A Multidimensional Unfolding Method Based on Bayes' Theorem

A Multidimensional Unfolding Method Based on Bayes' Theorem

June 1994 | G. D'Agostini
This paper presents a multidimensional unfolding method based on Bayes' Theorem, which is designed to estimate the true distribution of experimental data from distorted observed distributions. The method overcomes the limitations of traditional bin-to-bin corrections and matrix inversion methods by using an iterative procedure that updates the initial probability distribution of the true values. The key steps include: 1. **Introduction**: The problem of unfolding experimental distributions due to physics and detector effects is introduced, highlighting the need for a robust method. 2. **Bayes' Theorem**: The theorem is explained in the context of independent causes and effects, providing a framework for updating probabilities based on observed data. 3. **Unfolding an Experimental Distribution**: The method is detailed, including the calculation of expected event numbers for each cause, the estimation of the true number of events, and the determination of the final probabilities and efficiency. 4. **Estimation of Uncertainties**: The uncertainties in the unfolded distribution are analyzed, considering both statistical and systematic errors. 5. **Treatment of Background**: The method can naturally incorporate background contributions by adding them to the possible causes. 6. **Results**: The effectiveness of the method is demonstrated through simulations, showing that it converges to the true distribution with multiple iterations, especially when smoothing is applied between iterations. 7. **Conclusions**: The method is shown to be promising for multidimensional unfolding, with stable results and accurate uncertainty estimation. The paper concludes that the recursive application of Bayes' Theorem, combined with smoothing, provides a reliable and efficient way to unfold multidimensional distributions in experimental physics.This paper presents a multidimensional unfolding method based on Bayes' Theorem, which is designed to estimate the true distribution of experimental data from distorted observed distributions. The method overcomes the limitations of traditional bin-to-bin corrections and matrix inversion methods by using an iterative procedure that updates the initial probability distribution of the true values. The key steps include: 1. **Introduction**: The problem of unfolding experimental distributions due to physics and detector effects is introduced, highlighting the need for a robust method. 2. **Bayes' Theorem**: The theorem is explained in the context of independent causes and effects, providing a framework for updating probabilities based on observed data. 3. **Unfolding an Experimental Distribution**: The method is detailed, including the calculation of expected event numbers for each cause, the estimation of the true number of events, and the determination of the final probabilities and efficiency. 4. **Estimation of Uncertainties**: The uncertainties in the unfolded distribution are analyzed, considering both statistical and systematic errors. 5. **Treatment of Background**: The method can naturally incorporate background contributions by adding them to the possible causes. 6. **Results**: The effectiveness of the method is demonstrated through simulations, showing that it converges to the true distribution with multiple iterations, especially when smoothing is applied between iterations. 7. **Conclusions**: The method is shown to be promising for multidimensional unfolding, with stable results and accurate uncertainty estimation. The paper concludes that the recursive application of Bayes' Theorem, combined with smoothing, provides a reliable and efficient way to unfold multidimensional distributions in experimental physics.
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Understanding A Multidimensional unfolding method based on Bayes' theorem