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I am currently thinking a lot about the definition of probability and one thing that I am really not content with at all is the frequentist definition of probability as long run relative frequencies of "the same" random experiment, with which I mean the execution of the experiment under similar conditions. It is virtually never specified what exactly counts as similar conditions. So I wonder how an adherent to the frequentist notion of probability would best describe what is meant with the repetition of an experiment under similar conditions.

EDIT: I am currently reading https://arxiv.org/abs/quant-ph/0408058 and it also picks up the issue I am concerned with:

It is clearly essential, in any serious experiment, to standardize the tossing procedure in such a way as to ensure that the probability of heads is constant. This raises the question: how can we be sure that we have standardized properly? And, more fundamentally: what does it mean to say that the probability is constant?"

and further:

Frequentists are impressed by the fact that we infer probabilities from frequencies observed in finite ensembles. What they overlook is the fact that we do not infer probabilities from just any ensemble, but only from certain very carefully selected ensembles in which the probabilities are, we suppose, constant (or, at any rate, varying in a specified manner). This means that statistical reasoning makes an essential appeal to the concept of a single-case probability: for you cannot say that the probability is the same on every trial if you do not accept that the probability is defined on every trial. The only question is whether the single-case probabilities are to be construed as objective realities (“propensities”), or whether they should be construed in an epistemic sense.

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    $\begingroup$ Look to general probability theory: Two random variables can be equal in distribution, equal in probability, or equal almost surely. In statistics, an experiment is just a random variable. A confirmatory trial, for instance, seeks to recruit the same number of patients, with the same clinical criteria, and randomize treatment/control in the same fashion, although their outcomes may be disparate. $\endgroup$ – AdamO Mar 8 '18 at 15:51
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I am really not content with at all is the frequentist definition of probability as long run relative frequencies of "the same" random experiment, with which I mean the execution of the experiment under similar conditions. It is virtually never specified what exactly counts as similar conditions.

Well, you don't see the conditions, because there is no clear-cut conditions for this. As already noticed in comment by @AdamO, on theoretical grounds, we consider the experiments as random variables, and there are few ways how random variables can be equal. What we usually mean in here, is that the variables are independent and identically distributed, or exchangeble. The way how we planned our experiment, or gathered the data, determines what can we assume about it.

Usually the conservative theoretical assumptions are not exactly met in practice. That's because those are theoretical properties of random variables, they do not tell you anything about the actual, real-life, experiment. We design our experiments in such way, so that we could assume that they are "the same", but there is no single set of conditions to be met, so that we could be sure that they actually are "the same". When building an experiment, we obviously are not able to control all the possible factors in the environment, so that the different experiments would be identical, but we usually try making them as similar as possible and after the experiment we double-check if there is no any unusual differences between the data from the experiments. History of science is full of the examples of people who missed that their experiments were not identical, when they thought they were.

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I am not entirely sure what you mean by 'two random experiments are same'.

A rough overview of a random experiment is usually given right at the beginning of basic probability theory and if you take the most common example of flipping an unbiased coin, you can guess what it actually means.

The three main features of a random or probabilistic experiment that distinguishes it from a deterministic experiment are :

  • Several or all possible outcomes of the experiment are known in advance.

  • Any particular performance of the experiment (i.e. a trial) results in an outcome that is not known in advance.

  • The experiment can be repeated under similar conditions.

So consider the random experiment of tossing a fair coin twice. The sample space consisting of the set of all possible elementary events is then given by $\Omega=\{HH,HT,TH,TT\}$. Let us rename the elementary events by real numbers according to a definite rule. This rule is your random variable.

Suppose the rule is the count of heads $($#$H)=X$, say. Then clearly $X$ is a mapping such that the image of $\{HH\}$ under $X$ is $2$, that of $\{HT\}$ under $X$ is $1$ and so on. Note that we haven't discussed any probability so far. And in fact, even in the formal definition of a random variable the notion of probability is not strictly present. The fact that probability of an event can be justified empirically by repeating a random experiment a number of times is where the frequency definition comes in.

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  • $\begingroup$ What I mean with "the same" is what you refer to as repetition under similar conditions. My question is how to justify when the conditions are similar. $\endgroup$ – Sebastian Mar 8 '18 at 21:07

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