Statistical Inference of Spin Models using Boltzmann Machine Learning

Date: September 2, 2024

Overview

This project involves the development and implementation of statistical inference algorithms for spin models using the Boltzmann machine learning framework. The primary focus is on optimizing the log-likelihood to infer parameters of a Boltzmann machine from given data, applying both equilibrium and non-equilibrium models. The work also extends to modeling biological systems, specifically the neural activity in the brain of a salamander.

Objectives

1. Boltzmann Machine Learning Algorithm: Derive the learning rules for the Boltzmann machine by maximizing the log-likelihood function, using a gradient descent approach.
2. Parameter Inference in Equilibrium Models: Infer model parameters such as fields and couplings by monitoring the optimization process and adjusting the learning rate.
3. Non-equilibrium Modeling: Implement a non-equilibrium Boltzmann learning algorithm to capture dynamic behaviors in spin systems, focusing on temporal correlations.
4. Application to Biological Data: Apply the developed algorithms to infer neural interactions in the salamander brain, validating the model using correlation measures.

Theoretical Background

The Boltzmann machine learning framework is rooted in optimizing the Kullback-Leibler divergence between the data distribution and the model distribution. This is simplified to minimizing the negative log-likelihood:

$$ L(\theta) = -\frac{1}{M} \sum_{k} \log P_{\theta}(s^{(k)}) $$

where ( P_{\theta}(s^{(k)}) ) is the probability of the data given model parameters ( \theta ). Gradient descent is used to iteratively update the parameters:

$$ \theta^{(n+1)} = \theta^{(n)} - \eta \nabla L(P_{\theta}) $$

Key Components

• statistical_inference.py: Implements the core Boltzmann learning algorithms, including:

  • Creation of random initial states and fields.
  • Metropolis Monte Carlo methods for sampling spin states.
  • Gradient descent optimization for parameter learning.
  • Log-likelihood calculation for model evaluation.

• statistical_inference_glauber.py: Extends the inference to non-equilibrium models using Glauber dynamics:

  • Time series generation based on Glauber dynamics.
  • Calculation of temporal correlations.
  • Inference of symmetric and asymmetric couplings.

Methods

1. Gradient Descent Optimization:
   • Derivatives of the log-likelihood with respect to model parameters are computed and used to iteratively update the fields (h_i) and couplings (J_{ij}).
2. Monte Carlo Sampling:
   • Metropolis algorithm and Glauber dynamics are used to generate samples from the equilibrium and non-equilibrium distributions, respectively.
3. Parameter Inference:
   • Infer parameters by fitting the model to training data, monitoring the log-likelihood, and adjusting the learning rate to optimize performance.

Results

• Equilibrium Model Performance:
  • Inference quality depends on the size of the training data set; larger datasets yield better parameter estimates.
  • Learning rate impacts the convergence rate and accuracy of the model.
• Non-Equilibrium Model Performance:
  • Temporal correlations in the salamander's brain data were better captured using a non-equilibrium approach with asymmetric couplings.
  • Symmetric coupling models were faster to converge but less accurate in capturing detailed dynamics.

How to Run

1. Requirements:

   • Python 3.x
   • Numpy
   • Matplotlib
   • Numba

2. Execution:

   • Run the scripts in the following order to reproduce the results:

     python statistical_inference.py
     python statistical_inference_glauber.py

3. Data Input:

   • Data for salamander neural activity should be in the format specified in the scripts. Modify paths to the input files as necessary.

Conclusion

The developed algorithms demonstrate effective parameter inference for both equilibrium and non-equilibrium spin models. Applying these methods to biological data, such as neural activity in salamanders, shows that non-equilibrium models with asymmetric couplings are superior in capturing temporal correlations, though they require careful tuning of the learning rate and parameter initialization.

Future Work

• Explore alternative optimization methods to improve convergence rates.
• Extend the models to incorporate higher-order interactions or additional biological constraints.
• Apply the methods to other types of neural data to validate the generality of the approach.