Score-based structural learning for BIC, BDe, etc.
Score-based structural learning methods such as BIC (Bayesian Information Criterion) and BDe (Bayesian Information Criterion) will be the ones used to evaluate the goodness of models and select the best model structure by combining the complexity of the statistical model and the goodness of fit of the data. These methods are mainly based on Bayesian statistics and are widely used as information criteria for model selection.
1. the Bayesian Information Criterion (BIC):
BIC is expressed in the following form.
\[ BIC = -2 \cdot \log(L) + k \cdot \log(n) \]
where \(L\) is the likelihood, \(k\) is the number of parameters in the model, and (n) is the number of data points BIC is based on the concept of maximum likelihood estimation, which maximizes the maximum likelihood and introduces a model complexity penalty by constraining the number of parameters The smaller the BIC, the better the model.
2. BDe (Bayesian Information Criterion):
BDe is a Bayesian information criterion that selects a model based on the posterior probability distribution.
\[ BDe = \log(P(D|M)) – \frac{k}{2} \log(n) \]
where \(P(D|M)\) is the posterior probability that data \(D\) is generated according to model \(M\), \(k\) is the number of parameters in the model, and \(n\) is the number of data points BDe selects the model that maximizes the posterior probability, and since this method uses model selection from a Bayesian perspective, it can be more efficient than BIC when there are fewer data This method is useful when the number of data is smaller than BIC or when the true structure of the model is uncertain.
These score-based structural learning methods are used to find models that fit the data well while preventing overlearning during model selection. However, model selection, including these methods, is only one means of model selection, and the optimal method may differ depending on the data and problem to be applied.
Algorithms used for score-based structural learning such as BIC, BDe, etc.
Score-based structural learning will be a method that uses information criteria to select the structure of the model. Typical score-based structural learning algorithms include methods that optimize BIC and BDe. We describe those algorithms below. 1.
1. Algorithm to optimize BIC:
The algorithm to ensure that the model that minimizes BIC is selected mainly consists of the following steps
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- Definition of the model search space: Define the class of models to be subjected to structural learning (e.g., the structure of a Bayesian network).
- Calculation of the likelihood and number of parameters for each model: Calculate the likelihood and number of parameters for each model.
- Calculate the BIC: Calculate the BIC for each model based on the BIC formula.
- Select the model with the minimum BIC: Select the model with the minimum BIC.
2. Algorithm to optimize BDe:
Algorithms to ensure that the model that maximizes BDe is selected are commonly used in Bayesian network structural learning, for example. The general procedure is as follows
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- Define the model search space: Define the class of models to be subjected to structural learning.
- Set prior probabilities: Set prior probabilities for each model.
- Calculate the posterior probabilities for the data: Calculate the posterior probabilities for each model using Bayes’ theorem.
- Calculate BDe: Calculate BDe for each model based on the BDe formula.
Select the model with the largest BDe: Select the model with the largest BDe.
These methods are generally used in combination to select the optimal model by balancing the complexity of the model and the goodness of fit of the data in the process of structural learning. However, for large and complex models, efficient algorithms and approximation methods may be used because of the potentially high computational cost.
Application examples of score-based structural learning such as BIC, BDe, etc.
Score-based structural learning methods such as BIC and BDe have been applied mainly to structural learning of graphical models and Bayesian networks. Application examples are described below.
1. gene network analysis:
In the field of bioinformatics, BIC and BDe are used to estimate gene networks using gene expression data. This allows modeling of interactions and regulatory relationships among genes, testing biological hypotheses, and discovering new biological insights.
2. financial data analysis:
In the field of finance, score-based methods including BIC and BDe are used to model trends in stock prices and market indices. This includes anomaly detection, risk assessment, and portfolio optimization.
3. meteorological data analysis:
In meteorology, score-based structural learning is applied to construct weather networks using weather data such as temperature, humidity, and wind speed. This allows weather patterns and variations to be modeled for forecasting and anomaly detection.
4. medical data analysis:
In the medical field, score-based structural learning is used to build disease progression models and diagnostic networks using patients’ clinical data and diagnostic results. This is used to predict disease onset and treatment efficacy.
5. social network analysis:
In social network analysis, score-based methods are used to model the interactions of individual agents or nodes. This allows for understanding the characteristics of influential nodes and the network as a whole, as well as for analyzing information propagation and decision-making processes.
These applications show that score-based structural learning methods are widely used in various fields. However, due attention must be paid to the characteristics of the data and the assumptions behind them.
