“We cannot defy the laws of probability, because they capture important truths about the world.”
-Amos Tversky and Daniel Kahneman[1]
Science is often described as having a set of values or rules that guide scientists in their quest for knowledge. Through history the great figures of science, from Aristotle to Bacon, Galileo and Descartes, to Einstein, Feynman and Sagan, have championed these values[2]:
- Empiricism: Ideas are tested by observations.
- Testability: Hypotheses should make testable predictions.
- Strength of Evidence: Extraordinary claims require extraordinary evidence.
- Parsimony: Prefer the simplest theory that adequately explains the observations.
- Disinterestedness: Don’t become emotionally attached to your ideas. Remain unbiased and impartial.
These are often used to justify the methods and reasoning that scientists use to understand the world, but this leads to the question: what is the justification for these values themselves? Some philosophers of science would simply say that these are elemental beliefs, the foundational ideas that the rest of science is built on. After all, if you require all your principles to be justified by another principle, you quickly run into something called the regress problem in epistemology. Every time a justification is provided, you must again ask “why?”, and demand yet another justification for that reasoning, ad infinitum.
But probability theory provides a different way of looking at these values. In the context of probability theory, these values can be seen as more than just suggestions or guidelines for doing good science; they become precise mathematical rules that follow from the laws of probability, and often the reasoning behind them becomes more intuitive.
For example, in the context of probability theory, the maxims that “ideas are tested by observations” and “hypotheses should make testable predictions” become statements about the connection between observations and beliefs, a connection that probability theory describes with mathematical precision. We can write down the equation for Bayes’ rule and point to the part that represents our theory, or our experimental observation. We can do the math and see how that observation affects what we believe about the truth of our theory. We can see how a hypothesis that makes no testable predictions can’t even be plugged into the formula.
Similarly, “extraordinary claims require extraordinary evidence” can be seen as talking about prior probabilities. A prior probability is the probability of a hypothesis before you see the evidence for it. If a hypothesis is very unlikely to begin with (for example, “aliens stole my keys“) then it requires very strong supporting evidence before it would become probable. Hypotheses that are not so unlikely (like “my keys fell down behind the couch“) require less supporting evidence before they will become probable.
The suggestion to “prefer the simplest theory that adequately explains the observations”, often called Occam’s Razor, is a consequence of the fact that the probability of anything decreases if you add more detail to it. For example, the probability that you will win the lottery can never be less than the probability that you will win the lottery on a Thursday, since the first one happens whenever the second one does. If you add more detail to a idea you also make it less probable. (To think otherwise is an error of reasoning known as the conjunction fallacy.)
The value of disinterestedness, to “remain unbiased and impartial” is a reminder to listen to the evidence. We tend to feel the weight of evidence more strongly when it confirms our preconceived beliefs than when it opposes them, an effect called confirmation bias. This value tells us not to struggle against evidence in an attempt to preserve beliefs the we wish to hold, but to let evidence overturn our beliefs when it so dictates.
“If you know before you look, you cannot see for knowing.”
-Sir Terry Frost[3]
Probability theory doesn’t let us side-step the regress problem; nothing does. But looking at these scientific values through the lens of probability theory can give us another way of understanding why they work, and can clarify what they’re supposed to tell us about the knowledge-seeking process we call science.
