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Suppose I have some continuous data that's distributed according to

$$x_i \sim p(X|\theta)$$

for some unknown parameters $\theta$. We may draw some $N$ samples from density $p$. Then we can say that the data has $N$ degrees of freedom, which can be an important concept for coming up with estimators for the parameters in $\theta$. But what if the samples are not exchangeable, e.g. if they are auto-regressive:

$$x_i \sim p(X_i|\theta, X_{i-1})$$ with $$x_0 \sim \int p(X_i|\theta)p(\theta)d\theta$$

Then what is the way to calculate the degrees of freedom when the $\bf{X}$ samples are structured and not I.I.D.?

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The concept of degrees of freedom only applies to linear models, not to arbitrary random variables. This is because the degrees of freedom are defined as the dimension of a vector space. So, your question doesn't really make sense.

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  • $\begingroup$ That's good to know. Is there a term for the concept of what I'm describing however, which is not defined as merely a dimension, but incorporates probabilistic coupling between data samples? I've read that dof corresponds to the rank of a quadratic form, akin to the dimension of vector space as you state. Is there any statistical concept for probabilistic degrees of freedom around continuous properties of the quadratic form besides the rank, e.g. the determinant? $\endgroup$
    – user27886
    Sep 11, 2016 at 6:46
  • $\begingroup$ @user27886 I don't think so. $\endgroup$ Sep 11, 2016 at 15:32

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