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I have a discrete 7-element random vector $\vec{X}$ with probability mass function $P_{\vec{X}}(\color{blue}{x_1,x_2,x_3},x_4,x_5,x_6,\color{red}{x_7})$ that has a symmetry in its certain components.

The symmetry is such that: the first three components (the blue one) exhibits symmetry among themselves (black ones); the next three have symmetry in themselves too, and last one (red) is alone. (For further information about $P_{\vec{X}}$: $\vec{X}$ is a finite mixture of 8 components in seven dimensions. The seven variables in each of the eight components; are conditionally independent.)

Mathematically the symmetries are:

$$\begin{align} P_{\vec{X}}(\color{blue}{x_1,x_2,x_3},x_4,x_5,x_6,\color{red}{x_7})&=P_{\vec{X}}(\color{blue}{x_2,x_1,x_3,}x_4,x_5,x_6,\color{red}{x_7}) \\ &= P_{\vec{X}}(\color{blue}{x_3,x_2,x_1,}x_5,x_6,x_4,\color{red}{x_7}) \\ & \vdots \\ & \vdots \\ & \vdots \\ &= P_{\vec{X}}(\color{blue}{x_3,x_2,x_1,}x_6,x_5,x_4,\color{red}{x_7}). \end{align}$$

Now lets come to the problem:

By exploiting the symmetry in my pmf, I need to take samples from $P_{\vec{X}}(x_1, \cdots x_7)$. Does there exist a way that is computationally efficient (linear w.r.t no. of symmetries instead of exponential w.r.t no. of variables) over applying a naive sampling method that do not exploit inherent symmetry ? My actual problem is very high dimensional for instance $2^{100}$, so I desperately need to exploit the symmetry.

My attempt: 1) I took independent samples from each component and respect the mixing proportions by taking the proportional no. of samples from that component. However, still, the no. of variables (which are seven) are not decreasing.

2) Using MetroPolis Hastings algo is taking too much time? Perhaps,I am not aware of what should be the better proposal distribution.

Any leads would be highly appreciated.

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