Philosophical Prelude

Overview

Teaching: min
Exercises: min

Questions

Objectives

Philosophical Prelude

The fundamental nature of probability as used in applied statistical inference was described succinctly by John Stuart Mill as part of his larger program of characterizing inductive reasoning,^[Mill, John Stuart. 1882. A System of Logic: Raciocinative and Inductive. Eighth edition. Harper & Brothers, Publishers, New York. Part III, Chapter 18.]

We must remember that the probability of an event is not a quality of the event itself, but a mere name for the degree of ground which we, or some one else, have for expecting it.
\(\ldots\) Every event is in itself certain, not probable; if we knew all, we should either know positively that it will happen, or positively that it will not. But its probability to us means the degree of expectation of its occurrence, which we are warranted in entertaining by our present evidence.

Mill is saying that our estimate of the probability of an event fundamentally depends on how much we know. For example, if you ask me now to forecast the weather on a random day in Edinburgh next year, I’ll forecast 52% because it rains on average 191 days per year in Edinburgh.^[World Weather and Climate (2018) Edinburgh rainfall.] If I know the day is in December, I’ll make an estimate of 58% conditioned on knowing that it rains on average 18 of 31 days in December. Continuing to add information, if it’s December 7 and I have up-to-date radar, then my estimate of the chance of rain on December 8 or December 9 might be anything, depending on the current meteorological conditions.^[The current forecast from weather.com as of 11:30 pm December 7, 2018 is a 60% chance of rain on December 8 and a 10% chance of rain on December 9.]

Putting Mill’s view in more modern terms, probability is a relative measure of uncertainty conditioned on available information. In other words, probability involves statements about an agent’s or collection of agents’ knowledge of the world, not about the world directly.^[In philosophical terms, the nature of probability is epistemic (based on knowledge) rather than ontological (based on metaphysics) or deontic (based on belief).] This allows us to believe the world is deterministic, while still reasoning probabilistically based on available evidence. Apparently, this was Pierre-Simon Laplace’s position, as he wrote,^[Pierre-Simon Laplace. 1814. A Philosophical Essay on Probabilities. English translation of the 6th edition, Truscott, F.W. and Emory, F.L. 1951. Dover Publications. page 4.]

We may regard the present state of the universe as the effect of its past and the cause of its future. An intellect which at a certain moment would know all forces that set nature in motion, and all positions of all items of which nature is composed, if this intellect were also vast enough to submit these data to analysis, it would embrace in a single formula the movements of the greatest bodies of the universe and those of the tiniest atom; for such an intellect nothing would be uncertain and the future just like the past would be present before its eyes.

It also provides us the wiggle room to take sides with Albert Einstein, who wrote,^[Albert Einstein. 1926. Personal letter to Max Born. December 4.]

The theory [quantum mechanics] says a lot, but does not really bring us any closer to the secret of the “old one.” I, at any rate, am convinced that He does not throw dice.

Probability theory merely provides a mathematically and logically consistent approach to quantifying uncertainty and performing inductive inference.

Key Points