Metropolis-Hastings

Remarks:

- A sample produced by the above algorithm differs from an iid sample. For one thing, such a sample may involve repeated occurrences of the same value, since rejection of$Y_t$leads to repetition of$X^{(t)}$at time$t+1$(an impossible occurrence in absolutely continuous iid settings)
- Minimal regularity conditions on both$f$and the conditional distribution$q$for$f$to be the limiting distribution of the chain$X^{(t)}$:$\cup_{x\in \mathrm{supp}\, f}\mathrm{supp}\, q(y\mid x)\supset \mathrm{supp}\,,f$.
- $f$is the stationary distribution of the Metropolis chain: it satisfies the detailed balance property.$K(x,y) = \rho(x,y)q(y\mid x)+(1-r(x))\delta_x(y)\,.$

The MH Markov chain has, by construction, an invariant probability distribution

$f$

, if it is also an aperiodic Harris chain, then we can apply ergodic theorem to establish a convergence result.- A sufficient condition to be aperiodic: allow events such as$\{X^{(t+1)}=X^{(t)}\}$.
- Property of irreducibility: sufficient conditions such as positivity of the conditional density$q$.
- If the MH chain is$f$-irreducible, it is Harris recurrent.
- A somewhat less restrictive condition for irreducibility and aperiodicity.

In the following sections, let's introduce other versions of Metropolis-Hastings.

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