Chapter 1: Introduction to the Statistical Relevance Model

The statistical relevance (SR) model, chiefly formulated by Wesley Salmon in 1984, posits that, for any given explanation, the associated factors (explanans) must hold statistical significance in relation to the phenomenon being elucidated (explanandum), independent of any specific probability involved¹.

The core principle of the SR model can be summarized as follows: for a specific category A, an attribute C is considered statistically relevant to another attribute B if the probability of B under the condition of both A and C exceeds the probability of B under just A.

In simpler terms, if attribute C demonstrates statistical relevance to attribute B, it may serve as a valid explanation for B, provided that the probability of B given both A and C surpasses that of B given A alone. Thus, attribute B is effectively clarified through the positively relevant attribute C. According to this model, an explanation comprises two primary components:

1. The prior probability of B within A: P(B|A) = P
2. A structured partition of A concerning B, denoted as (A. C1, … A. Cn), along with the probability of B within each subset of the partition: P(B|A&Ci)

To illustrate this with a practical example, let's consider the reasons behind a 20-year-old female's pregnancy.

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Initially, we define a reference class of adolescent females, labeled as class A. This reference class is then partitioned into two distinct categories concerning the explanandum: B1 (those who have become pregnant) and B2 (those who have not).

Next, we divide the reference class A using a set of explanatory factors into classes Ci, which may include aspects such as religious beliefs, ethnicity, income levels, etc. An SR explanation would indicate that a specific set of attributes are explanatory by identifying the partition Ca to which the individual in question belongs, demonstrating that:

B iff P(B1/A&Ca) > P(B/A)

Now, let’s apply this framework to explore the reasons behind the success of scientific theories. Suppose we have a scientific theory x belonging to class A of scientific theories. A class of theories that are approximately true (C1) will show a higher statistical relevance to the class of successful theories (B) than those that are not approximately true (C2). This can be expressed as:

P(B/A&C1) > P(B/A&C2)

For instance, if we assume P(B/A&C1) = 0.4 and P(B/A&C2) = 0.04, it becomes evident that the success of scientific theory x is linked to its belonging to the class of approximately true theories C1, thereby supporting the realist explanation of success based on its proximity to the truth.

This video titled "What is a statistical model?" provides a foundational understanding of statistical models and their significance.

Consequently, the approximate truth of a theory serves as a relevant factor in explaining its success. It’s essential to note that while the first probability should exceed the second, it doesn't necessarily need to have a high value.

Additionally, we can formalize Laudan’s meta-induction argument, asserting that if numerous historical examples of non-referential theories were available, it would follow that:

P(B/A&C2) > P(B/A&C1)

As previously demonstrated, Laudan struggles to substantiate this claim, despite asserting that his list of non-referring theories could be expanded indefinitely. It is evident that a mere handful of non-referential theories cannot form a robust argument.

Furthermore, it is possible to formalize the realist belief in the existence of both observable and unobservable entities in a world independent of human perception. According to Laudan, for truth to explain success in science, the following must hold true:

P(B/A&C2) = 0, and P(B/A&C1) = 1.

From this discussion, two important observations emerge: first, Laudan's expectation of no successful non-approximately true theories is excessively stringent; second, adopting a more flexible understanding of explanation allows us to assert that approximate truth is indeed the best explanation for identifying successful theories.

Chapter 2: Caveats and Considerations

Two critical points merit consideration. Firstly, utilizing this statistical model does not imply that it is immune to criticism or that it is entirely irrefutable. Indeed, one can readily conceive potential counterexamples where statistical relevance fails to elucidate a specific fact.

Secondly, it is crucial to recognize that the statistical model was developed alongside a causal mechanical model (Railton, 1978; Salmon, 1984), wherein causality is acknowledged as a vital element. Unlike the latter, Salmon posits that statistical relevance carries minimal explanatory weight. Furthermore, this model clearly appeals to explanatory considerations, thus leaning towards abductive rather than inductive reasoning.

In summary, the SR model is a valuable framework for assessing whether the elements related to a specific explanation are statistically relevant to the phenomenon being elucidated.

The second video titled "MFML 075 - Statistical significance" delves into the concept of statistical significance and its implications in research and analysis.