# latent_algebra#

Contains mathematical functions relating the latent terms, P(given_label), P(given_label | true_label), P(true_label | given_label), P(true_label), etc. together. For every function here, if the inputs are exact, the output is guaranteed to be exact. Every function herein is the computational equivalent of a mathematical equation having a closed, exact form. If the inputs are inexact, the error will of course propagate. Throughout K denotes the number of classes in the classification task.

Functions:

 compute_inv_noise_matrix(py, noise_matrix, *) Compute the inverse noise matrix if py := P(true_label=k) is given. compute_noise_matrix_from_inverse(ps, ...[, py]) Compute the noise matrix P(label=k_s|true_label=k_y). compute_ps_py_inv_noise_matrix(labels, ...) Compute ps := P(labels=k), py := P(true_labels=k), and the inverse noise matrix. compute_py(ps, noise_matrix, ...[, ...]) Compute py := P(true_labels=k) from ps := P(labels=k), noise_matrix, and inverse_noise_matrix. compute_py_inv_noise_matrix(ps, noise_matrix) Compute py := P(true_label=k), and the inverse noise matrix. compute_pyx(pred_probs, noise_matrix, ...) Compute pyx := P(true_label=k|x) from pred_probs := P(label=k|x), noise_matrix and inverse_noise_matrix.
cleanlab.internal.latent_algebra.compute_inv_noise_matrix(py, noise_matrix, *, ps=None)[source]#

Compute the inverse noise matrix if py := P(true_label=k) is given.

Parameters:
• py (np.ndarray (shape (K, 1))) – The fraction (prior probability) of each TRUE class label, P(true_label = k)

• noise_matrix (np.ndarray) – A conditional probability matrix (of shape (K, K)) of the form P(label=k_s|true_label=k_y) containing the fraction of examples in every class, labeled as every other class. Assumes columns of noise_matrix sum to 1.

• ps (np.ndarray) – Array of shape (K, 1) containing the fraction (prior probability) of each NOISY given label, P(labels = k). ps is easily computable from py and should only be provided if it has already been precomputed, to increase code efficiency.

Examples

For loop based implementation:

# Number of classes
K = len(py)

# 'ps' is p(labels=k) = noise_matrix * p(true_labels=k)
# because in *vector computation*: P(label=k|true_label=k) * p(true_label=k) = P(label=k)
if ps is None:
ps = noise_matrix.dot(py)

# Estimate the (K, K) inverse noise matrix P(true_label = k_y | label = k_s)
inverse_noise_matrix = np.empty(shape=(K,K))
# k_s is the class value k of noisy label label == k
for k_s in range(K):
# k_y is the (guessed) class value k of true label y
for k_y in range(K):
# P(true_label|label) = P(label|y) * P(true_label) / P(labels)
inverse_noise_matrix[k_y][k_s] = noise_matrix[k_s][k_y] *                                                  py[k_y] / ps[k_s]

Return type:

ndarray

cleanlab.internal.latent_algebra.compute_noise_matrix_from_inverse(ps, inverse_noise_matrix, *, py=None)[source]#

Compute the noise matrix P(label=k_s|true_label=k_y).

Parameters:
• py (np.ndarray) – Array of shape (K, 1) containing the fraction (prior probability) of each TRUE class label, P(true_label = k).

• inverse_noise_matrix (np.ndarray) – A conditional probability matrix (of shape (K, K)) of the form P(true_label=k_y|label=k_s) representing the estimated fraction observed examples in each class k_s, that are mislabeled examples from every other class k_y. If None, the inverse_noise_matrix will be computed from pred_probs and labels. Assumes columns of inverse_noise_matrix sum to 1.

• ps (np.ndarray) – Array of shape (K, 1) containing the fraction (prior probability) of each observed NOISY label, P(labels = k). ps is easily computable from py and should only be provided if it has already been precomputed, to increase code efficiency.

Return type:

ndarray

Returns:

noise_matrix (np.ndarray) – Array of shape (K, K), where K = number of classes, whose columns sum to 1. A conditional probability matrix of the form P(label=k_s|true_label=k_y) containing the fraction of examples in every class, labeled as every other class.

Examples

For loop based implementation:

# Number of classes labels
K = len(ps)

# 'py' is p(true_label=k) = inverse_noise_matrix * p(true_label=k)
# because in *vector computation*: P(true_label=k|label=k) * p(label=k) = P(true_label=k)
if py is None:
py = inverse_noise_matrix.dot(ps)

# Estimate the (K, K) noise matrix P(labels = k_s | true_labels = k_y)
noise_matrix = np.empty(shape=(K,K))
# k_s is the class value k of noisy label labels == k
for k_s in range(K):
# k_y is the (guessed) class value k of true label y
for k_y in range(K):
# P(labels|y) = P(true_label|labels) * P(labels) / P(true_label)
noise_matrix[k_s][k_y] = inverse_noise_matrix[k_y][k_s] *                                          ps[k_s] / py[k_y]

cleanlab.internal.latent_algebra.compute_ps_py_inv_noise_matrix(labels, noise_matrix)[source]#

Compute ps := P(labels=k), py := P(true_labels=k), and the inverse noise matrix.

