Part 2 — An Advanced Thesis: Learning from Joint Distributions
In continuation on the discussions from Part 1, where I surmised that we truly do not need big data for training today’s model, I present a thesis on how we can consider using Joint Distributions to learn something about an asset or a class by learning about it’s functions across different modalities.
Consider a real-world example where you have a genomic dataset, but you lack sufficient statistical power at the individual gene level within a specific cell type for a particular species. One way to address this is to recognize that the same gene might be active (expressed) across different cell types. An even better insight is that this gene may have similar functions across different species as well. By leveraging this understanding, you can reformulate the problem to learn about the gene’s function across all cell types and species simultaneously. This collective learning approach enhances your ability to infer the gene’s function in a specific cell and species by integrating information from multiple cells and species by increasing the statistical strength as an aggregate.
Hence, you can learn gene functions from a joint distribution of cross-cell, cross-species datasets providing the highest variance, dispersion, and other statistical measures necessary for effective learning. This process can be deeply understood through mathematical formulations. I cover the foundational math underlying the process of joint distributions.
In this article, I examine the fundamental mathematics that can be applied to conduct an exploratory analysis of the distribution, gaining insights into the aggregate strength of the distribution across modalities.
This is a two-part series (for now)
Joint Probability Distribution
Consider a set of observations X={x1,x2,…,xn}, where each x_i represents a vector of features derived from different cells and species. We aim to model the joint distribution P(X), capturing the dependencies among these features.
The joint distribution P(X) can be expressed as:
For computational feasibility, we often assume that X follows a multivariate normal distribution:
where μ is the mean vector and Σ is the covariance matrix. The probability density function of a multivariate normal distribution is given by:
Covariance and Precision Matrices
The covariance matrix Σ captures the variance and covariances of the features. Its elements are defined as:
The inverse of the covariance matrix, Σ, is known as the precision matrix, which provides insights into the conditional dependencies between features:
If Θij=0, then x_i and x_j are conditionally independent given all other variables.
Conditional Distributions
To learn gene functions, we focus on the conditional distributions P(xi∣Pa(xi)), where Pa(xi) denotes the set of parent nodes of x_i in a probabilistic graphical model such as a Bayesian network.
Using properties of the multivariate normal distribution, the conditional distribution of x_i given Pa(xi) can be expressed as:
where:
- Σ_i,Pa(xi) is the covariance between x_i and its parents.
- Σ_Pa(xi),Pa(xi) is the covariance matrix of the parents.
- Σ_i∣Pa(xi) is the conditional variance of x_i given its parents.
Principal Component Analysis (PCA)
Principal Component Analysis (PCA) is a technique used to reduce the dimensionality of the data while retaining the most significant variance components. The covariance matrix Σ is decomposed into its eigenvalues and eigenvectors:
where W is the matrix of eigenvectors and Λ is the diagonal matrix of eigenvalues. The principal components are given by:
where Z is the transformed dataset in the principal component space.
Variance and Dispersion
To maximize the effectiveness of learning, we aim to maximize the variance and dispersion of the dataset. The total variance captured by the principal components is given by the trace of the covariance matrix:
where λ_i are the eigenvalues of Σ.
The dispersion can be quantified using the determinant of the covariance matrix ∣Σ∣, which measures the volume of the data distribution:
Information Theory and Mutual Information
To further understand the dependencies and information shared between features, we can use concepts from information theory. The mutual information between two features x_i and x_j quantifies the amount of information obtained about one feature through the other:
where H(x) is the entropy of x and H(xi,xj) is the joint entropy of x_i and x_j.
For a multivariate normal distribution, the entropy H(X) is given by:
Bayesian Networks and Markov Random Fields
In Bayesian networks, the joint probability distribution is factorized into conditional distributions. For a set of variables X={x1,x2,…,xn}, the joint distribution can be written as:
where Pa(x_i) are the parents of x_i in the network. In a Markov random field, the joint distribution is expressed in terms of clique potentials ψ_C:
where C is the set of cliques in the graph and Z is the partition function ensuring the distribution sums to one.
Gaussian Processes
Gaussian processes provide a powerful framework for learning and inference in high-dimensional spaces. A Gaussian process is a collection of random variables, any finite number of which have a joint Gaussian distribution. It is fully specified by a mean function m(x) and a covariance function k(x, x’):
The covariance function k(x, x’) encodes the relationship between different points in the input space. Given a set of training data D={(xi,yi)}, the posterior distribution over the function values at new points X∗ given by:
where K denotes the covariance matrix computed using the covariance function k.
Spectral Decomposition and Eigenvalue Analysis
For more complex and high-dimensional datasets, spectral decomposition and eigenvalue analysis are crucial. Given the covariance matrix Σ, its spectral decomposition is given by:
where Q is an orthogonal matrix of eigenvectors and Λ is a diagonal matrix of eigenvalues. The eigenvalues λ_i represent the variance explained by each principal component. To ensure numerical stability and improve computational efficiency, we often work with the logarithm of the eigenvalues:
This logarithmic transformation helps in handling the wide range of eigenvalues that may be present in high-dimensional data.
Advanced Techniques: Kernel Methods and Manifold Learning
Kernel methods allow us to handle non-linear relationships by mapping data into higher-dimensional spaces. Given a kernel function k(x,x′), the kernel matrix K is defined as:
Common kernel functions include the Gaussian (RBF) kernel:
and the polynomial kernel:
Manifold learning techniques such as Isomap, Locally Linear Embedding (LLE), and t-Distributed Stochastic Neighbor Embedding (t-SNE) aim to uncover the low-dimensional structure in high-dimensional data. For instance, Isomap constructs a graph based on pairwise distances and computes the geodesic distances between points to preserve the manifold structure.
Conclusion
By leveraging these advanced mathematical techniques, we can effectively learn gene functions from joint distributions, capturing the complex dependencies and variabilities inherent in cross-cell, cross-species datasets. This approach provides a robust framework for understanding the intricate relationships in biological systems and advancing our knowledge of gene functions.
Such a sophisticated understanding highlights the shift from the traditional need for vast amounts of data to a more nuanced approach where quality, variance, and advanced modeling techniques play pivotal roles. As we move forward, the continued integration of these methodologies will pave the way for more efficient and insightful discoveries in AI and beyond.
