Given i.i.d data $X_1, X_2, ..., X_n$ ~ certain distribution having certain parameter $\theta$, and we are then compute the maximum likelihood of $\theta$, namely $\theta_{MLE}$.

I'm just wondering, is $\theta_{MLE}$ invariant if we change the data order (permutate) for any distribution? My understanding so far tells me that it's always the case. But am i wrong ? is there a distribution on which this is not the case?


By exchangeability we mean

Formally, an exchangeable sequence of random variables is a finite or infinite sequence $X_1,X_2,\dots, X_n$ of random variables such that for any finite permutation $\sigma$ of the indices $1, 2, 3, \dots$, (the permutation acts on only finitely many indices, with the rest fixed), the joint probability distribution of the permuted sequence

$$ X_{\sigma(1)}, X_{\sigma(2)}, X_{\sigma(3)}, \dots $$

is the same as the joint probability distribution of the original sequence. In short, the order of the sequence of random variables does not affect its joint probability distribution.

If you have a sequence of independent and identically distributed random variables $X_1,X_2,\dots,X_n$, that say, each follow the distribution $f$, then by independence their joint distribution is

$$ g(x_1,x_2,\dots,x_n) = \prod_{i=1}^n f(x_i) $$

and since multiplication has the commutative property, so $a \times b = b \times a$, then obviously the distribution doesn't matter. Notice that this follows from independence alone, so they wouldn't even need to be identically distributed. Yet, exchangeability does not require independence, so while it is related to the concept of i.i.d., they are not synonyms.

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