i am trying to understand the differences between bayesian and frequentist statistics. I read that in frequentist statistics the unknown population parameter theta is considered a constant but in DerSimonian and Laird method the assign for example a Normal Probability distribution theta ~ N[μ,τ^2]. Can anyone clarify how a quantity is considered a constant if it has a probability distribution?
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The reason is what we mean by "probability". We have an intuition of what it means given our daily life experiences, but how do we define it specifically? In the frequentist viewpoint, it is the ratio of successes over total trials when the number of total trials is very big. However, the Bayesian viewpoint probability denotes the degree of confidence we have in an outcome, where 1 (or 0) means total certainty of one event (or its complementary).
I assume that by constant you mean that the value is fixed and it's therefore not a random variable. Also, I don't know the reference you provide, but still, we can elaborate a bit on why a probability for theta can make sense even if its value is actually fixed. In the Bayesian viewpoint, it is very straightforward: while the actual value of theta is fixed, you don't know it so its probability distribution simply reflects your beliefs about what its value is. In the frequentist's viewpoint, a probability distribution over theta might refer to how its estimation is distributed. What does it mean? Given a sample of data points, you can make an estimation of what the value of theta is. However, if you tried a different sample, your estimation would be slightly different. Therefore, the estimation of theta is also distributed according to a probability distribution that one can work out for a specific problem.
In summary, in the Bayesian viewpoint, the distribution of theta reflects mathematically your beliefs, while in the frequentist one it reflects how the estimation of theta varies over the different possible samples that you might use for its estimation.
