Technical Summary: From Numerical Proportions to Analogical Proportions Between Probabilities

Problem Statement

The paper addresses the theoretical and practical compatibility between analogical proportions and probabilities. Analogical proportions, expressed as "a is to b as c is to d" ($a:b::c:d$), are well-established for Boolean, nominal, and numerical values, often serving as the basis for classification methods (e.g., predicting a class for a new item based on three known analogical items). However, the relationship between these proportions and probabilistic representations remains under-explored.

The central research question is: If four profiles (represented as vectors of attributes) form an analogical proportion, do the probability distributions associated with these profiles also form an analogical proportion? The authors investigate this across two levels:

1. Probability values: Simple probabilities associated with specific attribute values.
2. Probability distributions: Full distributions describing the frequencies of possible values for a discrete attribute across different profiles.

Methodology

1. Theoretical Framework

The authors extend the definition of analogical proportions from discrete domains (Boolean, nominal) to continuous and probabilistic domains. They analyze two primary definitions for numerical analogical proportions:

• Arithmetic Proportion ($::_{ari}$): Defined by $a - b = c - d$.
• Geometric Proportion ($::_{geo}$): Defined by $a/b = c/d$ (with $0/0=1$).

They further investigate the Arithmetico-Geometric Proportion ($::_{arigeo}$), which requires both conditions to hold simultaneously.

The paper establishes several key theoretical properties:

• Arithmetic Definition: For probability distributions $a, b, c, d$ over a finite domain, $a:b::_{ari}c:d$ holds if $a_i - b_i = c_i - d_i$ for all components $i$. This definition preserves the Total Variation Distance ($TV(a,b) = TV(c,d)$) and guarantees the existence of a solution $d$ given $a, b, c$ under specific constraints.
• Geometric Definition: The pure geometric definition ($a_i/b_i = c_i/d_i$) is shown to be problematic for probability distributions because it does not guarantee that the resulting vector $d$ sums to 1 (normalization), even if $a, b, c$ are valid distributions.
• Arithmetico-Geometric Definition: This stricter definition ($a_i - b_i = c_i - d_i$ AND $a_i/b_i = c_i/d_i$) implies that for each component, the values must follow specific patterns (e.g., $a_i=b_i$ and $c_i=d_i$, or $a_i=c_i$ and $b_i=d_i$). Crucially, this definition ensures the invariance of the Kullback-Leibler (KL) divergence between pairs ($KL(a||b) = KL(c||d)$).

2. Experimental Validation

The authors empirically test the hypothesis that analogical proportions between profiles imply analogical proportions between their associated probability distributions.

• Datasets: Two large-scale datasets were used:
  1. MovieLens 100K: 100,000 ratings from 943 users on 1,682 movies. Profiles were constructed from user demographics (gender, age, occupation).
  2. US Traffic Accidents: ~7.7 million accidents across 49 states. Profiles were constructed from categorical variables (timezone, day/night, weather, wind).
• Procedure:
  • Profiles were defined as combinations of attribute values.
  • For each profile, a probability distribution was derived based on the frequency of the target variable (movie rating 1-5 or accident severity 1-4).
  • Valid analogical proportions ($a:b::c:d$) were identified among the profiles.
  • Analogical Dissimilarity (AD) was calculated for the associated distributions. AD measures the deviation from a perfect analogical proportion (specifically the arithmetic one, defined as $AD = \frac{1}{2n} \sum |(a_i - b_i) - (c_i - d_i)|$).
• Thresholds: Experiments were conducted with varying consistency thresholds (minimum number of instances per profile) to ensure statistical reliability.

Key Contributions

1. Formalization of Probabilistic Analogical Proportions: The paper provides rigorous definitions for analogical proportions between probability distributions, distinguishing between arithmetic, geometric, and arithmetico-geometric variants.
2. Theoretical Properties:
   • Demonstrated that the arithmetic definition preserves Total Variation Distance.
   • Demonstrated that the arithmetico-geometric definition preserves Kullback-Leibler divergence.
   • Proved that the pure geometric definition is generally insufficient for probability distributions due to normalization constraints.
3. Empirical Evidence: Through extensive experiments on real-world datasets, the authors show that when profiles form an analogical proportion, their associated probability distributions often form an arithmetic analogical proportion with low dissimilarity.
4. Extension to Continuous Densities: The paper briefly outlines how these definitions extend to continuous probability density functions and discusses parameter-based analogies for distributions like Gaussian or Binomial.

Results

• Arithmetic Proportion Validity: In the MovieLens and US Traffic Accidents datasets, a significant number of profile analogies resulted in distribution analogies with very low Analogical Dissimilarity (AD < 0.05). For example, the relationship between male and female ratings in one age group mirrored the relationship in another age group across rating distributions.
• Arithmetico-Geometric Rarity: When testing for arithmetico-geometric proportions (requiring both arithmetic and geometric conditions), the results showed much higher dissimilarity. The authors conclude that such strict analogies between probability distributions are rare in real-world data, likely due to the high constraints imposed by the geometric condition.
• Threshold Sensitivity: The validity of the inference rule (that profile analogy implies distribution analogy) appeared stronger with larger sample sizes (higher consistency thresholds), suggesting that noise in small datasets obscures the underlying proportional relationships.

Significance and Claims

The paper claims to provide a systematic study of the link between analogical reasoning and probability, moving beyond previous work that treated them separately.

• Theoretical Contribution: It establishes that analogical proportions can be meaningfully defined in the probabilistic domain, offering new tools for reasoning under uncertainty.
• Practical Implication: The results suggest that analogical inference can be extended to predict probability distributions for new profiles based on known analogical profiles, without needing to analyze the target population directly. This could enhance classification and transfer learning methods.
• Modest Outlook: The authors acknowledge that the relationship is not universally valid (some cases show significant distortion) and that the theoretical underpinnings of why this relationship holds in many real-world cases require further research. They note that the current work is a preliminary step, with future research needed to explore profiles with numerical attributes and to refine the theoretical explanations for the observed empirical validity.

The work is presented as an extension of previous conference papers [29, 30], reorganizing the theoretical content and introducing entirely new experimental sections to validate the hypothesis that "analogical proportions between profiles are associated with analogical proportions between probabilities."