Metropolis Hastings Sampler In Pigeons.jl: A How-To Guide

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Metropolis Hastings Sampler in Pigeons.jl: A How-To Guide

Hey guys! Today, we're diving into the awesome world of Metropolis Hastings samplers within the Pigeons.jl ecosystem. This is a fantastic topic, especially if you're keen on understanding and implementing parallel tempering techniques. The goal here is to provide a comprehensive guide that not only addresses the initial request but also gives you a solid foundation to build upon.

Why Metropolis Hastings?

First off, let's chat about why you might want to use a Metropolis Hastings sampler. Essentially, this algorithm is a Markov Chain Monte Carlo (MCMC) method for obtaining a sequence of random samples from a probability distribution for which direct sampling is difficult. Think of it as a way to explore a complex, high-dimensional space where you don't have a straightforward way to draw samples. It's particularly handy when the probability distribution is known only up to a constant factor.

In the context of Pigeons.jl, having a simple Metropolis Hastings sampler can be super beneficial, especially for educational purposes. It allows newcomers to grasp the core concepts of MCMC and parallel tempering without getting bogged down in complicated implementations. The beauty of starting with something simple is that you can gradually add complexity and optimizations as your understanding grows.

Moreover, Metropolis Hastings provides a flexible framework. You can tweak the proposal distribution to suit the specific problem you're tackling. This adaptability is crucial because the efficiency of the sampler heavily depends on how well the proposal distribution matches the target distribution. For instance, a random walk proposal might be suitable for some problems, while others might benefit from a more informed proposal.

When you're introducing someone to parallel tempering, starting with a Metropolis Hastings sampler is a smart move. Parallel tempering, also known as replica exchange Monte Carlo, involves running multiple Metropolis Hastings chains at different temperatures. The 'temperature' controls how easily the chain can escape local minima. By allowing chains to exchange states periodically, you can significantly improve the exploration of the sample space, especially for multimodal distributions.

So, to sum it up, Metropolis Hastings is a foundational algorithm that offers simplicity, flexibility, and educational value. It's an excellent starting point for anyone venturing into the realms of MCMC and parallel tempering with Pigeons.jl.

Addressing the Original Request

The initial request highlighted the desire for a simple Metropolis Hastings/Random Walk Proposal sampler within Pigeons.jl. The user mentioned that while SliceSampler works great out of the box, having a more basic sampler would be beneficial for introducing parallel tempering concepts. They also pointed out an example here that unfortunately errored.

Let's break down how we can address this. First, we need to create a basic Metropolis Hastings sampler. This involves defining a proposal distribution, calculating acceptance probabilities, and implementing the core MCMC loop. Then, we can integrate this sampler into the Pigeons.jl framework.

Here’s a step-by-step approach to creating such a sampler:

  1. Define the Target Distribution: This is the probability distribution you want to sample from. It could be any function that returns a probability density value.
  2. Choose a Proposal Distribution: This distribution is used to generate candidate samples. A common choice is a Gaussian distribution centered at the current state (i.e., a random walk proposal).
  3. Implement the Acceptance Probability Calculation: The acceptance probability determines whether to accept the proposed sample or stay at the current state. It's calculated using the Metropolis Hastings formula, which involves the ratio of the target distribution at the proposed and current states, as well as the ratio of the proposal distribution.
  4. Write the MCMC Loop: This loop iteratively generates candidate samples, calculates acceptance probabilities, and updates the state based on a random draw. The number of iterations determines the length of the Markov chain.
  5. Integrate with Pigeons.jl: This involves creating a new sampler type within the Pigeons.jl framework and ensuring it can be used with the existing tempering infrastructure.

Given that the provided example errored, it’s crucial to ensure that any new implementation is thoroughly tested and debugged. This includes checking for common issues such as numerical stability, proper initialization, and convergence.

By providing a well-documented and tested Metropolis Hastings sampler, we can empower users to explore parallel tempering techniques more easily. This not only addresses the original request but also enhances the educational value of Pigeons.jl.

Implementing a Basic Metropolis Hastings Sampler

Alright, let's get our hands dirty and walk through a basic implementation of the Metropolis Hastings sampler. Keep in mind, this is a simplified version to illustrate the core concepts. You can then adapt and extend it to fit your specific needs. We will use Julia for implementation.

Step 1: Define the Target Distribution

First, we need a target distribution to sample from. For simplicity, let's use a standard normal distribution. In Julia, you can define it as follows:

using Distributions

target(x) = pdf(Normal(0, 1), x)

Here, target(x) returns the probability density of x under a standard normal distribution. You can replace this with any other distribution you're interested in.

Step 2: Choose a Proposal Distribution

Next, we need a proposal distribution to generate candidate samples. A common choice is a Gaussian distribution centered at the current state. This is often referred to as a random walk proposal. Let's define a function to sample from this proposal:

using Random

proposal(x, step_size) = x + rand(Normal(0, step_size))

Here, proposal(x, step_size) generates a new candidate sample by adding a random value drawn from a normal distribution with mean 0 and standard deviation step_size to the current state x. The step_size parameter controls how far the proposal can move from the current state.

Step 3: Implement the Acceptance Probability Calculation

The heart of the Metropolis Hastings algorithm is the acceptance probability calculation. This determines whether to accept the proposed sample or stay at the current state. The acceptance probability is calculated as follows:

function acceptance_probability(x_new, x_current)
    target(x_new) / target(x_current)
end

In this function, x_new is the proposed sample, and x_current is the current state. The function returns the ratio of the target distribution at the proposed and current states. We take the minimum of this ratio and 1 to ensure the acceptance probability is always between 0 and 1.

