Probability in biology: overview of a comprehensive theory of probability in living systems

The Objective Bayesian Probability that an Unknown Positive Real Variable Is Greater Than a Known Is 1/2

If there are two dependent positive real variables x1 and x2, and only x1 is known, what is the probability that x2 is larger versus smaller than x1? There is no uniquely correct answer according to “frequentist” and “subjective Bayesian” definitions of probability. Here we derive the answer given the “objective Bayesian” definition developed by Jeffreys, Cox, and Jaynes. We declare the standard distance metric in one dimension, d(A,B)≡|A−B|, and the uniform prior distribution, as axioms. If neither variable is known, P(x2<x1)=P(x2>x1). This appears obvious, since the state spaces x2<x1 and x2>x1 have equal size. However, if x1 is known and x2 unknown, there are infinitely more numbers in the space x2>x1 than x2<x1. Despite this asymmetry, we prove P(x2<x1∣x1)=P(x2>x1∣x1), so that x1 is the median of p(x2|x1), and x1 is statistically independent of ratio x2/x1. We present three proofs that apply to all members of a set of distributions. Each member is distinguished by the form of dependence between variables implicit within a statistical model (gamma, Gaussian, etc.), but all exhibit two symmetries in the joint distribution p(x1,x2) that are required in the absence of prior information: exchangeability of variables, and non-informative priors over the marginal distributions p(x1) and p(x2). We relate our conclusion to physical models of prediction and intelligence, where the known ’sample’ could be the present internal energy within a sensor, and the unknown the energy in its external sensory cause or future motor effect.

Living Systems Escape Solipsism by Inverse Causality to Manage the Probability Distribution of Events

The external worlds do not objectively exist for living systems because these worlds are unknown from within systems. How can they escape solipsism to survive and reproduce as open systems? Living systems must construct their hypothetical models of external entities in the form of their internal structures to determine how to change states (i.e., sense and act) appropriately to achieve a favorable probability distribution of the events they experience. The model construction involves the generation of symbols referring to external entities. This paper attempts to provide a new view that living systems are an inverse-causality operator. Inverse causality (IC) is an algorithmic process that generates symbols referring to external reality states based on a given data sequence. For applying this logical model involving if–then entailments to living systems involving material interactions, the cognizers-system model was employed to represent the IC process; here, living systems were modeled as a subject of cognition and action. A focal subject system is described as a cognizer composed of sub-cognizers, such as a sensor, a signal transducer, and an effector. Analysis using this model proposes that living systems invented the “measurers” for conducting IC operations through their evolution.

Unification of Epistemic and Ontic Concepts of Information, Probability, and Entropy, Using Cognizers-System Model

Information and probability are common words used in scientific investigations. However, information and probability both involve epistemic (subjective) and ontic (objective) interpretations under the same terms, which causes controversy within the concept of entropy in physics and biology. There is another issue regarding the circularity between information (or data) and reality: The observation of reality produces phenomena (or events), whereas the reality is confirmed (or constituted) by phenomena. The ordinary concept of information presupposes reality as a source of information, whereas another type of information (known as it-from-bit) constitutes the reality from data (bits). In this paper, a monistic model, called the cognizers-system model (CS model), is employed to resolve these issues. In the CS model, observations (epistemic) and physical changes (ontic) are both unified as “cognition”, meaning a related state change. Information and probability, epistemic and ontic, are formalized and analyzed systematically using a common theoretical framework of the CS model or a related model. Based on the results, a perspective for resolving controversial issues of entropy originating from information and probability is presented.

Biologically inspired information theory: Adaptation through construction of external reality models by living systems

Higher animals act in the world using their external reality models to cope with the uncertain environment. Organisms that have not developed such information-processing organs may also have external reality models built in the form of their biochemical, physiological, and behavioral structures, acquired by natural selection through successful models constructed internally. Organisms subject to illusions would fail to survive in the material universe. How can organisms, or living systems in general, determine the external reality from within? This paper starts with a phenomenological model, in which the self constitutes a reality model developed through the mental processing of phenomena. Then, the it-from-bit concept is formalized using a simple mathematical model. For this formalization, my previous work on an algorithmic process is employed to constitute symbols referring to the external reality, called the inverse causality, with additional improvements to the previous work. Finally, as an extension of this model, the cognizers system model is employed to describe the self as one of many material entities in a world, each of which acts as a subject by responding to the surrounding entities. This model is used to propose a conceptual framework of information theory that can deal with both the qualitative (semantic) and quantitative aspects of the information involved in biological processes.

On some recent insights in Integral Biomathics

This paper summarizes the results in Integral Biomathics obtained to this moment and provides an outlook for future research in the field.