Variational Inference in Hyperlattices

Variational Inference in Probabilistic Hyperlattice Models

Variational inference is a technique for approximating the posterior distribution over the parameters of a probabilistic model, when the exact posterior is intractable or difficult to compute. The idea is to approximate the posterior with a simpler distribution that belongs to a tractable family, such as a Gaussian or a mixture of Gaussians, and to find the parameters of the approximate distribution that minimize the Kullback-Leibler (KL) divergence to the true posterior.

In the context of probabilistic hyperlattice models, variational inference can be used to learn the parameters of the model from data, by approximating the posterior distribution over the parameters with a simpler distribution that belongs to a tractable family. The approximate posterior can be represented as a hyperlattice with the same structure as the original model, but with simpler distributions at each node, such as Gaussians or Dirichlet distributions. The parameters of the approximate posterior can be learned by minimizing the KL divergence to the true posterior, using techniques such as stochastic gradient descent or expectation-maximization.

Variational inference has several advantages over other techniques for learning probabilistic models, such as MLE or Bayesian inference. It can handle large and complex datasets, it can scale to high-dimensional and sparse data, and it can provide a principled way to handle missing data and model selection. However, it also has some limitations, such as the choice of the approximate distribution, the difficulty of assessing the quality of the approximation, and the sensitivity to the initialization of the parameters.

Overall, variational inference is a powerful and flexible technique for learning probabilistic hyperlattice models, and can be used in a wide range of applications and domains.

Variational Inference for Probabilistic Hyperlattice Models in Common Lisp

There are several techniques for performing variational inference for probabilistic hyperlattice models in Common Lisp, including mean-field variational inference, stochastic variational inference, and variational autoencoders. Here’s an example of how you can implement mean-field variational inference for a hyperlattice model in Common Lisp:

(defun variational-inference-mean-field (data hyperlattice)
  "Perform mean-field variational inference for the hyperlattice model."
  (let ((counts (make-hash-table :test 'equal)))
    (dolist (datum data)
      (incf (gethash datum counts 0)))
    (dolist (element (hyperlattice-elements hyperlattice))
      (let ((probabilities (make-array (hyperlattice-dimensions element))))
        (dotimes (i (hyperlattice-dimensions element))
          (setf (aref probabilities i) (/ (gethash i counts 0) (length data))))
        (setf (hyperlattice-probabilities element) probabilities)))))

In this code, the variational-inference-mean-field function performs mean-field variational inference for the hyperlattice model. The function computes the frequency of each element in the data, and uses it to estimate the probabilities of the elements in the hyperlattice model.

Overall, this function provides a basic framework for performing mean-field variational inference for a probabilistic hyperlattice model in Common Lisp, and can be extended and customized for specific applications and domains.