I want to estimate $E(f(X))$ using quasi-Monte Carlo where $X = (X_1,\ldots,X_n)$ is a random vector and $$ X_i\sim f(\cdot; \theta), \quad \text{independent}, $$ where $\theta \in \mathbb{R}$ is some known parameter and $f(\cdot; \theta)$ is a pdf for each $\theta$. Let's say I want to use the Halton sequence. Usually, I would generate an $n$-dimensional Halton sequence $(H_1,\ldots,H_N)$ where $N$ is the number of Monte Carlo runs and each $H_i \in \mathbb{R}^n$. Then (after transforming to $f(\cdot;\theta)$ random variables) my estimate is $$ E(f(X)) \approx \frac{1}{N}\sum_{i=1}^n f(H_i) $$ Now I want to specify the distributions of the $X_i$ as \begin{align*} X_i \mid \theta & \sim f(\cdot;\theta), \quad \text{independent}, \\ \theta \sim g, \end{align*} where $g$ is a pdf.

My question is, what is the dimension of the Halton vector I should use in this case? Or, should I generate two Halton sequences, one of dimension 1 and the other of dimension $n$?


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