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FI-CBL: A Probabilistic Method for Concept-Based Learning with Expert Rules

Lev V. Utkin, Andrei V. Konstantinov, Stanislav R. Kirpichenko

TL;DR

FI-CBL tackles concept-based learning by combining patch-level embeddings with a transparent frequentist-Bayesian inference framework and by enabling explicit incorporation of expert rules. Patches are clustered to form a distribution over concepts via empirical counts, and posteriors for each concept value are obtained through Bayes’ rule, with a multinomial model supporting inference for new images. A key contribution is the principled, rule-driven update of priors and conditionals, allowing expert knowledge to steer probabilistic reasoning, which yields strong performance in small-data regimes and provides interpretable, auditable decision logic. The work demonstrates practical advantages in medical-imaging-like tasks and underscores FI-CBL’s potential for robust, explainable AI under data-scarce conditions.

Abstract

A method for solving concept-based learning (CBL) problem is proposed. The main idea behind the method is to divide each concept-annotated image into patches, to transform the patches into embeddings by using an autoencoder, and to cluster the embeddings assuming that each cluster will mainly contain embeddings of patches with certain concepts. To find concepts of a new image, the method implements the frequentist inference by computing prior and posterior probabilities of concepts based on rates of patches from images with certain values of the concepts. Therefore, the proposed method is called the Frequentist Inference CBL (FI-CBL). FI-CBL allows us to incorporate the expert rules in the form of logic functions into the inference procedure. An idea behind the incorporation is to update prior and conditional probabilities of concepts to satisfy the rules. The method is transparent because it has an explicit sequence of probabilistic calculations and a clear frequency interpretation. Numerical experiments show that FI-CBL outperforms the concept bottleneck model in cases when the number of training data is small. The code of proposed algorithms is publicly available.

FI-CBL: A Probabilistic Method for Concept-Based Learning with Expert Rules

TL;DR

FI-CBL tackles concept-based learning by combining patch-level embeddings with a transparent frequentist-Bayesian inference framework and by enabling explicit incorporation of expert rules. Patches are clustered to form a distribution over concepts via empirical counts, and posteriors for each concept value are obtained through Bayes’ rule, with a multinomial model supporting inference for new images. A key contribution is the principled, rule-driven update of priors and conditionals, allowing expert knowledge to steer probabilistic reasoning, which yields strong performance in small-data regimes and provides interpretable, auditable decision logic. The work demonstrates practical advantages in medical-imaging-like tasks and underscores FI-CBL’s potential for robust, explainable AI under data-scarce conditions.

Abstract

A method for solving concept-based learning (CBL) problem is proposed. The main idea behind the method is to divide each concept-annotated image into patches, to transform the patches into embeddings by using an autoencoder, and to cluster the embeddings assuming that each cluster will mainly contain embeddings of patches with certain concepts. To find concepts of a new image, the method implements the frequentist inference by computing prior and posterior probabilities of concepts based on rates of patches from images with certain values of the concepts. Therefore, the proposed method is called the Frequentist Inference CBL (FI-CBL). FI-CBL allows us to incorporate the expert rules in the form of logic functions into the inference procedure. An idea behind the incorporation is to update prior and conditional probabilities of concepts to satisfy the rules. The method is transparent because it has an explicit sequence of probabilistic calculations and a clear frequency interpretation. Numerical experiments show that FI-CBL outperforms the concept bottleneck model in cases when the number of training data is small. The code of proposed algorithms is publicly available.
Paper Structure (18 sections, 41 equations, 9 figures, 7 tables)

This paper contains 18 sections, 41 equations, 9 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Related work
  3. Background
  4. The model and its training
  5. The model inference
  6. Incorporating expert rules
  7. Expert rules and prior probabilities of concepts
  8. Expert rules and conditional probabilities
  9. Expert rules with uncertainty
  10. Illustrative example
  11. Concepts
  12. Training
  13. Inference
  14. Expert rules
  15. Prior probabilities
  16. ...and 3 more sections

Figures (9)

  • Figure 1: A general scheme of the image transformation to sets of clustered embeddings of patches: each image is divided into $s$ patches $\xi_{j}^{(i)}$; every patch is transformed into an embedding $e_{j}^{(i)}$; embeddings are clustered into $R$ clusters; probabilities of concepts are computed by having the image concept labels $\mathbf{c}_{i}$ and distributions of embeddings with corresponding concept values in clusters
  • Figure 2: An illustration of the concept-based description of images consisting of four patches and three clusters containing different patches
  • Figure 3: Examples of the modified MNIST dataset
  • Figure 4: The F1 measures of the proposed method and CBM as functions of the training set size for all concepts of the modified MNIST dataset
  • Figure 5: F1-measures as functions of $\beta$ for cases when any rule is not used (the curve with triangle markers), and when the rule “ IF $c^{(9)}=1$, THEN $c^{(0)}=1$” is used (the curve with circle markers) for the modified MNIST
  • ...and 4 more figures