Statistical Evidence Measured on a Properly Calibrated Scale for Multinomial Hypothesis Comparisons
“To a certain degree this scheme is typical for all theoretic knowledge: We begin with some general but vague principle, then find an important case where we can give that notion a concrete precise meaning, and from that case we gradually rise again to generality and if we are lucky we end up with an idea no less universal than the one from which we started. Gone may be much of its emotional appeal, but it has the same or even greater unifying power in the realm of thought and is exact instead of vague.”(, p. 6, emphasis added)
2. A Vague but General Understanding of Statistical Evidence
- Evidence as a function of changes in x/n for fixed n: If we hold n constant but allow x/n to increase from 0 up to ½, the evidence in favor of bias will at first diminish, and then at some point the evidence will begin to increase, as it shifts to favoring no bias. BBP(i) is illustrated in Figure 1a.
- Evidence as a function of changes in n for fixed x/n: For any given value of x/n, the evidence increases as n increases. The evidence may favor bias (e.g., if x/n = 0) or no bias (e.g., if x/n = ½), but in either case it increases with increasing n. Additionally, this increase in the evidence becomes smaller as n increases. For example, five tails in a row increase the evidence for bias by a greater amount if they are preceded by two tails, compared to if they are preceded by 100 tails. BBP(ii) is illustrated in Figure 1b.
- x/n as a function of changes in n (or vice versa) for fixed evidence: It follows from BBPs(i) and BBPs(ii) that in order for the evidence to remain constant, n and x/n must adjust to one another in a compensatory manner. For instance, if x/n increases from 0 to 0.05, in order for the evidence to remain the same, n must increase to compensate; otherwise, the evidence would go down following BBP(i). BBP(iii) is illustrated in Figure 1c.
3. A Precise but Non-general Definition of Statistical Evidence
4. First Generalization: From e1 to e2
5. Second Generalization: From e2 to e3
6. Measurement Calibration
“[There is] a notable dissimilarity between thermodynamics and the other branches of classical science….[Thermodynamics] reflects a commonality or universal feature of all laws…[it] is the study of the restrictions on the possible properties of matter that follow from the symmetry properties of the fundamental laws of physics. Thermodynamics inherits its universality…from its symmetry parentage.”(, pp. 2–3)
Conflicts of Interest
Area under LR
Volume under LR
Basic Behavior Pattern (for evidence)
Appendix A. The constants d.f. and b
|m = 3||m = 4|
|n||Max E||Min E||Diff||Ratio||Max E||Min E||Diff||Ratio|
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Vieland, V.J.; Seok, S.-C. Statistical Evidence Measured on a Properly Calibrated Scale for Multinomial Hypothesis Comparisons. Entropy 2016, 18, 114. https://doi.org/10.3390/e18040114
Vieland VJ, Seok S-C. Statistical Evidence Measured on a Properly Calibrated Scale for Multinomial Hypothesis Comparisons. Entropy. 2016; 18(4):114. https://doi.org/10.3390/e18040114Chicago/Turabian Style
Vieland, Veronica J., and Sang-Cheol Seok. 2016. "Statistical Evidence Measured on a Properly Calibrated Scale for Multinomial Hypothesis Comparisons" Entropy 18, no. 4: 114. https://doi.org/10.3390/e18040114