Abstract
Having considered the major components of scientific method in the preceding chapter, it is now time to consider one of the important aspects of science that, in general, is more or less ignored or played down in most expositions on scientific method: uncertainty. It is also one of the topics that Ph.D. students tend to demonstrate little interest in—as is often evidenced by their lack of knowledge about or reflection on the standard software routines they rely on for performing the statistical analyses that so much of their conclusions depend on.
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Notes
- 1.
I note regarding the last question that (Jeffreys 1983; 15) argues that human belief regarding uncertain phenomena is a function of two arguments and that a coherent theory of probability must account for this: “It is a fact that our degrees of confidence in a proposition habitually change when we make new observations or new evidence is communicated to us …. We must therefore be able to express it. Our fundamental idea will not be simply the probability of a proposition p, but the probability of p on data q.” As we will see shortly, this is the fundamental starting point for a so-called Bayseian approach to probability theory.
- 2.
The members of a set are typically listed in brackets, for example A = {x, y, z} is the set composed of the members or elements x, y and z. A ∪ B designates the union of A and B, the set containing all members of A and all members of B, and A ∩ B designates the intersection of A and B, the set containing elements that belong to both A and B and nothing else. For example, if A = {v, x, y, z} and B = {q, r, x, z} then A ∪ B = {q, r, v, x, y, z} and A ∩ B = {x, z}. A ⊆ B designates inclusion—that A is a subset of B (every member of A is a member of B); for example, the set A = {x} is a subset of B = {x, y}. ~A designates the complement of A, the set containing everything not belonging to A. Two sets are mutually exclusive if they have no members in common; the sets {x, y} and {q, w, z} are mutually exclusive.
- 3.
Although for sake of generality the axioms presented are based on conditional probabilities, they also cover simple unconditional probabilities as well if the conditioning event X is taken to be a tautology (a universal truth, such as Z \(\vee\) ~Z) and then removed.
- 4.
For a fascinating, challenging and rather unorthodox introduction to deduction, formal systems, mathematics and their relationship to ‘reality’, see the first two chapters in (Hofstadter 1989); this book also provides a detailed treatment of Gödel’s Theorem that was considered in the section on innate limitations of science in Chap. 2.
- 5.
Readers with a background in probability and statistics will see that Problem 1 can be solved by calculating the binomial probability: P(42 heads|100 tosses and P(head) = 0.5) = (100!/42!58!)(0.542)(1 − 0.5)58. Problem 2, which compares two hypotheses (the probability of a head is either 0.5 or 0.7) based on the data—42 heads and 58 tails—cannot be solved at this point in the exposition, but you should be able to solve it after reading the next section on the Bayesian paradigm. Here it should suffice to note that it is extremely unlikely that the coin is unfair (with probability of a head = 0.7) since only 42 heads resulted.
- 6.
To be more precise, we say that a sequence fn (n = 1,2,3, …) has the limit L as n goes to infinity if and only if, for every positive µ, no matter how small, there exists a number N such that, if n > N, |fn − L| < µ.
- 7.
The binomial probability of 63 or more heads is computed as: (100!/63!37!)(0.563)(1 − 0.5)37 + (100!/64!36!)(0.564)(1 − 0.5)36 + ··· + (100!/100!0!)(0.5100)(1 − 0.5)0 = 0.00602.
- 8.
Permutation: Distinct ordered arrangement of items. E.g. 3 items, A, B, C can be ordered in 6 ways/permutations (3!/1!1!1!) = 6, and items A, A, B, C can be ordered in 12 ways/permutations, (4!/2!1!1!) = 12. The general formula with a total of N items where N1 are of type 1, N2 are of type 2 … NR are of type R is: (N!/N1!N2!…NR!).
Combination: Order does not matter. N items taken R at a time lead to (N!/(N − R)!R!) combinations. E.g. the number of ways to choose 5 cards out of a standard deck of cards is (52!/(52 − 5)! × 5!). Note for R = 2 (only 2 types of items), the number of permutations equals the number of combinations; this is the case here with only green and red marbles.
- 9.
But there is yet another lesson about bias to be learned from this story, although it may not be directly applicable to data collection in the natural sciences. The size of the actual sample (2.4 million potential voters), huge though it was, was only about one-fourth of what was originally intended (more than 10 million potential voters were contacted). People who respond to surveys may be different from people who do not respond and thus may not be representative of the population they are assumed to represent.
- 10.
Although this is often the case in the natural sciences it is even more so in the social sciences, where it tends to be the rule rather than the exception. For example, this is very often the case in problems faced in economics or management, where decisions are to be made and where a decision maker typically has such prior knowledge based on experience (and intuition), often supported by expert evaluations and the like.
- 11.
A counter argument provided by some of those who deny the reasonableness of using subjective priors is that in the absence of a firmly established prior, it is necessary to list all the possible hypotheses and distribute probabilities among them, for example by ascribing the same probabilities to all of these using the principle of indifference (as we did with the marbles example). They then ask where such a list can come from—and argue that in principle the number of hypotheses can be infinite leading to a probability of zero for each, whereby the analysis cannot take place; (Chalmers 1999; 178).
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Pruzan, P. (2016). Uncertainty, Probability and Statistics in Research. In: Research Methodology. Springer, Cham. https://doi.org/10.1007/978-3-319-27167-5_8
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