Abstract
The concept of correlation is the building block of almost any Bayesian attempt to capture or explicate any interesting aspect of scientific reasoning in terms of probabilities. This paper discusses one particularly simple correlation measure which is highly significant for almost any such attempt within the philosophy of science or epistemology. In particular, it shows how this correlation measure is related to central attempts to capture essential aspects of scientific reasoning such as confirmation, coherence, and the explanatory power of hypotheses. This intimate connection between correlation and scientific reasoning necessitates answering the question of how correlation and truth are related. This paper proposes an answer to this question and outlines its consequences for epistemology and the philosophy of science.
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Notes
- 1.
Wheeler (2009) calls this the Wayne-Shogenji correlation measure. Wayne (1995) discusses whether \(\mathfrak{c}\mathfrak{o}\mathfrak{r}\mathfrak{r}\) can be taken to be a similarity measure and Shogenji (1999) interprets it as a coherence measure. In the following no such interpretation is presupposed. Since many philosophers before Shogenji and Wayne have used this or ordinally equivalent measures I refrain from following Wheeler in calling it the Wayne-Shogenji correlation measure. Some of these philosophers are Keynes (1921), Horwich (1982), and Milne (1996). I call this measure the Simple Correlation measure since it is considerably simpler than the measure of correlation for finitely many random variables X 1, …, X n that are usually discussed in the literature on probabilities, such as Watanabe’s Total Correlation measure C:
$$\displaystyle{C(X_{1},\ldots,X_{n}) =\sum _{x_{1}\in X_{1}}\ldots \sum _{x_{n}\in X_{n}}\Pr (x_{1} \cap \ldots \cap x_{n}) \times \log \left ( \dfrac{\Pr (x_{1} \cap \ldots \cap x_{n})} {\Pr (x_{1}) \times \ldots \times \Pr (x_{n})}\right ).}$$ - 2.
Fitelson expresses the worry most clearly:
Shogenji’s measure is based only on the n-wise (in)dependence of the set E. It is well known that a set E can be j-wise independent, but not i-wise independent, for any i ≠j […] Shogenji’s measure does not take into account the ‘mixed’ nature of the coherence (incoherence) of E (and its subsets), and it judges E as having the same degree of coherence (incoherence) as a fully independent (or fully dependent) set. (Fitelson 2003, 197)
- 3.
- 4.
For a more detailed discussion of Schupbach’s measure of coherence and how to render coherence measures sensitive to the correlation of all its subsets see Schupbach (2009).
- 5.
The original formulation of Popper’s (1959) measure of explanatory power is this: \(\mathit{EP}_{P}(H,E) = \frac{\Pr (E\vert H)-\Pr (H)} {\Pr (E\vert H)+\Pr (H)}.\)
- 6.
A sequence of pieces of evidence separates the set of possibilities W if and only if for every pair of worlds w i and w j  ∈ W (with w i ≠w j ) there is one piece of evidence in the sequence such that it is true in one of the possibilities and false in the other.
- 7.
Note that Theorem 1 restricts these claims: \(\mathfrak{c}\mathfrak{o}\mathfrak{r}\mathfrak{r}\) satisfies both conditions only almost surely: it only holds for every \(w \in {W}^{{\prime}}\) where is a \({W}^{{\prime}}\subseteq W\) with \({\Pr }^{{\ast}}({W}^{{\prime}}) = 1\). It does not necessarily hold for all w ∈ W.
- 8.
This shows that the correlation measure satisfies two of the three requirements on theory assessment functions put forward in Huber (2008).
- 9.
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Brössel, P. (2013). Correlation and Truth. In: Karakostas, V., Dieks, D. (eds) EPSA11 Perspectives and Foundational Problems in Philosophy of Science. The European Philosophy of Science Association Proceedings, vol 2. Springer, Cham. https://doi.org/10.1007/978-3-319-01306-0_4
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