empirical domain confined by observation. The scope of scientifically meaningful questions is extended to entities and processes inferred probabilistically (e.g., isolated worlds), contingent on their necessity within well-supported theories. Rational inference, not empirical observation alone, is elevated as the central criterion for what constitutes scientific meaning.

6. Algorithmic Perspective and Meta-Scientific Implications

The meta-theory elevates the scientific method to a computational algorithm optimizing over model space:

• Iterate data acquisition, modeling, and Bayesian ranking.
• Incorporate complexity, fundamental status, and falsifiability within the prior.
• Enumerate question sets and explicitly score candidate theories.

Science is formalized as repeated Bayesian induction, with theoretical concepts rendered as variables in an algorithmic process.

7. Foundational Formulas and Criteria

Central mathematical formulations underpinning the meta-theory include:

• Bayes’s theorem for theory ranking

$$P(T_i|D_t) = \frac{P(D_t|T_i)P(T_i)}{\sum_j P(D_t|T_j)P(T_j)}$$

• Falsifiability

$\exists\, D_t \quad \mathrm{s.t.}\quad P(D_t|T)=0$

• Marginalization

$P(\alpha|D_t) = \int P(\alpha, \pi|D_t)\; d\pi$

These expressions provide rigorous, quantitative handles for the otherwise qualitative steps of hypothesis formation and testing.

Conclusion

Bayesian meta-theory formally constitutes the mathematically rigorous logic underlying scientific inference, model selection, and theoretical progress. It operationalizes prior philosophical concepts with explicit definitions and procedural algorithms, resolves issues such as falsifiability and model complexity, and expands the scope of science to regions accessible only by rational inference within probabilistic models. This approach enables objective, quantitative comparison of theories and renders all steps of the scientific method susceptible to systematic mathematical analysis (Alamino, 2010).