Metropolis Algorithm for a high dimensional Bimodal distribution

I am using metropolis mcmc for an $$n=8$$ dimensional system on an (n-1)-sphere.

I was considering the 2d case, as it can be visualized. For this case, the probability density,up to a normalization, is $$$$p(x, y) = \exp{(-\beta(x^2 - y^2))},$$$$

with $$x^2 + y^2$$ = 1, on $$S^1$$. $$\beta$$ is some constant. For higher dimensions, the distribution is generalized to $$p(x_1, \cdots, x_n) = \exp{(-\beta \cdot f(x_1^2, \cdots, x_n^2))}$$, for some function $$f$$.

I use a Von-Mises Fisher proposal distribution; however, the acceptance rates are extremely low, around 0.009 and 0.01. Increasing the concentration parameter too much seems to cause the sampling to concentrate on one of the two peaks, either $$(0, -1)$$ or $$(0, 1)$$, but not both.

What proposal distribution is needed to properly "cover" both peaks? And how can this be generalized for a higher dimensional system, for the total of $$2^8$$ possible $$x_i \rightarrow -x_i$$ give the same probability?

EDIT: I have thought of a possible solution, though I don't know if there are nuanced issues. I'm thinking that when I choose a candidate, I then randomly construct a vector of 1's and -1's, say $$s = (s_1, \cdots, s_n)$$, where $$s_i = \pm 1$$. Then, the new candidate is $$x' = (s_1 \cdot x_1, s_2 \cdot x_2, \cdots, s_n \cdot x_n)$$.

• Since the distribution is symmetric, all that is needed is a simulation method on $(x_1^2,\ldots,x_n^2)$. May 26 at 7:04
• @Xi'an Do you mean to sample $(x_1^2,\cdots,x_n^2)$ rather than $(x_1,\cdots,x_n)$? If so, how would you then get information on the desired vector $(x_1,\cdots,x_n)$? May 26 at 7:39