Monte-Carlo
  • Introduction
  • AIS
  • Generate R.V.
    • Special Distribution
    • Copulas
    • Minimum of Two Exponential
  • Gibbs
    • Comparing two groups
    • Linear Regression
    • Simulation of Exp-Abs-xy
  • Markov Chain
  • MC Approximation
  • MC Integration
    • Rao-Blackwellization
  • MC Optimization
  • MCMC
    • MCMC diagnostics
  • Metropolis-Hastings
    • Metropolis
    • Independent MH
    • Random Walk MH
    • ARMS MH
  • PBMCMC
  • RJMCMC
  • Diagnosing Convergence
  • SMCTC
  • Tempering
    • Parallel Tempering
  • Misc
    • R vs. Julia
  • References
Powered by GitBook
On this page

Metropolis-Hastings

PreviousMCMC diagnosticsNextMetropolis

Last updated 6 years ago

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 YtY_tYt​ leads to repetition of X(t)X^{(t)}X(t) at time t+1t+1t+1 (an impossible occurrence in absolutely continuous iid settings)

  • Minimal regularity conditions on both fff and the conditional distribution qqq for fff to be the limiting distribution of the chain X(t)X^{(t)}X(t): ∪x∈supp fsupp q(y∣x)⊃supp ,f\cup_{x\in \mathrm{supp}\, f}\mathrm{supp}\, q(y\mid x)\supset \mathrm{supp}\,,f∪x∈suppf​suppq(y∣x)⊃supp,f.

  • fff is the stationary distribution of the Metropolis chain: it satisfies the detailed balance property.

    K(x,y)=ρ(x,y)q(y∣x)+(1−r(x))δx(y) .K(x,y) = \rho(x,y)q(y\mid x)+(1-r(x))\delta_x(y)\,.K(x,y)=ρ(x,y)q(y∣x)+(1−r(x))δx​(y).
  • A sufficient condition to be aperiodic: allow events such as {X(t+1)=X(t)}\{X^{(t+1)}=X^{(t)}\}{X(t+1)=X(t)}.

  • Property of irreducibility: sufficient conditions such as positivity of the conditional density qqq.

  • If the MH chain is fff-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.

The MH Markov chain has, by construction, an invariant probability distribution fff, if it is also an aperiodic , then we can apply to establish a .

Harris chain
ergodic theorem
convergence result