# Is it true that there is no “long-run relative frequency” for a theory being true?

I watched a lecture on Bayesian statistics which made the point that theories are either true or not. Thus, there can be no "long-run relative frequency" at which the theory is true, and Frequentist statistics cannot actually assess the probability that a theory is true (but have to rather compute the probability that data would be observed if a theory were true).

My instant reaction to this is that of course a there can be a frequency at which a theory is true. If I were to have a theory that "blue is a more popular colour than red", couldn't I count the number of people who prefer blue?

• I’m not following the second paragraph. The proportion of people who report preferring blue does not directly map onto the probability of the hypothesis “blue is a more popular color”. The former is at the level of individuals, the latter is a statement about a population. – Dave Mar 11 '18 at 16:49

I think you might misunderstand what is meant with long run frequency: Consider the case of a coin toss. According to the Frequentist notion of probability, a coin toss has a probability to come up heads because I can repeat it many times under similar conditions, which will yield an approximate estimate of the probability. One way to imagine this is that you make some random draws from a box that contains 0s and 1s (with replacement) and you don`t know the proportion in the box. (1 denotes tails, 0 denotes heads)

Furthermore a (Probability-) Freqentist will claim that it ONLY makes sense to talk of probabilities in cases where you can actually repeat your experiment under similar conditions!

So to come back to your case: What would the repetition under similar circumstances be in your example, or framed differently from which box are you drawing? Whether blue is a more popular colour than red does not depend on randomness at all, you will always obtain the same result (given the opinion of people does not change, but if it would, you would test a different hypothesis) You will always obtain the same answer: either yes or no. (So maybe you can even say that it has probability 1 or 0).

Maybe another example makes it clearer. Can a Frequentist sensically talk about the probability of a coin to have probability 0.7 to land heads? The answer is, it depends.

You might imagine a situation in which someone offers you a game: He will first draw a coin randomly from a box with various coins. And he will then toss the coin 10 times. If #heads >5 he wins, otherwise you win. In this case it makes sense for a Frequwntist to talk about the probability of the coin to have a probability of 0.7 to land heads, because the probability itself is a random variable, it depends on which coin is drawn from the box! The probability of the coin to have probability 0.7 is then simply the long run frequency of obtaining a coin that has the long run frequency of 0.7 to land heads.

But now imagine a different scenario. Someone offers you a similar game: He tosses 10 times with a given toss and if heads >5 he wins, otherwise you. (Which coin he tosses with is fixed) Then it does not make sense for a Frequentist to talk about the probability of the coin to have a probability of 0.7 to land heads! It is always the same coin, its probability to land heads is not random, it is fixed.

It is important to note, that for the Bayesian it does make sense to talk in a probabilistic way about the hypothesis "blue is more popular than red". Because if we claims: "I believe with a probability of 0.5 that blue is a more popular colour than red", then he simply expresses his own subjective ignorance on the subject.

Even if a parameter (such as the truth or falsity of your theory) is fixed, it might simply be that we don`t know which value it has, and it is this lack of knowledge Bayesians express with probability. The Bayesian does not demand that the experiment be repeatable under similar conditions for a probabilistic statement to make sense! You can always express your beliefs about something.

• In a sentence: frequentists care about long-run tendencies of observations, and don't think hypotheses have probabilities; Bayesians think hypotheses have probabilities, but don't care about long-run tendencies of observations. – Dan Hicks Mar 11 '18 at 17:18