BIOL 4330 Unit 2 1 2 Bayesian Analysis and Markov Chain Monte Carlo

Summary

The transcript explains Bayesian analysis, particularly its use in phylogenetic studies. It highlights how complex probability distributions are sampled using the Markov Chain Monte Carlo (MCMC) method to identify optimal evolutionary trees. The analogy of searching for the highest peak in a rugged landscape is used to illustrate how random starting points help explore possible outcomes. Bayesian analysis iteratively refines the model of evolution and parameters, improving accuracy while being computationally efficient. The process is compared with other methods like maximum likelihood and parsimony, emphasizing Bayesian analysis' speed and ability to provide support values.

Takeaways

• 📊 Statistical analysis requires understanding the distribution of outcomes, which can be simple or complex depending on the population.
• 🌳 For human height, a normal distribution is suitable, but phylogenetic analysis involves more complex, multi-dimensional probability distributions.
• 🔍 Markov Chain Monte Carlo (MCMC) is used to estimate complex probability distributions by simulating random walks through the search space.
• 🗺 The MCMC process starts with a random point and explores nearby areas, using an algorithm to move towards better scoring areas, akin to finding the highest peak in a rugged landscape.
• 📈 The process involves running the MCMC, pausing to re-estimate parameters, and repeating until a plateau of scores is reached, indicating the best overall solution.
• 🌐 Phylogenetic search space is multi-dimensional and large, requiring extensive sampling to ensure the best solutions are found.
• 🔄 The Bayesian analysis involves an iterative process of estimating parameters, running MCMC, and re-estimating parameters until convergence.
• 🌲 The 'burn-in' period discards initial trees with lower scores, focusing on saving trees with high scores that represent the best solutions.
• 📊 A consensus tree is made from the saved trees, with support values indicating how often certain relationships appear, providing a measure of confidence in the phylogeny.
• 🤔 Bayesian analysis is favored for its computational efficiency and ability to provide support measures, although other methods like maximum likelihood and parsimony are also discussed.

Q & A

What is the primary purpose of using statistical analysis in phylogenetic studies?

The primary purpose is to estimate the probability distribution of phylogenetic outcomes due to the complexity of phylogenies and the vast number of possible trees. Statistical analysis helps in identifying the most likely phylogenetic trees that represent evolutionary relationships.

Why are normal distributions insufficient for phylogenetic studies?

Normal distributions are insufficient for phylogenetic studies because phylogenies involve complex, multidimensional probability distributions, unlike simpler distributions like human height, which follow a normal distribution.

What is the Markov Chain Monte Carlo (MCMC) method, and why is it useful in phylogenetics?

MCMC is a statistical approach used to estimate probability distributions by sampling nearby regions in a complex search space. In phylogenetics, it helps simulate and estimate the likelihood of various phylogenetic trees and outcomes, even in highly rugged topologies with multiple local maxima.

How does the analogy of searching for the highest peak relate to phylogenetic tree optimization?

The analogy compares phylogenetic search space to a landscape with peaks. In MCMC, a random starting point is selected, and the algorithm explores surrounding areas to find the highest peak, much like trying to optimize a phylogenetic tree by improving its score step-by-step.

What is the 'burn-in period' in Bayesian phylogenetic analysis?

The burn-in period refers to the early stage of MCMC sampling where the trees with lower scores are discarded as the algorithm converges on better-scoring phylogenetic trees. During this phase, poorer solutions are filtered out.

How do Bayesian analyses differ from maximum likelihood and parsimony analyses?

Bayesian analyses not only search for the best trees based on a model of evolution but also provide a posterior probability for relationships, giving a measure of support for different parts of the tree. Maximum likelihood and parsimony do not automatically provide this support for relationships.

What is the role of the model of evolution in Bayesian phylogenetic analysis?

The model of evolution provides the framework for estimating the likelihood of different phylogenetic trees. Bayesian analysis refines the parameters of this model iteratively to improve tree scores, ultimately producing a tree that best fits the model.

