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'''Bayesian inference in [[Phylogenetics|phylogeny]]''' generates a posterior distribution for a parameter, composed of a [[phylogenetic tree]] and a model of evolution, based on the prior for that parameter and the likelihood of the data, generated by a multiple alignment. The Bayesian approach has become more popular due to advances in computational machinery, especially, [[Markov chain Monte Carlo]] algorithms. [[Bayesian inference]] has a number of applications in [[molecular phylogenetics]], for example, estimation of [[species]] phylogeny and species divergence times. | |||
==Basic Bayesian theory== | |||
Recall that for Bayesian inference: | |||
: <math> p(\theta | D) = \frac{p(D|\theta)p(\theta)}{p(D)}\ </math> | |||
The denominator <math>p(D)\ </math> is the ''marginal probability of the data'', averaged over all possible parameter values weighted by their prior distribution. Formally, | |||
: <math>p(D) = \int_\Theta p(D|\theta)p(\theta)d\theta\ </math> | |||
where <math>\Theta\ </math> is the parameter space for <math>\theta\ </math>. | |||
In the original [[Metropolis algorithm]], given a current <math>\theta\ </math>-value <math>x\ </math>, and a new <math>\theta\ </math>-value <math>y\ </math>, the new value is accepted with probability: | |||
:<math>h(y)/h(x) = \frac{p(D|y)p(y)}{p(D|x)p(x)}\ </math> | |||
==The LOCAL algorithm of Larget and Simon== | |||
The LOCAL algorithm begins by selecting an internal branch of the tree at random. The nodes at the ends of this branch are each connected to two other branches. One of each pair is chosen at random. Imagine taking these three selected edges and stringing them like a clothesline from left to right, where the direction (left/right) is also selected at random. The two endpoints of the first branch selected will have a sub-tree hanging like a piece of clothing strung to the line. The algorithm proceeds by multiplying the three selected branches by a common random amount, akin to stretching or shrinking the clothesline. Finally the leftmost of the two hanging sub-trees is disconnected and reattached to the clothesline at a location selected uniformly at random. ''This is the candidate tree''. | |||
Suppose we began by selecting the internal branch with length <math>t_8\ </math> (in Figure (a) (to be added)) that separates taxa <math>A\ </math> and <math>B\ </math> from the rest. Suppose also that we have (randomly) selected branches with lengths <math>t_1\ </math> and <math>t_9\ </math> from each side, and that we oriented these branches as shown in Figure(b). Let <math>m = t_1+t_8+t_9\ </math>, be the current length of the clothesline. We select the new length to be <math>m^{\star} = m\exp(\lambda(U_1-0.5))\ </math>, where <math>U_1\ </math> is a uniform random variable on <math>(0,1)\ </math>. Then for the LOCAL algorithm, the acceptance probability can be computed to be: | |||
:<math>\frac{h(y)}{h(x)} \times \frac{{m^{\star}}^3}{m^3}\ </math> | |||
===Assessing convergence=== | |||
Suppose we want to estimate a branch length of a 2-taxon tree under JC, in which <math>n_1</math> sites are unvaried and <math>n_2</math> are variable. Assume exponential prior distribution with rate <math>\lambda\ </math>. The density is <math>p(t) = \lambda e^{-\lambda t}\ </math>. The probabilities of the possible site patterns are: | |||
:<math>1/4\left(1/4+3/4e^{-4/3t}\right)\ </math> | |||
for unvaried sites, and | |||
:<math> 1/4\left(1/4-1/4e^{-4/3t}\right)\ </math> | |||
Thus the unnormalized posterior distribution is: | |||
: <math> h(t) = \left(1/4\right)^{n_1+n_2}\left(1/4+3/4{e^{-4/3t}}^{n_1}\right)\ </math> | |||
or, alternately, | |||
: <math> h(t) = \left(1/4-1/4{e^{-4/3t}}^{n_2}\right)(\lambda e^{-\lambda t})\ </math> | |||
Update branch length by choosing new value uniformly at random from a window of half-width <math>w\ </math> centered at the current value: | |||
:<math> t^\star = |t+U|\ </math> | |||
where <math>U\ </math>is uniformly distributed between <math>-w\ </math> and <math>w\ </math>. The acceptance | |||
probability is: | |||
:<math> h(t^\star)/h(t)\ </math> | |||
Example: <math>n_1 = 70\ </math>, <math>n_2 = 30\ </math>. We will compare results for two values of <math>w\ </math>, <math>w = 0.1\ </math> and <math>w = 0.5\ </math>. In each case, we will begin with an initial length of <math>5\ </math> and update the length <math>2000\ </math> times. (See Figure 3.2 (to be added) for results.) | |||
==Metropolis-coupled MCMC (Geyer)== | |||
If the target distribution has multiple peaks, separated by low valleys, the Markov chain may have difficulty in moving from one peak to another. As a result, the chain may get stuck on one peak and the resulting samples will not approximate the posterior density correctly. This is a serious practical concern for phylogeny reconstruction, as multiple local peaks are known to exist in the tree space during heuristic tree search under maximum parsimony (MP), maximum likelihood (ML), and minimum evolution (ME) criteria, and the same can be expected for stochastic tree search using MCMC. Many strategies have been proposed to improve mixing of Markov chains in presence of multiple local peaks in the posterior density. One of the most successful algorithms is the Metropolis-coupled MCMC (or <math>\mathrm{MC}^3\ </math>). | |||