Parameters:
• labels (np.ndarray) – A discrete vector of noisy labels, i.e. some labels may be erroneous. Format requirements: for dataset with K classes, labels must be in {0,1,...,K-1}.

• noise_matrix (np.ndarray) – A conditional probability matrix (of shape (K, K)) of the form P(label=k_s|true_label=k_y) containing the fraction of examples in every class, labeled as every other class. Assumes columns of noise_matrix sum to 1.

Return type:

Tuple[ndarray, ndarray, ndarray]

cleanlab.internal.latent_algebra.compute_py(ps, noise_matrix, inverse_noise_matrix, *, py_method='cnt', true_labels_class_counts=None)[source]#

Compute py := P(true_labels=k) from ps := P(labels=k), noise_matrix, and inverse_noise_matrix.

This method is ** ROBUST ** when py_method = 'cnt' It may work well even when the noise matrices are estimated poorly by using the diagonals of the matrices instead of all the probabilities in the entire matrix.

Parameters:
• ps (np.ndarray) – Array of shape (K, ) or (1, K) containing the fraction (prior probability) of each observed, noisy label, P(labels = k)

• noise_matrix (np.ndarray) – A conditional probability matrix ( of shape (K, K)) of the form P(label=k_s|true_label=k_y) containing the fraction of examples in every class, labeled as every other class. Assumes columns of noise_matrix sum to 1.

• inverse_noise_matrix (np.ndarray of shape (K, K), K = number of classes) – A conditional probability matrix ( of shape (K, K)) of the form P(true_label=k_y|label=k_s) representing the estimated fraction observed examples in each class k_s, that are mislabeled examples from every other class k_y. If None, the inverse_noise_matrix will be computed from pred_probs and labels. Assumes columns of inverse_noise_matrix sum to 1.

• py_method (str (Options: [”cnt”, "eqn", "marginal", "marginal_ps"])) – How to compute the latent prior p(true_label=k). Default is “cnt” as it often works well even when the noise matrices are estimated poorly by using the matrix diagonals instead of all the probabilities.

• true_labels_class_counts (np.ndarray) – Array of shape (K, ) or (1, K) containing the marginal counts of the confident joint (like cj.sum(axis = 0)).

Return type:

ndarray

Returns:

py (np.ndarray) – Array of shape (K, ) or (1, K). The fraction (prior probability) of each TRUE class label, P(true_label = k).

cleanlab.internal.latent_algebra.compute_py_inv_noise_matrix(ps, noise_matrix)[source]#

Compute py := P(true_label=k), and the inverse noise matrix.

Parameters:
• ps (np.ndarray) – Array of shape (K, ) or (1, K). The fraction (prior probability) of each observed, NOISY class P(labels = k).

• noise_matrix (np.ndarray) – A conditional probability matrix (of shape (K, K)) of the form P(label=k_s|true_label=k_y) containing the fraction of examples in every class, labeled as every other class. Assumes columns of noise_matrix sum to 1.

Return type:

Tuple[ndarray, ndarray]

cleanlab.internal.latent_algebra.compute_pyx(pred_probs, noise_matrix, inverse_noise_matrix)[source]#

Compute pyx := P(true_label=k|x) from pred_probs := P(label=k|x), noise_matrix and inverse_noise_matrix.

This method is ROBUST - meaning it works well even when the noise matrices are estimated poorly by only using the diagonals of the matrices which tend to be easy to estimate correctly.

Parameters:
• pred_probs (np.ndarray) – P(label=k|x) is a (N x K) matrix with K model-predicted probabilities. Each row of this matrix corresponds to an example x and contains the model-predicted probabilities that x belongs to each possible class. The columns must be ordered such that these probabilities correspond to class 0,1,2,… pred_probs should have been computed using 3 (or higher) fold cross-validation.

• noise_matrix (np.ndarray) – A conditional probability matrix (of shape (K, K)) of the form P(label=k_s|true_label=k_y) containing the fraction of examples in every class, labeled as every other class. Assumes columns of noise_matrix sum to 1.

• inverse_noise_matrix (np.ndarray) – A conditional probability matrix (of shape (K, K)) of the form P(true_label=k_y|label=k_s) representing the estimated fraction observed examples in each class k_s, that are mislabeled examples from every other class k_y. If None, the inverse_noise_matrix will be computed from pred_probs and labels. Assumes columns of inverse_noise_matrix sum to 1.

Returns:

pyx (np.ndarray) – P(true_label=k|x) is a (N, K) matrix of model-predicted probabilities. Each row of this matrix corresponds to an example x and contains the model-predicted probabilities that x belongs to each possible class. The columns must be ordered such that these probabilities correspond to class 0,1,2,… pred_probs should have been computed using 3 (or higher) fold cross-validation.