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Suppose I have some random variable X with probability distribution P(.;theta). Suppose I have a single sample x from this distribution.

Does it make any sense to ask for the derivative of x with respect to theta?

I believe it does not, but am having trouble convincing some of my colleagues.

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2 Answers

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It does not because $x$ is just some element of your state space. You could conceivably choose $x$ as your sample point for any or all of the $\theta$'s. So what it would make sense to differentiate with respect to $\theta$ is $$\mathbb{P}_{\theta}(X=x)$$ as long as you have a discrete distribution. For a continuous distribution you could substitute $x$ with some interval.

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What might make sense is if $X$ is a function $g(U,\theta)$ of some underlying random variable $U$ (with distribution not depending on $\theta$) and $\theta$, where the differentiable function $g$ is chosen so that $g(U,\theta)$ has the given (continuous) distribution. Then you could say $\dfrac{dX}{d\theta} = \dfrac{\partial g}{\partial \theta}(U,\theta)$. Of course this depends on the "implementation", i.e. the choice of $g$ and $U$, rather than just on the distribution of $X$.

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