http://wiki.christophchamp.com/index.php?action=history&feed=atom&title=Metropolis-coupled_Markov_chain_Monte_CarloMetropolis-coupled Markov chain Monte Carlo - Revision history2026-04-30T06:55:20ZRevision history for this page on the wikiMediaWiki 1.26.2http://wiki.christophchamp.com/index.php?title=Metropolis-coupled_Markov_chain_Monte_Carlo&diff=1547&oldid=prevChristoph: Added more info. + "References"2005-12-30T00:37:33Z<p>Added more info. + "References"</p>
<table class='diff diff-contentalign-left'>
<col class='diff-marker' />
<col class='diff-content' />
<col class='diff-marker' />
<col class='diff-content' />
<tr style='vertical-align: top;' lang='en'>
<td colspan='2' style="background-color: white; color:black; text-align: center;">← Older revision</td>
<td colspan='2' style="background-color: white; color:black; text-align: center;">Revision as of 00:37, 30 December 2005</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l1" >Line 1:</td>
<td colspan="2" class="diff-lineno">Line 1:</td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>'''Metropolis-coupled Markov chain Monte Carlo''' (or '''MCMCMC''' or '''(MC)<sup>3</sup>''') is a variant of [[Markov chain Monte Carlo]] in which multiple chains are run in parallel, each with a different "temperature". Only the information from the cold chain is recorded. Periodically, trees between chains may be swapped.</div></td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>'''Metropolis-coupled Markov chain Monte Carlo''' (or '''MCMCMC''' or '''(MC)<sup>3</sup>''') is a variant of [[Markov chain Monte Carlo]] in which multiple chains are run in parallel, each with a different "temperature" <ins class="diffchange diffchange-inline">(Geyer, 1991)</ins>. Only the information from the cold chain is recorded. Periodically, trees between chains may be swapped.</div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline"><blockquote></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline">"Some of these chains are '[[Heated chains|heated]]' by raising the [[posterior probability]] to a power &beta;. For example, if ''f'' (&psi;|'''''X''''') is the posterior probability density distribution of the phylogenetic parameters, then a heated version of the posterior distribution is ''f'' (&psi;|'''''X''''')<sup>&beta;</sup>. Here, &beta;(0 < &beta; < 1) is the heat value of the chain. Heating a Markov chain increases the acceptance probability of new states. A heated chain tends to accept more states than a cold chain, allowing a heated chain to more readily cross valleys in the landscapes of trees." (Altekar ''et al.'', 2004).</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins class="diffchange diffchange-inline"></blockquote></ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>(MC)<sup>3</sup> can be used to empirically determine the posterior probability distribution of trees, branch lengths, and substitution parameters.</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>(MC)<sup>3</sup> can be used to empirically determine the posterior probability distribution of trees, branch lengths, and substitution parameters.</div></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l24" >Line 24:</td>
<td colspan="2" class="diff-lineno">Line 28:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>* [[Metropolis-Hastings]]</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>* [[Metropolis-Hastings]]</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>* [[Heated chains]]</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>* [[Heated chains]]</div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">== References ==</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">* Altekar G, Dwarkadas S, Huelsenbeck JP, and Ronquist F (2004). Parallel Metropolis coupled Markov chain Monte Carlo for Bayesian phylogenetic inference. ''Bioinformatics''.</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color:black; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">* Geyer CJ (1991). Markov chain Monte Carlo maximum likelihood. In Keramidas (ed.), ''Computing Science and Statistics: Proceedings of the 23rd Symposium on the Interface''. Fairfax Station: Interface Foundation, pp. 156-163.</ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>[[Category:Academic Research]]</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>[[Category:Academic Research]]</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>[[Category:Statistics]]</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>[[Category:Statistics]]</div></td></tr>
</table>Christophhttp://wiki.christophchamp.com/index.php?title=Metropolis-coupled_Markov_chain_Monte_Carlo&diff=1543&oldid=prevChristoph: Started article2005-12-30T00:05:53Z<p>Started article</p>
<p><b>New page</b></p><div>'''Metropolis-coupled Markov chain Monte Carlo''' (or '''MCMCMC''' or '''(MC)<sup>3</sup>''') is a variant of [[Markov chain Monte Carlo]] in which multiple chains are run in parallel, each with a different "temperature". Only the information from the cold chain is recorded. Periodically, trees between chains may be swapped.<br />
<br />
(MC)<sup>3</sup> can be used to empirically determine the posterior probability distribution of trees, branch lengths, and substitution parameters.<br />
<br />
<br />
; Temperature : A variable used in Metropolis-coupled Markov chain Monte Carlo. The temperature affects the likelihood of acceptance of a proposed tree and also affects the likelihood that two chains will accept a proposed tree swap.<br />
<br />
== Practice ==<br />
* Only one chain is sampled<br />
* The other chains are heated (i.e. they can take bigger steps)<br />
* Chains can swap states<br />
* Allows crossing of valleys<br />
* This process of linking heated chains is called "Metropolis-coupling"<br />
* The full-fledged Bayesian analysis is MCMCMC!<br />
* The heated chains use powers of the likelihood ratio in their acceptance ratio<br />
* We must '''only''' use the main chain to calculate posterior probabilities.<br />
* In reality, all these factors involve very large numbers. It's not uncommon to throw away thousands of trees as part of the burn-in and calculate the posterior from millions of trees.<br />
* Heated chains: usually 4-5.<br />
* [[MrBayes]] by Huelsenbeck is the main program in current use<br />
<br />
== See also ==<br />
* [[Markov chain]]<br />
* [[Markov chain Monte Carlo]]<br />
* [[Metropolis-Hastings]]<br />
* [[Heated chains]]<br />
<br />
[[Category:Academic Research]]<br />
[[Category:Statistics]]</div>Christoph