Linked Questions

6
votes
4answers
4k views

Is “ independent and identically distributed” an assumption or a fact ?

This is in the context of two random variables. A frequent assumption (e.g. of the error term in ANOVA) is of independent and identically distributed random variables. There is a question on this site ...
10
votes
4answers
663 views

Suppose $X_1, X_2, \dotsc, X_n$ are i.i.d. random variables. When is the sequence expected to decrease for the first time?

As suggested in the title. Suppose $X_1, X_2, \dotsc, X_n$ are continuous i.i.d. random variables with pdf $f$. Consider the event that $X_1 \leq X_2 \dotsc \leq X_{N-1} > X_N$, $N \geq 2$, thus $...
9
votes
2answers
281 views

How is $P(D;\theta) = P(D|\theta)$?

I started reading about maximum likelihood estimator and Bayesian statistics recently. I understand that given a statistical model $(X, (P_\theta))$ where, $\theta$ belongs to a large parameter space $...
8
votes
3answers
244 views

Where can I read about the justification for the use of parametric probability distributions?

I would like to find a reference, preferably free on the internet, where I can read about the theoretical or practical justification for the use of parametric / analytic probability distributions. By ...
4
votes
1answer
487 views

Correct understanding of De Finetti`s representation theorem

I am currently interestend in understanding De Finetti`s representation theorem. As I am only familiar with Frequentist thinking I have some problems to understand its meaning. I have already read the ...
5
votes
1answer
446 views

Why are words in a document for bag-of-words model exchangeable but not independent?

I've been watching a talk (section between 07:20-08:00) given by Michael Jordan and I'm getting confused between independence and exchangeability. He says that "If we have a document and you ...
4
votes
1answer
113 views

Are two coin flips conditionally independent if we know that the coin is biased towards heads?

Suppose Alice (A) and Bob (B) each flip the same, potentially-biased coin. Then, P(A=H) < P(A=H | B=H), because Bob's flip increases our suspicion that the coin is biased towards heads. Now ...
3
votes
1answer
178 views

De Finetti: equivalence a.s. according to which measure

In Zen`s answer at What is so cool about de Finetti's representation theorem? that is concerned with De Finetti's 0 -1 representation theorem, he says that "De Finetti's law of large numbers" ...
5
votes
1answer
149 views

Do the parameters that arise in de Finetti`s representation theorem follow the rules of probability?

I recently stumbled upon de Finetti`s (pretty cool) representation theorem (What is so cool about de Finetti's representation theorem?). I wondered whether the RV $\Theta$ that arises in this ...
1
vote
0answers
66 views

A Contradiction in Bayesian Statistics?

In Bayesian statistics, we usually assume that data $X_{1},...,X_{n}$ are independent to each other. But the predictive distribution shows that they are not independent because we have $p(X_{n+1}|X_{1}...