Examples of implementations of score-based structural learning such as BIC, BDe, etc.
Implementing a score-based structural learning method depends on the specific application and the model used, but in general, the following steps are followed. Here is a simple example of structural learning of a Bayesian network using BIC. Python is used here.
First, install the necessary libraries.
pip install pgmpyThe next example will be an example of Bayesian network structure learning using BIC.
from pgmpy.estimators import BicScore
from pgmpy.estimators import HillClimbSearch
from pgmpy.models import BayesianModel
import pandas as pd
# Dummy data generation
data = pd.DataFrame(data={'A': [1, 0, 1, 0, 1],
'B': [0, 1, 1, 1, 0],
'C': [1, 0, 1, 0, 1],
'D': [0, 1, 0, 1, 1]})
# Structural Learning of Bayesian Networks Using BIC
hc = HillClimbSearch(data, scoring_method=BicScore(data))
best_model = hc.estimate()
# Display of trained models
print("Structure of the trained Bayesian network:")
print(best_model.edges())In this example, the pgmpy library is used; BicScore is the class used to compute the BIC, and HillClimbSearch is the class used to learn the structure of the Bayesian network.
It is important to load data and modify the model according to the specific dataset and model, and other score-based methods (e.g. BDe) can be implemented using similar procedures.
Challenges and measures for BIC, BDe, and other score-based structural learning
There are several challenges with score-based structural learning methods. The following describes the main challenges and how they are addressed.
1. over-complexity of models:
Challenge: Score-based methods penalize model complexity, but may select overly complex models.
Solution: Use cross-validation or other means to reconcile the trade-off between model complexity and performance. Tuning of hyperparameters and introduction of regularization terms may also be considered.
2. convergence to a locally optimal solution:
Challenge: Structural learning has a very large search space and may converge to a locally optimal solution.
Solution: Sometimes, starting from several initial values, or combining meta-heuristics, a more extensive search may be attempted.
3. problem of computational efficiency:
Challenge: Score-based structural learning is computationally expensive due to the large search space and can be very slow for large data sets and complex models.
Solution: Approximation methods, parallel computing, distributed computing, and other methods can be used to improve computational efficiency, and there are also methods for restricting the search space.
4. selection of prior distributions:
Challenge: Bayesian methods have difficulty in selecting an appropriate prior distribution because the choice of the prior distribution affects the results.
Solution: It is important to select a prior distribution based on domain knowledge and data characteristics to ensure robustness of the results, and sensitivity analysis could be performed to evaluate the impact of the prior distribution.
Reference Books and Reference Information
For more detailed information on Bayesian inference, please refer to “Probabilistic Generative Models” “Bayesian Inference and Machine Learning with Graphical Models” and “Nonparametric Bayesian and Gaussian Processes.
A good reference book on Bayesian estimation is “The Theory That Would Not Die: How Bayes’ Rule Cracked the Enigma Code, Hunted Down Russian Submarines, & Emerged Triumphant from Two Centuries of C“
“Think Bayes: Bayesian Statistics in Python“
“Bayesian Modeling and Computation in Python“
1. “Probabilistic Graphical Models: Principles and Techniques” by Daphne Koller and Nir Friedman
– Abstract: Provides a comprehensive overview of the fundamentals and applications of probabilistic graphical models, including Bayesian and Markov networks.
2. “Bayesian Networks and Decision Graphs” by Finn V. Jensen and Thomas D. Nielsen
– Abstract: Explains the basic concepts of Bayesian networks and practical methods for inference and learning.
3. “Graphical Models, Exponential Families, and Variational Inference” by Martin J. Wainwright and Michael I. Jordan
– Abstract: Comprehensive coverage of the theoretical foundations and optimisation techniques of graphical models.
4. “Causation, Prediction, and Search” by Peter Spirtes, Clark Glymour, and Richard Scheines
– Abstract: Focuses on causal inference and Bayesian network learning methods.
5. “Machine Learning: A Probabilistic Perspective” by Kevin P. Murphy
– Abstract: Comprehensive coverage of machine learning in general, with detailed coverage of Bayesian models.
6. “Learning Bayesian Networks” by Richard E. Neapolitan
– Abstract: An introduction to Bayesian networks with a special focus on learning Bayesian networks.
– 論文: “A Bayesian Method for the Induction of Probabilistic Networks from Data” by Gregory F. Cooper and Edward Herskovits (the paper underlying the BDe score).
– Online resources: lectures on Bayesian networks and probabilistic graphical models offered on Coursera.
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