Learning Parameters of Hyperlattices

Learning Parameters of Probabilistic Hyperlattice Models in Common Lisp

There are several techniques for learning the parameters of a probabilistic hyperlattice model in Common Lisp, including maximum likelihood estimation (MLE), Bayesian inference, and variational inference. Here’s an example of how you can implement MLE and Bayesian inference for learning the parameters of a hyperlattice model in Common Lisp:

(defun learn-parameters-mle (data hyperlattice)
  "Learn the parameters of the hyperlattice model from data using maximum likelihood estimation."
  (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)))))

(defun learn-parameters-bayesian (data hyperlattice prior-alpha prior-beta)
  "Learn the parameters of the hyperlattice model from data using Bayesian inference with a Beta prior."
  (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))
          (let ((alpha (+ prior-alpha (gethash i counts 0)))
                (beta (+ prior-beta (- (length data) (gethash i counts 0)))))
            (setf (aref probabilities i) (beta-binomial-pdf i alpha beta))))
        (setf (hyperlattice-probabilities element) probabilities)))))

(defun beta-binomial-pdf (k alpha beta)
  "Compute the probability mass function of the Beta-Binomial distribution with parameters alpha and beta, evaluated at k."
  (* (binomial-coefficient (- alpha 1) (- k 1))
     (binomial-coefficient (- beta 1) (- (- alpha k) 1))
     (binomial-coefficient (- alpha beta) (- 1) (/ 1.0 alpha))
     (binomial-coefficient (- alpha beta) (- 1) (/ 1.0 beta))))

(defun binomial-coefficient (n k)
  "Compute the binomial coefficient of n choose k."
  (if (or (< k 0) (> k n))
      0
      (let ((result 1))
        (dotimes (i k)
          (setf result (* result (/ (- n i) (+ 1 i)))))
        result)))

In this code, the learn-parameters-mle function learns the parameters of the hyperlattice model from data using maximum likelihood estimation. 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.

The learn-parameters-bayesian function learns the parameters of the hyperlattice model from data using Bayesian inference with a Beta prior. The function computes the posterior distribution over the parameters of the hyperlattice model using Bayes’ rule, and updates the prior distribution with the observed data. The beta-binomial-pdf function computes the probability mass function of the Beta-Binomial distribution, which is used as the prior distribution over the parameters of the hyperlattice model. The binomial-coefficient function computes the binomial coefficient of n choose k, which is used in the computation of the Beta-Binomial distribution.

Overall, these functions provide a basic framework for learning the parameters of a probabilistic hyperlattice model in Common Lisp, and can be extended and customized for specific applications and domains.