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The frequentist interpretation of probability states that the probability of a currently unrealized event occurring is the limit of its relative (usually temporal) frequency in a large number of historical "trials of the event". More prosaically, if I flip any coin a large number of times and find that the number of heads has approached 50%, then I should expect that my next flip will land on heads 50% of the time.

Do the "trials of the event" have to be precisely identical and if they're not, does that make real world (i.e. physical, non-mathematical) application of the frequentist interpretation rigorously invalid?

Obviously, there are many possible causes that prevent any temporal sequence of real world trials from being precisely identical. Foremost, I will be unable to exactly control the angular momentum I impart to the coin for every trial. In fact, I can't say this for sure, but I suspect that even "differential" changes in imparted angular momentum combined with the chaotic behavior of the coin-universe system will lead to "unpredictable" outcomes. Such behavior in fact is probably what we mean by "random" experiment vs. Laplacian determinism even w/o recourse to quantum mechanics. Another example of the impossibility of precisely identical trials is the effect of the gravitational pull of Jupiter since it has moved in space across two trials.

So, rigorously, why should I be able to say anything at all about the probability of the next event given that I haven't replicated the exact experiment.

Pragmatically, I realize that the frequentist approach "just works" for many real world applications (such as flipping a given coin). This is probably due to the multitude of effects offsetting each other, but I am interested in the narrow philosophical underpinnings of the interpretation.

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  1. The frequentist interpretation also allows for some frequency with which the observed frequencies differ from the ideal. In particular, you should expect in N trials of a "perfectly unbiased" coin to observe a variance of up to N/2 (a standard deviation of sqrt(N/2), equivalently) from the norm for a collection of N samples, with some probability — and a variance of more than this also with some probability. This is to say: you cannot be certain that the norm of the sample will even be particularly near to the idealised mean, though it is likely to be so on average.

  2. Probability theory is just a model; or perhaps more accurately a meta-model. By abstracting away details which we think have a negligeable impact on outcomes — such as the effect of the position of Jupiter on your coin-flip — we arrive at a model which is practically useful, although perhaps not as comprehensive as possible. Its success comes from the fact that such fine details do not seem to play any significant role in outcomes. As such, probability theory is a success story of applied mathematics, which is to say a success story of quantitative epistomology.

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It is probably an inapt description to say that the events should be either precisely or imprecisely the same, but rather it would be better to say that future events or states must satisfy this or that property. Thus in a coin flip we wouldn't characterize these events by their similarity to any single event, but simply ask whether each event is a coin flip (and more precisely, whether it is a coin flip that results in heads). This too can see some ambiguity, but nothing that is special to probability theory--any ambiguity that I see arising from these considerations are ambiguities in judging whether a property is satisfied.

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