Geometric Ergodicity of the Random Walk Metropolis with Position-Dependent Proposal Covariance
We consider a Metropolis–Hastings method with proposal <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">N</mi><mo>(</mo><mi>x</mi><m...
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| Format: | Article |
| Language: | English |
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MDPI AG
2021-02-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/9/4/341 |
| _version_ | 1797412608200933376 |
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| author | Samuel Livingstone |
| author_facet | Samuel Livingstone |
| author_sort | Samuel Livingstone |
| collection | DOAJ |
| description | We consider a Metropolis–Hastings method with proposal <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">N</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>h</mi><mi>G</mi><msup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo></mrow></semantics></math></inline-formula>, where <i>x</i> is the current state, and study its ergodicity properties. We show that suitable choices of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow></semantics></math></inline-formula> can change these ergodicity properties compared to the Random Walk Metropolis case <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">N</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>h</mi><mo>Σ</mo><mo>)</mo></mrow></semantics></math></inline-formula>, either for better or worse. We find that if the proposal variance is allowed to grow unboundedly in the tails of the distribution then geometric ergodicity can be established when the target distribution for the algorithm has tails that are heavier than exponential, in contrast to the Random Walk Metropolis case, but that the growth rate must be carefully controlled to prevent the rejection rate approaching unity. We also illustrate that a judicious choice of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow></semantics></math></inline-formula> can result in a geometrically ergodic chain when probability concentrates on an ever narrower ridge in the tails, something that is again not true for the Random Walk Metropolis. |
| first_indexed | 2024-03-09T05:05:35Z |
| format | Article |
| id | doaj.art-fa3b5449c86040baa4ea8aedf2ad4b05 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-09T05:05:35Z |
| publishDate | 2021-02-01 |
| publisher | MDPI AG |
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| series | Mathematics |
| spelling | doaj.art-fa3b5449c86040baa4ea8aedf2ad4b052023-12-03T12:56:06ZengMDPI AGMathematics2227-73902021-02-019434110.3390/math9040341Geometric Ergodicity of the Random Walk Metropolis with Position-Dependent Proposal CovarianceSamuel Livingstone0Department of Statistical Science, University College London, London WC1E 6BT, UKWe consider a Metropolis–Hastings method with proposal <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">N</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>h</mi><mi>G</mi><msup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo></mrow></semantics></math></inline-formula>, where <i>x</i> is the current state, and study its ergodicity properties. We show that suitable choices of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow></semantics></math></inline-formula> can change these ergodicity properties compared to the Random Walk Metropolis case <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">N</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>h</mi><mo>Σ</mo><mo>)</mo></mrow></semantics></math></inline-formula>, either for better or worse. We find that if the proposal variance is allowed to grow unboundedly in the tails of the distribution then geometric ergodicity can be established when the target distribution for the algorithm has tails that are heavier than exponential, in contrast to the Random Walk Metropolis case, but that the growth rate must be carefully controlled to prevent the rejection rate approaching unity. We also illustrate that a judicious choice of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow></semantics></math></inline-formula> can result in a geometrically ergodic chain when probability concentrates on an ever narrower ridge in the tails, something that is again not true for the Random Walk Metropolis.https://www.mdpi.com/2227-7390/9/4/341Monte CarloMCMCMarkov chainscomputational statisticsbayesian inference |
| spellingShingle | Samuel Livingstone Geometric Ergodicity of the Random Walk Metropolis with Position-Dependent Proposal Covariance Mathematics Monte Carlo MCMC Markov chains computational statistics bayesian inference |
| title | Geometric Ergodicity of the Random Walk Metropolis with Position-Dependent Proposal Covariance |
| title_full | Geometric Ergodicity of the Random Walk Metropolis with Position-Dependent Proposal Covariance |
| title_fullStr | Geometric Ergodicity of the Random Walk Metropolis with Position-Dependent Proposal Covariance |
| title_full_unstemmed | Geometric Ergodicity of the Random Walk Metropolis with Position-Dependent Proposal Covariance |
| title_short | Geometric Ergodicity of the Random Walk Metropolis with Position-Dependent Proposal Covariance |
| title_sort | geometric ergodicity of the random walk metropolis with position dependent proposal covariance |
| topic | Monte Carlo MCMC Markov chains computational statistics bayesian inference |
| url | https://www.mdpi.com/2227-7390/9/4/341 |
| work_keys_str_mv | AT samuellivingstone geometricergodicityoftherandomwalkmetropoliswithpositiondependentproposalcovariance |