7

According to the late Popper, among others, probability is the propensity of a set of conditions to produce certain long run relative frequencies. Therefore if we say that a certain set of conditions have the probability of heads of 0.5, we thereby express that repeatedly conducting an experiment according to these conditions will yield a long run frequency of heads of 0.5. This does not mean that the propensity is the same as the long run frequency, but rather that the propensity is what causes the long run frequency to behave in that manner.

"In this interpretation, the generating conditions are considered as endowed with a propensity to produce the observed frequencies. As Popper put it: ‘But this means that we have to visualise the conditions as endowed with a tendency or disposition, or propensity, to produce sequences whose frequencies are equal to the probabilities; which is precisely what the propensity interpretation asserts.’" (Popper quoted by Gillies in "Philosophical Theories of Probability").

I wonder whether the propensity interpretation implicitly utilizes something like the principle of indifference. I will illustrate my thought process by the means of a concrete example:

I can characterize a coin toss according to conditions C, such as:

  • the coin is tossed by a person with no intention to influence its outcome.
  • the coin is symmetric.
  • the coin is tossed on earth.
  • etc.

Now assume now that I have data D of 1000 such experiments that were conducted under the above explicated conditions. My best estimate of the propensity is then presumable the relative frequency of heads in these 1000 experiments. So P(heads|condition C)= number of heads/1000 := h/n

So assume now that I am in a situation that can be characterized according to the above conditions C and I want to determine the single case probability for the situation. My "best guess" for the propensity then seems to be h/n, if I have not further relevant information about how the coin toss is conducted apart from the fact that it was conducted under conditions C.

But what is actually done when h/n is calculated? Every data point in the data set D follows the characterization C. But obviously the conditions for each toss beyond that were not equal. Otherwise the results would have been the same for all 1000 experiments. The conditions varied in some way that was not measured. However when I calculate h/n I weigh the outcome of each experiment equally. It seems to me however that this is something very similar to the principle of indifference. Weighing each experiment equally corresponds to a believe that each experiment was equally "entitled".

The same argument would hold e.g. if I was to determine the weight of an object. I might weigh a specific object 1000 times. My best guess would then be that the true weight of the object is the sum of all measurements divided by 1000. But this again utilizes the principle of indifference in a similar manner than in the above example. Because I have no reason whatsoever to prefer one measurement over the other I weigh them equally.

Would you agree that this is the principle of indifference in disguise or am I profundly confused about the matter?

I want to add another question that was inspired by the comments. In many examples, such as roulette or coin tosses I think the propensity of a situation only makes sense because the conditions C do not completely describe the situation: in the coin toss example, the exact angle and motion etc. are not specified. The set of generating conditions is in some sense incomplete. So it seems to me that propensity in such situations does not actually refer to a physical property. The fact that the same conditions can result in different outcomes is merely due to the fact that the conditions to not completely characterize the situation. Is this correct?

9

One problem is in "if we say that a certain set of conditions have the probability of heads of 0.5, we thereby express that repeatedly conducting an experiment according to these conditions will yield a long run frequency of heads of 0.5". That is frequentism, not propensity probability. Propensities are not long term frequencies, they are hypothetical physical causes of the observed stable frequencies, objective dispositions, tendencies. Unlike frequentist probabilities they make sense even for individual events, not just for series of trials as in the frequentist interpretation, nor are they subjective, as in dependent on information available to the subject. Our only way of measuring them may be running long series of trials but propensities themselves are over and above what we measure or even our measuring of them altogether, they are objectively active out there.

This makes it clear that another problem is "the conditions for each toss beyond that were not equal. Otherwise the results would have been the same for all 1000 experiments". Propensities need not produce the same results in equal conditions (think of quantum double slit experiment). But, absent information on changes, one can only predict stable frequencies assuming such equal conditions. The term is ceteris paribus, other things being equal. The principle of indifference in this case does not figure into the conception of propensity, it rather expresses a condition on when the simple frequency prediction is applicable under the propensity theory. This is similar to simplifying assumptions of classical mechanics, such as no friction, etc., or to any other idealization. When we predict that a rock thrown into the window will break it we implicitly assume that a sudden gust of wind will not blow it off course. But even if it did the thrown rock's propensity to break the window remains unaffected.

  • 1
    Ok thank you for your remarks. I will modify my question. I did not intend to express that propensity is long run frequency, but that the propensity will show itself in the form of the long run frequency. Furthermore I am not sure whether I completely understand your second comment. While I agree that the same conditions might not produce the same results in some areas of physics, i think that this suggestion holds in situations such as the toss of a coin, doesn`t it? – Sebastian Mar 27 '18 at 22:42
  • 1
    Furthermore I would disagree that they are independent of information available to the subejct. Certainly there are many different descriptions of the conditions under which a coin was tossed, depending on the information of the observer? The problem might be that there are many different versions of propensity theory and we might talk about different accounts. (Or I do not really understand the propensity interpretation yet) – Sebastian Mar 27 '18 at 23:34
  • 2
    @0rangetree It seems to me that "complete description" does not exist even in classical situations, there are always infinitely many ceteris paribus clauses that can not all be explicated. So we have to settle for talk about properties under incomplete descriptions. Incomplete description then becomes an idealized surrogate of a "physical situation" that exhibits some propensity the same way incomplete descriptions in mechanics may still determine acting forces. – Conifold Mar 28 '18 at 2:40
  • 2
    Of course, subject may not have even enough information even for that, in which case one can resort to traditional subjective probabilities, which are roughly weighted averages of propensities over various determinate situations compatible with the available information. But those are not "objective propensities" that Popper analogizes to physical forces. – Conifold Mar 28 '18 at 2:40
  • 2
    @0rangetree The question is whether it is "truly random" under the incomplete description. The source of randomness in coin tosses is random background noise which is ineliminable, all we can do is make assumptions that it is unbiased. The supposed difference in quantum cases is that randomness remains even if we had a complete description of the entire universe, but once we close the system to a manageable size to determine propensities one is as "truly random" as the other. – Conifold Mar 28 '18 at 19:21

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Not the answer you're looking for? Browse other questions tagged or ask your own question.