1

I have a density that can be written as follows,

$$ p(\mathbf{z}) = \frac{1}{H_f}\prod_iq_i(\mathbf{z}) $$

I want to sample from the density $p(\mathbf{z})$ using samples from each of $q_i(\mathbf{z})$, and there are a bunch of algorithms that let me do this, according to the answer to this question here - Previous similar question

In those solutions, we are required to draw samples from $q_i(\mathbf{z})$. In the problem I am faced with, my $q_i(\mathbf{z})$ has the following form

$$ q_i(\mathbf{z}) = [h_i(\mathbf{z})]^{\alpha_i} $$ where $\alpha_i \in \mathbb{R}^{+}$ so the overall density can be written as $$ p(\mathbf{z}) = \frac{1}{H_f}\prod_i[h_i(\mathbf{z})]^{\alpha_i} $$ I know how to sample from $h_i(\mathbf{z})$ without MCMC techniques - $h_i(\mathbf{z})$ is an exponential family distribution that I have chosen such that I can sample from it easily

My question is as follows

  1. How do I sample from the unnormalised density $[h_i(\mathbf{z})]^{\alpha_i}$, given I can sample from $h_i(\mathbf{z})$

2.Can I still use the kind of techniques in the above reference?

3.Are there more modern approaches that aren't direct followups from the papers in that question?

Pablo
  • 111
  • 3
  • 2
    The question is unclear to me because, mathematically,$$g(q_1(z)^{\alpha_1},\ldots,q_n(z)^{\alpha_n})=f(q_1(z),\ldots,q_n(z))$$ – Xi'an Apr 13 '21 at 12:59
  • Fair point, I will edit the question and I hope this clarifies – Pablo Apr 13 '21 at 13:19
  • 1
    I still cannot make sense of this question, because your mathematical formulation does not appear to describe what is usually known as a mixture and it is too general to permit a solution. – whuber Apr 13 '21 at 13:27
  • I shall give an exact example and hope that is workable – Pablo Apr 13 '21 at 13:29
  • 2
    @whuber I have rewritten the question to try and resolve your problems with it. – Pablo Apr 13 '21 at 13:44
  • If $h(z)$ is an exponential family density, so is $h^{\alpha_i}(z)$. – Xi'an Apr 13 '21 at 13:50
  • Oh - that doesn't make sense to me, I'm afraid - can you clarify? If $h_{i}^{\alpha_i}(\mathbf{z}) = (\exp(\eta^{T}\psi(\mathbf{z}) - \Phi(\mathbf{\eta}))\nu(\mathbf{z}))^{\alpha_i} = \exp(\alpha_i\eta^{T}\psi(\mathbf{z}) - \alpha_i\Phi(\mathbf{\eta}))(\nu(\mathbf{z})^{\alpha_i})$, its surprising to me that this density is normalised, and the terms in the exponential don't appear to fit the normal structure. Can you explain to me an intuitive reason why this holds? If it does, I suppose the question is whether the technique for the original ExpFam density carry across to the new one? – Pablo Apr 13 '21 at 14:04

0 Answers0