Why is it important to re-estimate parameters during the Bayesian analysis?

Re-estimating parameters during Bayesian analysis allows for the adjustment of the model of evolution to better fit the data. This iterative process helps in converging toward the best phylogenetic tree with improved accuracy.

What does it mean when the analysis reaches a plateau phase?

The plateau phase occurs when repeated iterations of the Bayesian analysis no longer improve the tree scores, indicating that the best solutions have been found, and further iterations will not lead to better trees.

How does a consensus tree in Bayesian analysis provide support values for relationships?

In Bayesian analysis, after saving the trees with good scores, a consensus tree is created by comparing all saved trees. Relationships found in most trees are given higher support values, while those found in fewer trees are given lower values or collapsed into polytomies.

Outlines

00:00

📊 Understanding Probability Distributions in Phylogenetic Analysis

This paragraph introduces the concept of analyzing complex probability distributions in phylogenetic trees. It begins by comparing simpler distributions like human height to the more complex multi-dimensional probability spaces in phylogenetic analysis. To navigate this complexity, Markov Chain Monte Carlo (MCMC) is introduced as a technique to estimate and sample probability distributions in rugged search topologies. The analogy of a blind person searching for a peak in a rugged landscape is used to explain how MCMC samples areas with good solutions and repeats the process to ensure accuracy.

05:00

🧬 Refining Models in Phylogenetic Analysis Through Iteration

This paragraph explains the iterative process used to refine evolutionary models in phylogenetic analysis. Starting with a random model, scores are calculated and gradually improve as the analysis is repeated. The model is continually informed by posterior probabilities, and the process is repeated until scores plateau, indicating the best solution has been found. Rugged and simple topologies are contrasted, with an example of rugged peaks in the Superstition Mountains and a smoother topology like Mount Fuji, to illustrate how MCMC explores different possibilities.

10:00

🏔️ Searching for the Best Phylogenetic Tree: Multiple Iterations and Burn-in Periods

This section delves into the process of searching for the best phylogenetic tree by running multiple iterations and saving trees with good scores. It discusses the 'burn-in' period where initial bad trees are discarded, and the process repeats multiple times (four to eight times typically) to ensure the best trees are found. A consensus tree is eventually created from all saved trees, where relationships are scored based on how frequently they appear across the saved trees. This consensus tree provides support values for different branches, which help indicate the reliability of the relationships.

15:01

🌲 Using Bayesian Analysis for Efficient Tree Searches

This paragraph highlights the advantages of Bayesian analysis over other methods like maximum likelihood and parsimony in phylogenetic studies. Bayesian analysis is praised for being computationally efficient and providing a built-in support measure for relationships within the trees. The process is particularly useful when maximum likelihood is too computationally intense. The paragraph emphasizes that Bayesian analysis allows researchers to better understand phylogenetic relationships by using a statistical approach to evaluate results.

🧠 Optimizing Models of Evolution with Bayesian Analysis

This final paragraph discusses how Bayesian analysis optimizes the model of evolution by refining parameters iteratively. It notes that while maximum likelihood and parsimony focus on finding the best tree based on set criteria, Bayesian analysis modifies both the model of evolution and its parameters to reach the best solution. This iterative refinement continues until a plateau is reached, ensuring the best possible phylogenetic model. The conclusion emphasizes the need to understand the principles of various phylogenetic methods to achieve the best overall phylogeny.

Keywords

💡Normal Distribution

A normal distribution is a probability distribution that is symmetrical around its mean, where most occurrences take place close to the mean, and fewer occurrences happen as you move away from it. In the video, it's used as an example of a simple statistical model that works well for traits like human height, but it contrasts with the complex distributions required for phylogenetic analysis.

💡Phylogeny

Phylogeny refers to the evolutionary relationships between different species or organisms, often represented in the form of a tree. The video highlights that phylogenies are more complex and have much more intricate probability distributions compared to simpler traits like height, which adds complexity to their analysis.

💡Search Space

The search space refers to the range of possible outcomes or solutions that an algorithm must explore. In the context of phylogenies, this search space is described as 'rugged' and 'multi-dimensional,' meaning there are many possible solutions that need to be explored to find the best evolutionary tree.

💡Markov Chain Monte Carlo (MCMC)

MCMC is a statistical method used to estimate the probability distribution of complex systems. In the video, it is applied to phylogenetic analysis, allowing for efficient sampling of the search space to identify the most probable evolutionary trees by starting at random points and optimizing step-by-step.

💡Local Maximum

A local maximum is a peak in a probability landscape where the solution found is better than its immediate neighbors but might not be the best overall (global maximum). The video uses this concept to explain how MCMC might settle on a good but not the best solution when searching through phylogenies.

💡Posterior Probability

Posterior probability is the updated probability of a model or hypothesis given new data. In Bayesian analysis of phylogenies, the posterior probability is used to update the model of evolution as the algorithm finds better trees, ensuring that the analysis improves over time.

💡Burn-in Period

The burn-in period refers to the initial phase of a Bayesian or MCMC analysis, during which early, less accurate results are discarded. The video describes this as the period where trees with lower scores are eliminated, and only high-quality trees are kept for further analysis.

💡Model of Evolution

The model of evolution is a framework used to describe how genetic sequences change over time. In the video, the speaker explains how different models are used to estimate parameters that guide the search for the best evolutionary trees during the Bayesian analysis.

💡Support Values

Support values are measures of confidence in the relationships found in a phylogenetic tree. In the video, these values represent the percentage of times a particular relationship or grouping of species is found in the best-scoring trees, offering a measure of how reliable the tree is.

💡Maximum Likelihood

Maximum likelihood is a statistical method used to find the tree that best explains the observed genetic data. The video contrasts this with Bayesian analysis, explaining that while maximum likelihood is highly accurate, it can be computationally intense, making Bayesian methods a faster alternative.

Highlights

Understanding distribution of outcomes is crucial for statistical analysis, particularly when analyzing complex systems like phylogenies.

Normal distributions work well for simple populations, such as human height, but phylogenies require much more complex probability distributions.

Markov Chain Monte Carlo (MCMC) is used to simulate and estimate the probability distribution for complex phylogenetic spaces.

The analogy of optimizing elevation in a rugged landscape is used to explain how MCMC searches for the highest scoring phylogenetic trees.

By sampling local areas, MCMC efficiently searches complex, multi-dimensional spaces to find local and potentially global maximums.

In rugged search topologies, multiple high-scoring solutions can be found, and MCMC helps in identifying the best ones.

Random starting points and iterative sampling allow MCMC to explore vast phylogenetic spaces and avoid being trapped in local maxima.

The posterior probability distribution from MCMC informs models of evolution, helping to refine and improve subsequent analyses.

Scores typically improve rapidly at the start of an MCMC analysis, plateauing as the best solutions are found.

The 'burn-in period' involves discarding early trees with lower scores and saving those with higher scores as the analysis progresses.

MCMC helps create consensus trees by comparing all high-scoring trees and assigning support values based on the frequency of common relationships.

Bayesian analysis provides support values, which are useful for gauging the reliability of different phylogenetic relationships.

Compared to maximum likelihood, Bayesian analysis is computationally less intense but still provides reliable statistical results.

Bayesian analyses optimize both the model of evolution and the tree parameters, making it versatile for large datasets.

A key advantage of Bayesian analysis is its ability to incorporate statistical support measures directly into the tree-building process.

Transcripts

so as an example we've set up this

idea of having some understanding of a

distribution

of outcomes for us to be able to

do a statistical analysis now this is a

very simple these

normal distributions whether they're

flat or very pronounced

are fairly simple distributions and for

some

populations that works really really

well if we're looking at height

in humans fall is a fairly normal

distribution that's a fairly

simple thing to analyze now

when we are dealing with phylogenys our

probability

distributions are much much more complex

and this is because phylogenys are

complex and there are so many

of them and so although it's not ideal

maybe the best way to think about it is

like a

three-dimensional but in reality it's

multi-dimensional

space and so in this we might have some

areas that are pretty good

these local peaks and the higher we are

here the the better the

tree scores are but this is a very

rugged and

difficult topology and so because

phylogenys can have such a complex

probability of outcomes we and because

we don't know what that might be ahead

of time

we need a way that we can estimate and

look

at this probability outcome this search

space

and to do that we use a process called

markov chain monte carlo now

you don't need to know all the details

about it it's a fairly advanced

statistical approach we're going to give

you the bird's eye overview

so it's a way to quickly simulate and

estimate what the probability

distribution is

okay and to do this very simply what you

do is you pick a random

point to start at and then you

use some sort of an algorithm to sample

areas that are nearby

and i'm going to use the analogy of

a landscape let's say that you're

looking

at to find the highest peak you're

optimizing

elevation and you're trying to find the

highest peak so what you do is you

randomly drop an individual off

and then because you can't look around

right so

in phylogeny if you've got an individual

phylogen you can't just look at the

phylogeny look at all of the phylogenys

next to it see which ones are better you

have to test them

and so in this analogy let's say you

have a blind individual

and you're trying to help that blind

individual find the highest peak in an

area

so you drop them off you have them do

some

explorations right next to them so they

take a step

north south east west and if one of

those steps makes them

a little bit higher in elevation that's

the new place

and then they go from that one they do

north south east west and that's a new

place and so by doing

a very quick rough algorithm we could

potentially

continually go up in elevation until we

reach the highest point at that point

they would keep going north southwest

gate lower and we would sample this area

and sample and sample and sample

that would be a local maximum it's no

guarantee that it's overall the global

maximum

and so we would randomly start another

one and maybe we randomly start here we

go up up up and then we sample this peak

and so this is a very fast method

for sampling very rugged search

topologies and we're using this

three-dimensional search topology as an

analogy but in reality remember

that phylogenetic search space is a

multi-dimensional space and a very very

large one that we might need to spend

some time and effort

sampling but our markov chain monte

carlo

is an approach to sample the probability

distribution

and do so in a way that is fairly

certain of guaranteeing finding us all

of the really

best ones whether it's a you know very

rugged space like this we have a few

very good answers or it's a little bit

more spread out

and we'd have a sampling distribution

that was a little bit more spread out

and so this is again as long as you know

the details

which is what we we pick a random start

we look around it if we find a better

answer nearby

that becomes our new one we do that over

and over and over again until we've

sampled the search topology

fairly rapidly okay now

in addition to this markov chain monte

carlo for sampling the probability

search space we then use those

probabilities to

to calculate a uh model of evolution

and what we see when we do this for most

analyses

is that we start off with a model of

evolution

and we find a tree we then allow that

tree using this markov chain monte carlo

search strategy we allow that tree to

then inform that becomes our posterior

probability

and it informs our model of evolution we

re-estimate the parameters for that

model of evolution and then we run the

analysis again

again and we do that over and over and

over again

and what we find is that when we start

randomly we usually don't have a really

great score

but our scores and usually will improve

rather rapidly

until we are getting really similar

scores and as we

do this over and over again right

allowing our posterior probability

to then inform the

parameters for our next round of that

analysis

and we get scores out and we usually

will find a plateau and this plateau

represents

a peak in scores where we found a very

best overall solution

and if it's a very rugged search

topology we might get slightly different

trees during an another random event but

the scores are going to be very similar

if it's a very let me just i'm going to

google i'll bring in some scores

some uh possible over here um

i grew up in an area uh in arizona where

the nearby

mountains were called the superstition

uh mountains

and i choose them as an example because

they have this very rugged search

topology if you're trying to find the

highest peak in the superstitions

it could be difficult this is just the

most famous set of peaks but they're

actually others

so here let's use this one let's say

you're trying to find the highest peak

in something that looks like this

and obviously there are multiple

candidates and so this might take a lot

of searching but with enough we could be

relatively

certain that we found it and we might

get a set of answers that are very

similar from that one and that one as

far as what the highest peak is

a very different search topology would

be something like this

and i picked mount fuji because no

matter where you started in the in the

surroundings around mount fuji you would

be going up the slope to find the very

best

overall one and so if we have a very

small set of phylogenies that are all in

one kind of general vicinity with all

very similar

answers it would be a fairly simple one

no matter where we started we would end

up

recovering that but with markov chain

monte carlo we can also recover

all the best answers our population of

best answers

even if our search topology is very very

rugged so we do

random beginning points and then we save

all of the trees

that are in this very good score area we

discard

the phylogenesis that have lower scores

and this is called the burn-in period

now during this burn-in period and

actually even during the

plateau period we also allow our model

of evolution to vary

so let's do a step-by-step

walkthrough and i'm going to i'll type

it out here but let's do a step by step

walk through of what we do for a

bayesian analysis

number one we estimate

our parameters

for the model of evolution

and we do this i'll put this here it's a

little bit long but we do this based

on our initial

prior probability

now remember we don't know anything at

all about the search topology we don't

know if it's mount fuji we don't know if

it's superstition mountains

we have no idea and so our best estimate

at the beginning let's make

so our neighbor joining tree although

it's not the

probably the very best overall set of

relationships it gets us in the ballpark

and that helps us then estimate our

parameters for our model of evolution

and then we run some iterations we use

markov chain monte carlo we use that

model of evolution with the parameters

estimated from it to get a

answer and then we pause

so number two run

some analyses

and we'll put mcmc this markov chain

monte carlo as we run those analyses

and then at that point we pause

and re-estimate

our parameters so maybe as we're running

this analysis our scores for our trees

parameters

are getting a little bit better

and that's great if they're getting a

little bit better that means our

estimation of phylogeny is probably

better

and then we can re-estimate our

parameters and then we

repeat

and as long as our scores are getting

better and better and better we continue

to do that

then at some point our scores will reach

kind of this plateau phase where no

matter how many times we go through

these steps of repeating re-estimating

our

parameters for our model of evolution

getting the tree scores out of it

we get this burn-in or the burn-in

period is gone right so that says trees

are getting better and better and better

we reject all of those trees and then we

start saving all these trees that have

really good scores and many of these

trees might be

repetitions of ones we found before

we're finding the trees over and over

and over again

but we're kind of dialing in our model

of evolution

and then we repeat the whole thing

usually four times sometimes six or

eight times repeat the entire process

four to we'll say four to eight times

you can do more but after a while you're

kind of getting uh

limited returns for your efforts so

repeat this in prior

prior entire process four to eight times

and each of these times when we started

we're doing these

random starts we save all of these trees

so the end result of this is this

large large trees hundreds of thousands

of trees

that we've saved that are all pretty

good scores as we've looked at our

entire search to policy

we then make a consensus tree for all of

these trees that we've we've

found so basically we compare all of the

trees we look at

what relationships they have in common

and what relationships are different and

we and we put numbers on those

and these numbers become our support

value so for instance

if this relationship between pleodorana

and

clam idomonas these two species

if that is found in every single one of

those good trees that we've saved then

we put 100 there representing 100

of the time if a set of relationships so

these three species together is only

found there and 82 percent of the trees

then we put that number there and if a

set of relationships is not found in the

majority of the trees we collapse that

node down

and we make it a polytomy so the nice

thing

about the maximum

i'm sorry the bayesian analysis is that

we end up with support values

as part of the process of doing the tree

we'll look at other support values in

our next unit that are used for other

analyses but a maximum likelihood

analysis

gives you a tree but it doesn't tell you

if there are some of those relationships

that are better supported than others

a parsimony analysis the same thing you

get an end result and says this is the

best tree overall based on this

parsimony criteria

but it doesn't tell you if some of those

relationships are better supported than

others

a bayesian analysis does that so it's a

another time saving a part of a bayesian

analysis

so not only are bayesian analyses

overall faster than maximum likelihood

analyses

but they also give us a support measure

at the end of the analysis so that's a

huge

advantage for bayesian analysis okay

so

review of some things we've talked about

already

we can use known phylogenies like from

bacteria and determine whether parsimony

maximum likelihood or bayesian analyses

are best

and we can also use simulated data

and we've talked about areas like long

branch attraction where parsimony fails

we talked about areas like rate

heterogeneity also called heterochronic

where maximum likelihood might fail

but overall if we're in kind of a

majority of these trees we get

pretty good results from even neighbor

joining but certainly maximum likelihood

parsimony and bayes analyses are going

to give us good results

but because of statistical

considerations many people want to use

either maximum likelihood analysis but

that can be computationally intense and

so bayesian analyses is a little bit of

a less computationally intense way

to analyze and then we can look at

results

so here's a result where different

methods you know part weighted parsimony

versus parsimony

find the very best tree and notice that

in

most considerations given enough data

we're going to find the very best trees

but some methods perform better overall

like maximum likelihood even neighbor

joining if we have

correct models of evolution okay so

the take-home message again is that we

may want to use a variety of methods

if we want to use statistical approaches

we might do a maximum likelihood unless

it's too computationally intense for the

size of our data

and then we might want to do bayesian

but also compare it to a parsimony

analysis and our best overall phylogeny

might represent a consensus

among methods and so this congruence

among methods is a very good way and if

we have areas that

are not congruent then maybe we want to

reevaluate or gather more data to

determine if we can resolve that or

maybe we say well

i'm going to use the maximum likelihood

result or the bayesian result

because i want to do a falsifiability

test i want to

use those statistical approaches to be

able to do that

and so here's our big picture overview

of all of the methods that we've talked

about

now originally i put the bayesian

analysis in this

category of using nucleotide sites and

we certainly do that we're not using

distance data so it's very clearly in

this right hand column

but whether it goes in a clustering

algorithm or an optimality criterion is

a little bit different

in both parsimony and maximum likelihood

we really are

optimizing we're doing this by doing a

very intensive search of lots and lots

of trees

and we're really optimizing the score

based on either a parsimony criterion

or this likelihood score that we gave

you an overview of

in a bayesian analysis we're doing

something slightly different which is

why i put it up here but with a question

mark

we are really optimizing the model of

evolution

and the parameters for that model by

doing this iterative approach where we

start with an unknown distribution

maybe we give a neighbor joining tree we

then allow the analysis as we progress

to inform it

and to then go back in and re-evaluate

or

change the parameters for our model of

evolution and then see if that gives us

a better tree

if it does great we keep them if not and

we allow at the next round to

change those parameters for the model of

evolution so in reality a bayesian

analysis is

optimizing our model of evolution and

our parameters for that model of

evolution

and then allowing that to inform our

overall outcome

so it's not really maybe a better place

for bayesian analysis is kind of in the

gray area between these two things

because we're not really finding the

best tree based on a single optimality

criteria

we're allowing our analyses to inform

the parameters

change our model of evolution or at

least the parameters for that model of

evolution

and then reiterate to it again and again

and again until we don't get any better

we've kind of

reached a plateau phase where we're not

getting better scores no matter how many

times

we do this iterative approach so

review this make sure you're familiar

with these albeit somewhat of an

overview of methods for some of these

you should know how to map trees map

characters onto a tree in a parsimony

analysis and find a score for that tree

and then just understand the basic

principles for these more

computationally intense methods maximum

likelihood and statistical methods

for bayesian analysis