In this algorithm, <math>m\ </math> chains are run in parallel, with different stationary distributions <math>\pi_j(.)\ </math>, <math>j = 1, 2, \ldots, m\ </math>, where the first one, <math>\pi_1 = \pi\ </math> is the target density, while <math>\pi_j\ </math>, <math>j = 2, 3, \ldots, m\ </math> are chosen to improve mixing. For example, one can choose incremental heating of the form: | |||
: <math> \pi_j(\theta) = \pi(\theta)^{1/[1+\lambda(j-1)]}, \ \ \lambda > 0, </math> | |||
so that the first chain is the cold chain with the correct target density, while chains <math>2, 3, \ldots, m</math> are heated chains. Note that raising the density <math>\pi(.)</math> to the power <math>1/T\ </math> with <math>T>1\ </math> has the effect of flattening out the distribution, similar to heating a metal. In such a distribution, it is easier to traverse between peaks (separated by valleys) than in the original distribution. After each iteration, a swap of states between two randomly chosen chains is proposed through a Metropolis-type step. Let <math>\theta^{(j)}\ </math> be the current state in chain <math>j\ </math>, <math>j = 1, 2, \ldots, m\ </math>. A swap between the states of chains <math>i\ </math> and <math>j\ </math> is accepted with probability: | |||
:<math> \alpha = \frac{\pi_i(\theta^{(j)})\pi_j(\theta^{(i)})}{\pi_i(\theta^{(i)})\pi_j(\theta^{(j)})}\ </math> | |||
At the end of the run, output from only the cold chain is used, while those from the hot chains are discarded. Heuristically, the hot chains will visit the local peaks rather easily, and swapping states between chains will let the cold chain occasionally jump valleys, leading to better mixing. However, if <math>\pi_i(\theta)/\pi_j(\theta)\ </math> is unstable, proposed swaps will seldom be accepted. This is the reason for using several chains which differ only incrementally. (See Figure3.3 (to be added)). | |||
An obvious disadvantage of the algorithm is that <math>m\ </math> chains are run and only one chain is used for inference. For this reason, <math>\mathrm{MC}^3\ </math> is ideally suited for implementation on parallel machines, since each chain will in general require the same amount of computation per iteration. | |||
{{nofootnotes|date=April 2011}} | |||
== References == | |||
*{{cite book |author=Geyer, C.J. |chapter=Markov chain Monte Carlo maximum likelihood |editor=Keramidas, E.M. |title=Computing Science and Statistics: Proceedings of the 23rd Symposium of the Interface |publisher=Interface Foundation |location=Fairfax Station VA |year=1991 |isbn= |pages=156–163 |oclc=26603816}} | |||
*{{cite journal |author=Yang Z, Rannala B |title=Bayesian phylogenetic inference using DNA sequences: a Markov Chain Monte Carlo Method |journal=Mol. Biol. Evol. |volume=14 |issue=7 |pages=717–24 |date=July 1997|pmid=9214744 |url=http://mbe.oxfordjournals.org/cgi/pmidlookup?view=long&pmid=9214744}} | |||
*{{cite journal |author=Larget B, Simon DL |title=Markov chain Monte Carlo algorithms for the Bayesian analysis of phylogenetic trees |journal= |volume=16 |issue=6 |pages=750–9 |date=June 1999 |format=PDF |url=http://mbe.oxfordjournals.org/content/16/6/750.full.pdf+html}} | |||
*{{cite journal |author=Huelsenbeck JP, Ronquist F |title=MRBAYES: Bayesian inference of phylogenetic trees |journal=Bioinformatics |volume=17 |issue=8 |pages=754–5 |date=August 2001|pmid=11524383 |url=http://bioinformatics.oxfordjournals.org/cgi/pmidlookup?view=long&pmid=11524383}} | |||
*{{cite journal |author=Ronquist F, Huelsenbeck JP |title=MrBayes 3: Bayesian phylogenetic inference under mixed models |journal=Bioinformatics |volume=19 |issue=12 |pages=1572–4 |date=August 2003|pmid=12912839 |url=http://bioinformatics.oxfordjournals.org/cgi/pmidlookup?view=long&pmid=12912839}} | |||
[[Inferencia bayesiana en filogenia|es:Español]] | |||
{{Phylogenetics}} | |||
[[Category:Bioinformatics]] | |||
[[Category:Computational phylogenetics]] | |||
[[Category:Applications of Bayesian inference|Phylogeny]] | |||
Revision as of 01:43, 24 July 2013
Bayesian inference in phylogeny generates a posterior distribution for a parameter, composed of a phylogenetic tree and a model of evolution, based on the prior for that parameter and the likelihood of the data, generated by a multiple alignment. The Bayesian approach has become more popular due to advances in computational machinery, especially, Markov chain Monte Carlo algorithms. Bayesian inference has a number of applications in molecular phylogenetics, for example, estimation of species phylogeny and species divergence times.
