Why do Bayesian Networks Work with Abstract Framework Analysis? A Philosophic Perspective (Section 1)
Bertrand Russell had envisioned the representation of abstract thought through the means of atoms, described as factual ground truths within a larger thought framework representing piece of a larger factual proposition, true or untrue according to an individual perspective. In one of Bertrandâs lecture discussing the concept defined as Atomism, the belief that a factual statement can be partitioned it into compositional logic segments of smaller factual propositions. Russell states that larger factual statements can be divided into segments with a simplistic approach but the smaller representations created from the factual statement are built with a level of complexity themselves (Russell 1918). Therefore an atom within it itself is derived from the existence of compounded fact observed to exist in totality, true or false. Obtaining a factual proposition within a factual statement as a point of absolute true or absolute false, has a level of vagueness associated with the construction of it. Russell deems this quantifiably of vagueness as irrelevant in the analysis of abstract thought. This idea of vagueness appearing as a relative probability of error falls along the lines of Quineâs argument that the existence of a discrete objectification of a factual truth can be defined by continuation of evidence and further add to the claim for the model of abstract thought that Quine envisioned within the âweb of beliefâ.Â
 Critique of theories and the characterization behind quantifying abstract belief networks and scientific theory found in Quineâs âTwo Dogmas of Imperialismâ, reject the âdistinct boundaryâ between the synthetic characterization of a network, defined as the continuation of observable factors within the network, and analytic characterization of a network, the ability to quantify of an objective propositional truth within a working network. Quine, idealized the most pragmatic model for the quantification of belief networks and scientific theory, involves the retraction of this synthetic and analytic boundary with network characterization (Quine, 1951). The sum of the synthetic and analytic characterization lend itself well as a logical bedrock for formal analytical representations abstract frameworks of belief and theory. The removal of synthetic characterization would uproot the analytical nature of understanding the translation of logic, and would call for a systematic redefinition of representation within models for abstract frameworks. In this way, the analytical characterization of abstract frameworks is disjoint and incomplete without the synthetic characterization (Carlson 2015). Quine later went on to describe theory this as a âweb of beliefâ in later works (Quine 1978).
Brian Skyrms and Karel Lambert, outline the foundation for the principles that should embody an empirical framework representation of Quineâs propose âweb of belief â on the grounds of a coherent belief representation in âLaws of Nature: Essays on the Philosophical, Scientific and Historical Dimensionsâ. The laws of necessity laid out in the beginning portion of this work elaborate on the core components of the âweb of beliefâ Quine envisioned in his later work. Skyrms and his counterpart argued that the proposed method for building this âweb of beliefâ empirically, involve the ability to encase degree of belief within a closed and finite field of values through methods of Boolean algebra, with non-negative correlations to the degrees of belief within all aspects of the framework. The robust beliefs,  seen as the factual ground truths in an initial network, act as the core structure to the observable network. Experiments within this network are proposed applications of evidence applied to factual atoms, creating the new core structure through actions of âsuccessful inductionâ on the likely outcomes of the probabilistic network over random correlation. (Pearl 1988) Skyrms further elaborates on the foundational pillars of inference models which are coherent with the definition of inferential continuation within empirical representations and do not disrupt the nature of the Kolmogov axioms. These axioms, stated within chapter four, are some of the axioms that lay the groundwork for the construction of node representation and combinational logic characterizations of abstract reasoning inferential models such as the Bayesian Network.
Further elaborating on the ideas brought forth by Skyrms and Lambert within inferential models dealing with abstract reasoning, Pearl introduces the idea of the need for a model that does not explicitly enumerate all possible states of a substantially large subset of uncertain probable worlds but rather a model which can capture with maximal accuracy the characteristic behaviors within a âweb of beliefâ with a certain amount of likelihood. The visible interaction of the application of proposed transition between the current state and the next state, only plausibly computational under a restrictive independence assumption for efficiently local detached computation. The updating occurring between each state implied by the definition of preset sub-functions that are applied to the observable state with a certain amount of strength m. In the case of Bayesian reasoning, the strength m is defined by the conditional probability where m has a direct implication on the truth value of the inferential piece derived from pieces of given knowledge (Pearl 1988). These pieces embody the concept of a rule network, or a belief network in probability theory, similar to concepts laid out by Brian Skyrms and Karel Lambert. Through these foundations, the observable evidence acting in a manner in which there is a creation of an altered truth, made through a local inference decision making system (Pearl 1988).
This chain of inference, leading to ideas within the empirical representation of reasoning that led to the formation of thought in using systems such as the Bayesian network. The philosophical perspective of abstract frameworks maps well the the structure of the Bayesian network and this is why those who are proponents of Bayesianism in Philosophy, argue that the Bayesian network is proposed as a great approximation methods for abstract reasoning. Â
References:
Carlson, Matthew. âLogic and the Structure of the Web of Belief.â Journal for the History of Analytical Philosophy, vol. 3, no. 5, 2015, doi:10.15173/jhap.v3i5.28.
Pearl, Judea. 1988. Probabilistic Reasoning in Intelligent Systems. San Mateo, CA: Morgan Kaufman.
Russell, Bertrand. âTHE PHILOSOPHY OF LOGICAL ATOMISM [with Discussion].â The Monist, vol. 28, no. 4, 1918, pp. 495â527. JSTOR, www.jstor.org/stable/27900704. Accessed 11 Mar. 2021.
Skyrms, Brian and Karel Lambert, 1995. âThe Middle Ground: Resiliency and Laws in the Web of Belief,â in Friedel Weinert, ed.,Laws of Nature: Essays on the Philosophical, Scientific and Historical Dimensions. Berlin: De Gruyter
Quine, Willard Van Orman and J. S. Ullian, 1978. The Web of Belief. New York: Random House.
Quine, Willard Van Orman. Â 1951, 1953. Â âTwo Dogmas of Empircisim.â The Philosophical Review 60 (1): 20-43. Â Reprinted in From a Logical Point of View, Cambridge MA: Harvard University Press, 1953.
