Quadratic Likelihood Functional for Non-Asymptotic Parameter Estimation
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Solution Overview
Problem
Statistical estimation methods, particularly maximum likelihood (ML), face limitations in the non-asymptotic regime where the number of parameters is comparable to or larger than the number of data points, leading to issues such as reduced efficiency, inability to estimate covariance matrices, amplification of noise as signal, and computational inefficiencies.
Innovation Solution
The introduction of a quadratic likelihood functional (QLF) replaces the log-likelihood function, offering a Fisher consistent and computationally efficient alternative that is less sensitive to low-probability events and noise, allowing for effective parameter estimation and image reconstruction without requiring data binning.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If maximum likelihood (ML) method is used for parameter estimation, then asymptotic efficiency is improved, but in the non-asymptotic regime where the number of parameters is comparable to or larger than the number of data points, the method suffers from reduced efficiency, inability to estimate covariance matrices, and amplification of noise as signal
Solution Approach 1:
The patent changes the functional form of the likelihood function from logarithmic to quadratic. Specifically, it replaces the log-likelihood function L(θ) with a quadratic likelihood functional Q(θ) = -∫[f(x|θ) - g(x)]²dx, where f(x|θ) is the modeled probability density function and g(x) is the observed data distribution. This parameter change in the objective function transforms the optimization landscape, making it more suitable for non-asymptotic regimes by providing better numerical stability and avoiding the pathological behavior of the log-likelihood when parameters outnumber data points.
2Measurement precision
If log-likelihood function is used, then Fisher consistency is achieved, but computational efficiency deteriorates due to the non-quadratic nature and sensitivity to low-probability events
Solution Approach 1:
The patent employs a quadratic functional form that is computationally cheaper to evaluate and optimize compared to the log-likelihood function. The quadratic form Q(θ) = -∫[f(x|θ) - g(x)]²dx allows for simpler gradient calculations and more stable convergence properties, making it a 'computationally disposable' approach that can be efficiently evaluated multiple times during iterative optimization without the heavy computational burden associated with log-likelihood calculations, especially in high-dimensional parameter spaces.
3Adaptability or versatility
If the number of parameters is comparable to or larger than the number of data points, then model flexibility is improved, but noise amplification and over-interpretation of data occur
Solution Approach 1:
The quadratic likelihood functional provides a form of regularization through its mathematical structure. By using Q(θ) = -∫[f(x|θ) - g(x)]²dx, the method inherently cushions against overfitting in high-dimensional regimes. The quadratic form penalizes large deviations more smoothly than the log-likelihood, providing a built-in protective mechanism that prevents noise amplification before it can occur, especially when the number of parameters approaches or exceeds the number of data points.
Data Source
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AI summary
A method and system are provided for constructing a model of a target object in a computer processor by receiving an input signal from a source, the input signal containing data describing the target object and a plurality of parameters, the input signal having a noise portion; selecting a group of initial parameters, estimating a nonparametric probability distribution function (pdf) comprising a linear combination of a set of square-integrable basis functions, computing a quadratic likelihood functional (QLF), evaluating a fit of the initial parameters to the data based on the QLF, iteratively optimizing the QLF by selecting a new group of parameters and evaluating the fit of the new group of parameters into a predetermined condition is achieved. Once an acceptable fit is achieved, an output of a model of the target object constructed using optimized parameters can be displayed.