Step 4: Write the MCMC Loop

Now, let's put everything together in the MCMC loop:

function metropolis_hastings(n_samples, initial_state, step_size)
    samples = zeros(n_samples)
    x_current = initial_state
    samples[1] = x_current
    
    for i in 2:n_samples
        x_new = proposal(x_current, step_size)
        acceptance_ratio = acceptance_probability(x_new, x_current)
        
        if rand() <= acceptance_ratio
            x_current = x_new
        end
        
        samples[i] = x_current
    end
    
    return samples
end

In this function, n_samples is the number of samples to generate, initial_state is the starting point for the chain, and step_size is the step size for the proposal distribution. The function returns an array of samples drawn from the target distribution.

Step 5: Run the Sampler

Finally, let's run the sampler and see what we get:

n_samples = 10000
initial_state = 0.0
step_size = 1.0

samples = metropolis_hastings(n_samples, initial_state, step_size)

using StatsPlots

histogram(samples, bins=50, normalize=true, label="Metropolis Hastings Samples", title="Sampling from Standard Normal")

x = range(-4, 4, length=100)
y = pdf.(Normal(0, 1), x)
plot!(x, y, label="Standard Normal PDF")

This code runs the Metropolis Hastings sampler with 10,000 samples, an initial state of 0.0, and a step size of 1.0. It then plots a histogram of the samples along with the true probability density function of the standard normal distribution.

Important Considerations

  • Tuning the Step Size: The choice of step_size can significantly impact the efficiency of the sampler. If the step size is too small, the chain will move very slowly and explore the space inefficiently. If the step size is too large, the acceptance probability will be low, and the chain will get stuck. Experiment with different values to find a good balance.
  • Burn-in Period: The initial samples from the chain may not be representative of the target distribution. It's common to discard the first few thousand samples as a 'burn-in' period.
  • Convergence Diagnostics: It's essential to check whether the chain has converged to the target distribution. There are various diagnostic tools available, such as trace plots, autocorrelation plots, and Gelman-Rubin statistics.

By following these steps, you can implement a basic Metropolis Hastings sampler in Julia. This provides a solid foundation for understanding more advanced MCMC techniques and integrating them into Pigeons.jl.

Integrating with Pigeons.jl

Now that we have a basic Metropolis Hastings sampler, let’s think about how to integrate it with Pigeons.jl. The goal is to create a sampler type that can be used within the existing tempering infrastructure. This involves defining a new struct for the sampler and implementing the necessary methods to interact with the Pigeons.jl framework.

Step 1: Define the Sampler Struct

First, we need to define a struct to represent our Metropolis Hastings sampler. This struct should hold any parameters that are needed to configure the sampler, such as the step size and the proposal distribution.

struct MetropolisHastingsSampler{T <: Real}
    step_size::T
end

Here, MetropolisHastingsSampler is a struct that holds the step_size parameter. The {T <: Real} syntax indicates that step_size must be a real number.

Step 2: Implement the Sampling Step

Next, we need to implement a method that performs a single sampling step. This method should take the current state and the sampler as input and return the next state.

function sample_step(sampler::MetropolisHastingsSampler, x_current)
    x_new = proposal(x_current, sampler.step_size)
    acceptance_ratio = acceptance_probability(x_new, x_current)
    
    if rand() <= acceptance_ratio
        return x_new
    else
        return x_current
    end
end

This function takes a MetropolisHastingsSampler instance and the current state x_current as input. It generates a new candidate sample using the proposal function, calculates the acceptance probability, and returns the new state if it's accepted, or the current state if it's rejected.

Step 3: Integrate with Tempering Infrastructure

To integrate the sampler with the Pigeons.jl tempering infrastructure, you may need to adapt the sampler to work with the specific API and data structures used by the package. This might involve implementing methods for initializing the sampler, updating the sampler's state, and exchanging states between different chains.

Since Pigeons.jl provides a flexible framework for defining samplers, you can leverage the existing tools and abstractions to create a seamless integration.

Example

Here’s an example of how you might use the MetropolisHastingsSampler within a tempering context:

# Assuming you have a tempered model defined

# Create a Metropolis Hastings sampler
mh_sampler = MetropolisHastingsSampler(1.0)

# Initialize the state
initial_state = 0.0

# Run the sampler for a certain number of iterations
n_iterations = 1000

state = initial_state
for i in 1:n_iterations
    state = sample_step(mh_sampler, state)
    # Do something with the state (e.g., store it, update the model)
end

This example shows how to create a MetropolisHastingsSampler, initialize the state, and run the sampler for a certain number of iterations. You can adapt this code to fit your specific tempering setup.

By following these steps, you can integrate your Metropolis Hastings sampler with Pigeons.jl and leverage its powerful tempering infrastructure.

Final Thoughts

So, there you have it! We've covered the basics of Metropolis Hastings samplers and how to implement one in Julia, with a focus on integrating it into the Pigeons.jl framework. Remember, this is just a starting point. Feel free to experiment with different proposal distributions, step sizes, and convergence diagnostics to optimize the performance of your sampler.

By understanding these fundamental concepts, you'll be well-equipped to tackle more advanced MCMC techniques and explore the exciting world of parallel tempering. Happy sampling, and feel free to reach out if you have any more questions or need further assistance!