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Can I just check I'm right about something: suppose that I've got a probabilistic system $x=s(\theta,w)$ which is fundamentally defined in terms of a set of deterministic parameters $\theta$ and a stream of random bits (which we'll call a particular instance $w$) and I'd like to infer things about the distribution of $x$ (eg, mean) for various parameter values (assuming the bit streams the system encounters are uniformly distributed). For a given set of parameters $\theta_i$ you can estimate the distribution by running the system from scratch on independently generated pseudo-random sequences $w_1,\dots,w_n$ and taking the resulting $x$ values as samples from the distribution. It's very important that the $w_i$s are independent (as much as PRNG output can be).
Now, I'm right in saying that if I'm estimating the distributions of $x$ for each set of parameters in $\theta_0, \dots, \theta_m$, it doesn't matter (statistics-wise) if I reuse the same "set of independent pseudo-random sequences" for the distribution estimation of $x$ for each $\theta_i$?