Let Val(X) be a set of states, and let T1, . . . , Tk be a set of kernels, each of which satisfies the detailed balance equation relative to some stationary distribution π. Let p1, . . . , pk be any distribution over the indexes 1, . . . , k. Prove that the mixture Markov chain T , which at each step takes a step sampled from Ti with probability pi, also satisfies the detailed balance equation relative to π.
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