Basic Bayesian theory
Recall that for Bayesian inference:
The denominator is the marginal probability of the data, averaged over all possible parameter values weighted by their prior distribution. Formally,
where is the parameter space for .
In the original Metropolis algorithm, given a current -value , and a new -value , the new value is accepted with probability:
The LOCAL algorithm of Larget and Simon
The LOCAL algorithm begins by selecting an internal branch of the tree at random. The nodes at the ends of this branch are each connected to two other branches. One of each pair is chosen at random. Imagine taking these three selected edges and stringing them like a clothesline from left to right, where the direction (left/right) is also selected at random. The two endpoints of the first branch selected will have a sub-tree hanging like a piece of clothing strung to the line. The algorithm proceeds by multiplying the three selected branches by a common random amount, akin to stretching or shrinking the clothesline. Finally the leftmost of the two hanging sub-trees is disconnected and reattached to the clothesline at a location selected uniformly at random. This is the candidate tree.
Suppose we began by selecting the internal branch with length (in Figure (a) (to be added)) that separates taxa and from the rest. Suppose also that we have (randomly) selected branches with lengths and from each side, and that we oriented these branches as shown in Figure(b). Let , be the current length of the clothesline. We select the new length to be , where is a uniform random variable on . Then for the LOCAL algorithm, the acceptance probability can be computed to be:
Assessing convergence
Suppose we want to estimate a branch length of a 2-taxon tree under JC, in which sites are unvaried and are variable. Assume exponential prior distribution with rate . The density is . The probabilities of the possible site patterns are:
for unvaried sites, and
Thus the unnormalized posterior distribution is:
or, alternately,
Update branch length by choosing new value uniformly at random from a window of half-width centered at the current value:
where is uniformly distributed between and . The acceptance probability is:
Example: , . We will compare results for two values of , and . In each case, we will begin with an initial length of and update the length times. (See Figure 3.2 (to be added) for results.)
Metropolis-coupled MCMC (Geyer)
If the target distribution has multiple peaks, separated by low valleys, the Markov chain may have difficulty in moving from one peak to another. As a result, the chain may get stuck on one peak and the resulting samples will not approximate the posterior density correctly. This is a serious practical concern for phylogeny reconstruction, as multiple local peaks are known to exist in the tree space during heuristic tree search under maximum parsimony (MP), maximum likelihood (ML), and minimum evolution (ME) criteria, and the same can be expected for stochastic tree search using MCMC. Many strategies have been proposed to improve mixing of Markov chains in presence of multiple local peaks in the posterior density. One of the most successful algorithms is the Metropolis-coupled MCMC (or ).
In this algorithm, chains are run in parallel, with different stationary distributions , , where the first one, is the target density, while , are chosen to improve mixing. For example, one can choose incremental heating of the form:
![{\displaystyle \pi _{j}(\theta )=\pi (\theta )^{1/[1+\lambda (j-1)]},\ \ \lambda >0,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2e9d8c159f93b004aa99ba2ce2249578daf6db1e)
so that the first chain is the cold chain with the correct target density, while chains are heated chains. Note that raising the density to the power with has the effect of flattening out the distribution, similar to heating a metal. In such a distribution, it is easier to traverse between peaks (separated by valleys) than in the original distribution. After each iteration, a swap of states between two randomly chosen chains is proposed through a Metropolis-type step. Let be the current state in chain , . A swap between the states of chains and is accepted with probability:
At the end of the run, output from only the cold chain is used, while those from the hot chains are discarded. Heuristically, the hot chains will visit the local peaks rather easily, and swapping states between chains will let the cold chain occasionally jump valleys, leading to better mixing. However, if is unstable, proposed swaps will seldom be accepted. This is the reason for using several chains which differ only incrementally. (See Figure3.3 (to be added)).
An obvious disadvantage of the algorithm is that chains are run and only one chain is used for inference. For this reason, is ideally suited for implementation on parallel machines, since each chain will in general require the same amount of computation per iteration.
References
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The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
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Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang - One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang - One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang