Bayes: Markov chain Monte Carlo

Summary

The video script introduces Markov chain Monte Carlo (MCMC) techniques, crucial for approximating complex Bayesian posterior models that are mathematically intractable. It explains the limitations of simple Bayesian models and the necessity of MCMC when dealing with multiple parameters in regression models. The script outlines the Monte Carlo method, highlighting its random sampling approach from the posterior distribution, and contrasts it with MCMC, which generates a dependent chain of samples to explore the parameter space. The video uses the Beta-Binomial model to illustrate how MCMC can provide a good approximation of the posterior, despite the complexity of the model.

Takeaways

• 🧩 The video provides an introduction to Markov chain Monte Carlo (MCMC) simulation techniques, which are used for approximating complex Bayesian posterior models.
• 📚 The script discusses the transition from simple Bayesian models to more complex models with multiple parameters, which can complicate the process of identifying the posterior model.
• 🔍 In complex models, calculating the overall plausibility of observing data across all possible parameter values can be difficult or impossible, necessitating the use of approximation techniques like MCMC.
• 🎲 MCMC techniques involve simulating a sample of parameter values (theta) to approximate features of the posterior distribution, overcoming the challenge of directly calculating the normalizing constant in Bayes' Rule.
• 👣 The Monte Carlo approach is presented as a special case of MCMC, where a random sample is drawn independently from the posterior distribution, which can be used to approximate the posterior when direct calculation is not feasible.
• 🏛 The origin of Monte Carlo methods is traced back to the 1940s and their development for understanding neutron travel in the context of nuclear weapons projects at Los Alamos.
• 🔑 The script illustrates the Monte Carlo method with an example of a Beta-Binomial model, demonstrating how to approximate the posterior distribution using simulated data pairs.
• 🔄 Markov chain Monte Carlo (MCMC) is introduced as a more sophisticated method for approximating the posterior when direct sampling is not possible, involving a chain of dependent values that build upon the previous value.
• 🔗 The Markov property is explained, stating that the future value in the chain depends only on the present value, not on the entire history of values, which is key to MCMC's efficiency.
• 🌐 The video script uses a visual example of a Markov chain to show how the chain explores the sample space and eventually provides a good approximation of the posterior distribution, even though the chain values are not directly drawn from it.
• ✨ The video concludes by emphasizing the power of MCMC techniques to approximate complex posterior models mathematically, highlighting their utility in Bayesian modeling.

Q & A

What is the main purpose of using Markov chain Monte Carlo (MCMC) simulation techniques?

The main purpose of using MCMC techniques is to approximate Bayesian posterior models that are too complex to specify mathematically, particularly when dealing with models that have multiple parameters and are defined in more complex settings.

Why do we need to move beyond simple Bayesian models to more complex ones?

We need to move beyond simple Bayesian models to more complex ones because they allow us to handle models with multiple regression coefficients (theta_1, theta_2, ..., theta_k) and to analyze data in more sophisticated and realistic ways.

What is the challenge when trying to identify the posterior model in complex settings?

The challenge in complex settings is that the posterior can be too complicated to identify from the product of the prior and likelihood alone. It requires calculating a normalizing constant in the denominator of Bayes’ Rule, which involves integrating across all possible parameter values, a task that can be extremely difficult or impossible.

What is the Monte Carlo approach and how does it relate to MCMC?

The Monte Carlo approach is a special case of Markov chain Monte Carlo that involves producing a random sample of size N from the posterior probability density function (pdf) f of theta given y. Each theta value in the sample is independent of the others and is drawn directly from the posterior pdf. It serves as an introduction to MCMC, which is a more sophisticated method for generating samples when direct sampling from the posterior is not feasible.

How does the Markov chain in MCMC differ from a random sample in a Monte Carlo simulation?

In a Markov chain, each value (theta i) is drawn from a model that depends on the previous value (theta i-1), creating a chain of dependence. This is in contrast to a random sample in a Monte Carlo simulation, where each sample is independent of the others. The Markov chain satisfies the Markov property, meaning the future state depends only on the current state, not on the sequence of events that preceded it.

Why is it necessary to use a dependent sample like a Markov chain when approximating the posterior?

A dependent sample like a Markov chain is necessary because it allows the simulation to explore the posterior distribution more efficiently, especially when the posterior is complex and cannot be directly sampled. Over time, the Markov chain explores the parameter space in a way that reflects the posterior distribution, despite not being drawn directly from it.

What is the Markov property in the context of MCMC?

The Markov property in the context of MCMC refers to the characteristic of a Markov chain where the future state (theta (i + 1)) is dependent only on the current state (theta i) and is independent of all previous states. This property is crucial for the chain to explore the parameter space effectively.

How does the length of the MCMC chain affect the quality of the approximation?

The length of the MCMC chain (N) is crucial for the quality of the approximation. A longer chain allows for more exploration of the parameter space, which typically results in a better approximation of the posterior distribution. However, it also means that the chain needs to be run for a sufficient number of iterations to overcome any initial bias or lack of diversity in the early samples.

What are some potential issues with using MCMC to approximate complex posterior models?

Potential issues with using MCMC include the dependency of chain values, which means the sample is not truly random, and the fact that the values are drawn from a model that is not the posterior pdf f. Additionally, the convergence of the chain to the target distribution must be carefully checked to ensure the approximation is valid.

How can one check the quality of the approximation provided by an MCMC simulation?

The quality of the approximation can be checked by comparing the distribution of the MCMC sample to the known posterior distribution, if available. Additionally, diagnostic tools such as trace plots, autocorrelation plots, and convergence statistics (like the Gelman-Rubin statistic) can be used to assess the quality and convergence of the MCMC simulation.

What is the significance of the 'mathemagic' mentioned in the script in the context of MCMC?

The term 'mathemagic' is used to describe the somewhat counterintuitive yet powerful process by which MCMC techniques can provide a good approximation of the posterior distribution. Despite the dependent nature of the Markov chain and the fact that it is not drawn directly from the posterior, under the right conditions, the chain's distribution converges to the posterior, allowing for accurate approximations.

Outlines

00:00

📚 Introduction to Markov Chain Monte Carlo (MCMC)

This paragraph introduces the concept of Markov chain Monte Carlo (MCMC) simulation techniques. It starts by emphasizing the importance of moving beyond basic Bayesian models to more complex ones involving multiple parameters. The paragraph explains the limitations of using Bayes' Rule in complex models, where calculating the posterior can be extremely difficult due to the need to integrate across all possible parameter values. The solution proposed is to use MCMC techniques to approximate the Bayesian posterior models that are too complex to specify mathematically. The process involves simulating a sample of theta values and using this sample to approximate features of the posterior. The paragraph also introduces the Monte Carlo approach as a special case of MCMC, which has a historical background related to the nuclear weapons project at Los Alamos National Laboratory. The Monte Carlo method is illustrated with an example of approximating a posterior in a Beta-Binomial model by simulating a random sample from the prior and then narrowing down to the pairs that match the observed data.

05:00

🔍 Exploring Monte Carlo and Markov Chain Monte Carlo Techniques

The second paragraph delves deeper into the Monte Carlo method, explaining how it can be used to approximate the posterior model by simulating pairs of pi values and corresponding data points. It details a process where from a set of prior plausible values of pi, Binomial data points are simulated, and then the pairs that match the observed outcome are selected to approximate the posterior. The paragraph demonstrates that even without knowing the exact posterior, Monte Carlo methods can provide a close approximation, as shown in a Beta-Binomial model example. However, it also points out the limitations of Monte Carlo when the posterior is too complex to sample directly from. This is where MCMC becomes essential, as it can handle more sophisticated Bayesian models. MCMC produces a sample of parameter values where each value in the chain is dependent on the previous one, satisfying the Markov property. The paragraph concludes by building intuition about how a Markov chain operates and its potential to explore the sample space of possible values effectively.

10:01

🔄 Understanding the Markov Chain Process in MCMC

The final paragraph focuses on the Markov chain process within MCMC, illustrating how it explores the sample space of possible values for a parameter, such as pi in the Beta-Binomial model. It discusses how the chain traverses different regions of the sample space, with the hope that by the end of the simulation, it will have visited various values with a frequency and range that reflects the posterior model. The paragraph addresses potential skepticism regarding the use of a dependent sample from a model other than the posterior to approximate it. It explains that despite the chain values being drawn from a conditional pdf that is not the posterior pdf, when the chain is long enough and the process is efficient, the Markov chain will behave like a random sample from the posterior. This behavior allows the distribution of the Markov chain sample to converge and provide a good approximation of the posterior. The paragraph concludes with an example of a Markov chain simulation that provides an excellent approximation of the Beta(12,8) posterior, despite none of the chain values being drawn from it, showcasing the power of MCMC techniques.

Keywords

💡Markov chain Monte Carlo (MCMC)

MCMC is a simulation technique used in Bayesian statistics to approximate complex posterior distributions that are difficult to calculate analytically. The video's theme revolves around explaining the necessity and process of using MCMC when dealing with models that have multiple parameters and complex settings. An example from the script involves using MCMC to approximate the posterior in a regression model with multiple coefficients, theta_1 to theta_k.

💡Bayesian models

Bayesian models are statistical models that incorporate prior knowledge or beliefs into the analysis and update this knowledge as evidence or data is observed. The video discusses transitioning from fundamental Bayesian models to more complex ones, where the posterior distribution becomes too complicated to identify directly from the prior and likelihood, thus necessitating the use of MCMC techniques.

💡Posterior model

In Bayesian analysis, the posterior model represents the updated beliefs about the parameters after considering the observed data. The script explains that while it's straightforward to identify the posterior in simple models using Bayes' Rule, in more complex models, it becomes necessary to approximate the posterior using MCMC due to its complexity.

💡Regression model

A regression model is a statistical approach for estimating the relationships between variables. In the context of the video, a regression model of Y is mentioned, defined by multiple regression coefficients. This serves as a practical example where MCMC becomes useful for approximating the posterior distribution of these coefficients in complex scenarios.

💡Normalizing constant

The normalizing constant is a factor in Bayesian analysis that ensures the posterior distribution sums or integrates to one, representing a probability distribution. The video script explains that calculating this constant becomes complicated in complex models, which is one of the reasons MCMC is used to approximate the posterior distribution.

💡Monte Carlo methods

Monte Carlo methods are a class of random sampling techniques used to approximate complex problems by generating random samples. The video script describes Monte Carlo as a special case of MCMC, where a random sample from the posterior distribution is generated independently for each parameter value, which is used to approximate the posterior in simpler cases.

💡Sample size or chain length (N)

In MCMC, the sample size or chain length (N) refers to the number of simulated parameter values in the sample. The script mentions that the MCMC sample size is crucial as it determines the number of theta values simulated, which in turn affects the quality of the approximation of the posterior distribution.

💡Markov property

The Markov property is a characteristic of a Markov chain where the future state depends only on the current state and not on the sequence of events that preceded it. The video script uses this property to explain how each value in the MCMC chain is dependent on the previous value, which simplifies the sampling process in complex models.

💡Conditional pdf (probability density function)

In the context of MCMC, the conditional pdf is the probability distribution from which each value in the chain is drawn, given the previous value. The script explains that this pdf is not the posterior distribution itself, but through the Markov chain process, it can produce a sample that approximates the posterior.

💡Convergence

Convergence in MCMC refers to the point at which the sample distribution produced by the chain closely resembles the target posterior distribution. The video script emphasizes that, with an efficient implementation and a sufficiently long chain, the Markov chain will converge to the posterior distribution, allowing for a good approximation.

💡Beta-Binomial model

The Beta-Binomial model is a Bayesian hierarchical model that combines a Beta distribution for the prior and a Binomial distribution for the likelihood. The script uses this model as an example to illustrate how MCMC can be used to approximate the posterior distribution of the success probability (pi) when direct calculation is not feasible.

Highlights

Introduction to Markov chain Monte Carlo (MCMC) simulation techniques for Bayesian models with multiple parameters.

Complex Bayesian models may require the calculation of a normalizing constant, which can be difficult or impossible.

MCMC techniques are used to approximate Bayesian posterior models that are too complex to specify mathematically.

The process involves simulating a sample of theta values to approximate features of the posterior.

Monte Carlo methods are a special case of MCMC, producing a random sample from the posterior pdf.

Monte Carlo methods were originally developed in the 1940s for nuclear weapons projects at Los Alamos National Laboratory.

In the Beta-Binomial model example, a Monte Carlo sample of pi values approximates the posterior pdf.

Direct sampling from the posterior is not always possible, necessitating the use of MCMC methods for more complex models.

MCMC methods produce a sample of parameter values where each value depends on the previous one, forming a Markov chain.

The Markov property ensures that the future value of the chain is independent of all past values, given the present value.

MCMC chains explore the sample space of possible parameter values, gradually converging to the posterior distribution.

Despite the dependent nature of MCMC samples, they can provide a good approximation of the posterior when the chain is long enough.

The distribution of an MCMC sample will converge to the posterior, allowing for accurate approximations in complex models.

An example of a Beta-Binomial model demonstrates how an MCMC chain of 10,000 pi values can approximate the true posterior.

MCMC techniques are a powerful tool for approximating complex Bayesian posterior models that cannot be easily specified.

The success of MCMC in approximating the Beta(12,8) posterior showcases the potential of these methods in Bayesian statistics.

MCMC methods provide a practical solution for integrating across all possible parameter values in complex Bayesian models.

The Markov chain's exploration of the parameter space is a key feature that enables the approximation of the posterior distribution.

Transcripts

In this video, we'll get a quick  introduction to Markov chain Monte Carlo,  

or MCMC simulation techniques.

Let’s first consider why we should care.  During this course, we’ve been studying  

nice Bayesian models. Mainly, we’ve  been able to identify a posterior model  

from the prior model and likelihood function  by chugging their formulas through Bayes’ Rule.  

BUT we’ll soon move beyond fundamental Bayesian  models, and onto models with multiple parameters  

in more complex settings. For example, we might be  interested in a regression model of Y, where this  

regression model is defined by multiple regression  coefficients theta_1, theta_2, up to theta_k.  

We can apply the Bayesian philosophy in these  more advanced settings, but it gets complicated.

In more simple model settings, we’ve been  able to utilize the proportionality here,  

or to identify the posterior from the  product of the prior and likelihood alone.  

However, in more complex settings, the posterior  can be too complicated to identify from this  

product. For example, this product won’t be  the kernel of a simple Beta or Gamma model.

In such cases, specifying the posterior  requires that we actually calculate this  

normalizing constant in the denominator of  Bayes’ Rule, that is, that we calculate the  

overall plausibility of observing data  y across all possible parameter values.  

Yet this calculation also gets complicated  in more complex model settings.  

To calculate the overall plausibility of observing  data y across all possible parameter values,  

we have to integrate across all of  these possible parameter values.  

That is, we have to do this  complicated multivariate integral.  

Actually doing this calculation can be  prohibitively difficult, if not impossible.

Yet there’s hope! It’s true that in Bayesian  modeling, we can’t always specify or know  

the posterior model. BUT when we can’t  know something, we can approximate it!  

To this end, we’ll use Markov chain Monte  Carlo (or MCMC) techniques to approximate  

Bayesian posterior models that are otherwise  too complicated to specify mathematically.  

The basic idea is this. We first  simulate a sample of theta values,  

theta 1, theta 2, on up to theta N where  N is our MCMC sample size or chain length.  

We then use this sample to approximate  features of the posterior f of theta given y.  

It turns out that this step 2 will be  straightforward. The bigger challenge is  

in step 1, actually simulating the sample of theta  values. There are various approaches to this task.  

We’ll start with the Monte Carlo approach, which  is a special case of Markov chain Monte Carlo.  

Monte Carlo techniques have a bit of a dubious  history. They were developed in the 1940’s to  

better understand neutron travel for the  nuclear weapons project at the Los Alamos  

National Laboratory. “Monte Carlo” was  the code name for this top secret work,  

a choice said to be inspired by the opulent  Monte Carlo casino in the French Riviera.

In terms of our goal of simulating a posterior  model, Monte Carlo methods produce a random sample  

of size N from the posterior pdf  f of theta given y. Specifically,  

in this sample of theta 1, theta 2, on up to  theta N values, each theta value is independent  

of the others, and each theta value is drawn  from the posterior pdf f of theta given y.

Consider an example. In the  familiar Beta-Binomial model,  

suppose we start with a Beta(4,6) prior for  pi, and we collect Binomial data on 10 trials.  

Upon observing 8 successes in those 10 trials,  our posterior model for pi is a Beta(12, 8).

Now, for illustrative purposes, suppose we weren’t  familiar with the Beta-Binomial model and weren’t  

able to specify this posterior. In this case,  we could use Monte Carlo methods to approximate  

the posterior. There are many approaches  that will produce a Monte Carlo sample.  

Consider just one. First, after setting  the random number seed for reproducibility,  

we can take a random sample of 100,000 pi  values from the Beta(4, 6) prior model.

Then, from each of these  prior plausible values of pi,  

we can simulate a Binomial data point from 10  trials with the corresponding probability of  

success pi. Using just the known prior  and conditional model for the data,  

these first 2 steps produce 100,000  pairs of pi values and data points y.

To approximate the posterior model  that balances the prior with the  

observed outcome of Y = 8 successes in 10 trials,  

we want to narrow in on just those pairs  that have a matching data point, y equals 8.

In this case, only 3729 of the 100,000 simulated  pairs had data points matching Y = 8. Thus this  

technique produced a Monte Carlo sample of 3729 pi  values, the first three of which are shown here.

Finally, we can use this Monte Carlo  sample to approximate the posterior.

For example, the distribution  of our Monte Carlo pi values  

provides an approximation of the posterior pdf.

When we compare this approximation  to the actual Beta(12,8) posterior  

that we were trying to approximate, the  results are nearly indistinguishable.  

Now, it’s important to remember that the  major point of Monte Carlo in practice is  

to approximate something we don’t actually know,  thus we don’t usually have this benefit of being  

able to check the quality of the approximation  as we’ve done here. That said, our success in  

this example provides some peace of mind that  Monte Carlo approximation can actually work.

Now, this is all great. The Monte  Carlo sample is nice and random,  

and we’ve seen that it can provide a  close approximation of our posterior.  

BUT, remember that the whole motivation for this  conversation is that we only need approximation  

tools when the posterior is really complicated.  And when the posterior is really complicated,  

we typically can’t directly sample from it as we  did in our example. And, that means that when we  

can’t sample directly from the posterior,  we can’t implement this Monte Carlo method.

This is where Markov chain Monte Carlo comes in.  MCMC methods can scale up for more sophisticated  

Bayesian models. Like Monte Carlo, MCMC  methods produce a sample of parameter values.  

Yet a Markov chain grows one value at a  time. From theta 1 to theta 2 to theta 3  

on up to theta N where, again,  capital N is the chain length.

Each value in the chain here is drawn from  a model that depends on the previous value.  

For example, theta (i + 1) is drawn from a  model with pdf g that depends upon data y and  

the previous chain value theta i. This evokes the  chain terminology -- each chain value depends upon  

the previous value in the chain, which depends  upon the previous value, and so on. This chain  

of dependence also satisfies the Markov property.  That is, given the present chain value, theta i,  

the future value, theta (i +1) is  independent of all other past values.  

Or in mathematical notation, if you tell me all  of the past chain values theta 1 on up to theta i,  

the model of the next value theta (i + 1)  depends only on the previous value theta i.

To build some intuition, consider a Markov  chain for our Beta(12,8) posterior. Let’s  

start with a short chain or sample of  just 20 pi values. As it grows in length,  

this chain traverses the sample space of possible  pi values, that sample space being on the y axis.  

Each step or "iteration" of the chain depends  upon the previous step -- thus you’ll notice  

here that sometimes the chain trends upward  or downward before correcting its path.

Further, notice that in just  the first 20 iterations,  

the chain here largely explores values  of pi between roughly 0.45 and 0.8.

After 200 iterations, the Markov chain  has started to explore new territory.  

The hope here is that, by the end of the  simulation, the chain will have visited  

various pi values with a frequency and range  that’s reflective of the posterior model.

But you might be skeptical here. MCMC methods can  scale up for more sophisticated Bayesian models,  

but there’s a cost. First, remember  that the chain values are dependent,  

each value in the chain depending upon the  previous value through the conditional pdf g.

Not only that, but this pdf g from which we draw  the chain values is not the posterior pdf f.

As such, a Markov chain theta 1 up to theta  N is not a random sample from the posterior.

This might all seem strange that we’re  trying to approximate the posterior model  

using a dependent sample from  something other than the posterior.  

But when done efficiently and when the chain is  long enough, this strange, dependent Markov chain  

will at least mimic or behave like a random sample  from the posterior. As such, the distribution of a  

Markov chain sample will converge to and provide a  good approximation of the posterior. For example,  

check out our final Markov chain simulation of  10,000 pi values for our Beta-Binomial model.  

This chain provides an excellent  approximation of the Beta(12,8) posterior,  

despite the fact that none of the chain values  were actually drawn from this posterior model.  

In short, by mathemagic, Markov chain Monte  Carlo techniques can help us approximate  

posterior models that are otherwise too  complicated to specify mathematically.