Source code for cleanlab.internal.latent_algebra

# Copyright (C) 2017-2022  Cleanlab Inc.
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"""
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.
"""

import warnings
import numpy as np
from typing import Tuple

from cleanlab.internal.util import value_counts, clip_values, clip_noise_rates


[docs]def compute_ps_py_inv_noise_matrix( labels, noise_matrix ) -> Tuple[np.ndarray, np.ndarray, np.ndarray]: """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.""" ps = value_counts(labels) / float(len(labels)) # p(labels=k) py, inverse_noise_matrix = compute_py_inv_noise_matrix(ps, noise_matrix) return ps, py, inverse_noise_matrix
[docs]def compute_py_inv_noise_matrix(ps, noise_matrix) -> Tuple[np.ndarray, np.ndarray]: """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.""" # 'py' is p(true_labels=k) = noise_matrix^(-1) * p(labels=k) # because in *vector computation*: P(label=k|true_label=k) * p(true_label=k) = P(label=k) # The pseudo-inverse is used when noise_matrix is not invertible. py = np.linalg.inv(noise_matrix).dot(ps) # No class should have probability 0, so we use .000001 # Make sure valid probabilities that sum to 1.0 py = clip_values(py, low=1e-6, high=1.0, new_sum=1.0) # All the work is done in this function (below) return py, compute_inv_noise_matrix(py=py, noise_matrix=noise_matrix, ps=ps)
[docs]def compute_inv_noise_matrix(py, noise_matrix, *, ps=None) -> np.ndarray: """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: .. code:: python # 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] """ joint = noise_matrix * py ps = joint.sum(axis=1) if ps is None else ps inverse_noise_matrix = joint.T / ps # Clip inverse noise rates P(true_label=k_s|true_label=k_y) into proper range [0,1) return clip_noise_rates(inverse_noise_matrix)
[docs]def compute_noise_matrix_from_inverse(ps, inverse_noise_matrix, *, py=None) -> np.ndarray: """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. 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: .. code:: python # 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] """ joint = (inverse_noise_matrix * ps).T py = joint.sum(axis=0) if py is None else py noise_matrix = joint / py # Clip inverse noise rates P(true_label=k_y|true_label=k_s) into proper range [0,1) return clip_noise_rates(noise_matrix)
[docs]def compute_py( ps, noise_matrix, inverse_noise_matrix, *, py_method="cnt", true_labels_class_counts=None ) -> np.ndarray: """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)``). Returns ------- py : np.ndarray Array of shape ``(K, )`` or ``(1, K)``. The fraction (prior probability) of each TRUE class label, ``P(true_label = k)``.""" if len(np.shape(ps)) > 2 or (len(np.shape(ps)) == 2 and np.shape(ps)[0] != 1): w = "Input parameter np.ndarray ps has shape " + str(np.shape(ps)) w += ", but shape should be (K, ) or (1, K)" warnings.warn(w) if py_method == "marginal" and true_labels_class_counts is None: msg = ( 'py_method == "marginal" requires true_labels_class_counts, ' "but true_labels_class_counts is None. " ) msg += " Provide parameter true_labels_class_counts." raise ValueError(msg) if py_method == "cnt": # Computing py this way avoids dividing by zero noise rates. # More robust bc error est_p(true_label|labels) / est_p(labels|y) ~ p(true_label|labels) / p(labels|y) py = inverse_noise_matrix.diagonal() / noise_matrix.diagonal() * ps # Equivalently, # py = (true_labels_class_counts / labels_class_counts) * ps elif py_method == "eqn": py = np.linalg.inv(noise_matrix).dot(ps) elif py_method == "marginal": py = true_labels_class_counts / float(sum(true_labels_class_counts)) elif py_method == "marginal_ps": py = np.dot(inverse_noise_matrix, ps) else: err = "py_method {}".format(py_method) err += " should be in [cnt, eqn, marginal, marginal_ps]" raise ValueError(err) # Clip py (0,1), s.t. no class should have prob 0, hence 1e-5 py = clip_values(py, low=1e-5, high=1.0, new_sum=1.0) return py
[docs]def compute_pyx(pred_probs, noise_matrix, inverse_noise_matrix): """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.""" if len(np.shape(pred_probs)) != 2: raise ValueError( "Input parameter np.ndarray 'pred_probs' has shape " + str(np.shape(pred_probs)) + ", but shape should be (N, K)" ) pyx = pred_probs * inverse_noise_matrix.diagonal() / noise_matrix.diagonal() # Make sure valid probabilities that sum to 1.0 return np.apply_along_axis( func1d=clip_values, axis=1, arr=pyx, **{"low": 0.0, "high": 1.0, "new_sum": 1.0} )