Are There Objective Probabilities?

Apparently, the All Souls exam is the hardest in the world. Whether that’s true or not, the questions are interesting, and answering them is a good mental challenge. In this series I intend to do that – without looking at anything external.

The first question is taken from the paper Philosophy I, here.

Let me begin by describing two opposed answers to this question, based on intuition. I will then combine these intuitions into a view which honours the right in both, while avoiding any confusions.

Suppose, at time t, there is a box and it contains, with probability 0.5, a marble.

Intuition 1: Objectively, measuring the state of the world at t, the marble is under Box 1. Before opening any boxes, I conclude that the marble lies under Box 1 with probability 0.5. After opening box one, however, I conclude that the marble lies under Box 1 with probability 1.

Nothing about the world has changed except that I have observed something. The probability has changed because of this observation. But an objective thing is something about which my observations and subjective state of knowledge cannot possibly make any difference. Therefore, there cannot be any such thing as an objective probability.

Intuition 2: At time t, if it is true that the probability of the marble being under a given box is 0.5, then if I guessed randomly, I would be correct one out of two times on average.

This simple observation gives me objective, usable knowledge about the world. Consider a game in which there are 100 boxes, each with 0.5 probability of containing a marble. I have to guess how many of these boxes actually contain a marble – call my guess N. I guess, then I lose £|(N-X)|.

In this game, my dominant strategy – i.e. the strategy which minimizes |N-X|, my loss – is to guess ‘50’ every time. This is an empirically demonstrable result – someone who disbelieves this is welcome to play the game and guess ‘1’ or ‘2’ instead of ‘50’ and observe the outcome. If an intelligent alien changed my brain such that I mistakenly believed that the probability was, e.g., 0.01, then that would not make the probability 0.01. The fact that the probability is 0.5 is completely independent of whether or not any particular mind believes that it is 0.5. But this is the definition of an objective truth. Therefore, probability must be objective.

How to resolve this paradox?

We have proven, informally, that:

  • Probability is mind-independent, i.e. probabilities do not depend on someone believing in them to be true (objective)
  • Estimated probabilities can change based on the state of knowledge of the observer (subjective)

Objective things are mind-independent. A rock does not depend on someones observing that rock; it exists whether or not someone is there to observe it. Moreover, if a rock is an igneous rock, then it is an igneous rock regardless, once again, of whether someone is there to observe it.

However: my estimation that the rock is igneous, depends on my state of knowledge about the world; or, in Bayesian terms, myprior. The probability estimate can change as I discover new facts, e.g. that the rock has a glassy texture or displays layering. Moreover, the fact that I believe that the probability is X, is not a fact about the world, it is a fact about my mind.

Does that mean that anything goes? Obviously not. The probability of a fair coin landing heads is 0.5. Provided that I know all the facts, I cannot change this probability by changing my beliefs. If I could, then we would witness people winning the lottery, or poker, at will. Clearly, this is not the case.

To sum up: probabilities are (a) true or false, (b) regardless of what I believe about this probabilities. Estimated probabilities (c) depend on the prior knowledge of the mind in question. Moreover, (d) the fact that the probability is X is a fact about my mind.

Based on these facts, it is wrong to conclude either that probability is subjective or that probability is objective. For example, traits a and b are features of things we call objective, but traits c and d are features of things that we call subjective.

We know all the facts. Telling us that it is subjective or objective gives us no additional information, nor does it lead to any differing expectations about reality. (Like the old question: if a tree falls in the woods, does it make a sound? It depends on whether your definition of sound includes ‘someone hearing it’, or whether it’s just ‘acoustic vibrations’. That’s it. There’s no factual disagreement here, it’s all at the level of language rather than the world.)

Probability has conceptual features from both classes, so some composite label would be best, if a label were necessary. As it is, neither ‘subjective’ nor ‘objective’ will do.