TARGETED MAXIMUM LIKELIHOOD ESTIMATION

• US 20090016470A1
• Filed: 10/04/2007
• Published: 01/15/2009
• Est. Priority Date: 07/13/2007
• Status: Active Grant

• Current Assignee: Regents of the University of California (University of California)
• Original Assignee: Regents of the University of California (University of California)
• Inventors: van der Laan, Mark
• Granted Patent: US 8,438,126 B2
• Time in Patent Office: Days
• US Class Current: 375/341
• CPC Class Codes: G06F 17/18 for evaluating statistical data , e.g. average values, frequency distributions, probability functions, regression analysis forecasting specially adapted for a specific administrative, business or logistic context G06Q10/04

Abstract

A method for obtaining an estimator for a distribution pertaining to a dataset is provided. In an illustrative embodiment, the method includes obtaining a dataset; determining a question pertaining to the data; determining an initial estimator descriptive of a distribution of the data; and selectively modifying the initial estimator based on the question, yielding a targeted estimator in response thereto. In a more specific embodiment, selectively modifying the initial estimator includes applying an additional equation and/or a fluctuation function to the initial estimator to yield the targeted estimator, wherein the additional equation or fluctuation function parameter (ε) causes the initial estimator to fluctuate or change as a function of the parameter. The fluctuation function is chosen so that when the parameter ε is set to zero, the initial estimator is not fluctuated. The targeted estimator and a fluctuation function may be employed in an accompanying targeted Bayesian method that involves mapping a prior distribution of a target feature to a posterior distribution of the target feature.

Claims

1. A method for obtaining an estimator comprising:

determining data;

determining an initial estimator descriptive of a first distribution of the data;

determining a target feature pertaining to the data; and

selectively modifying the initial estimator based on the target feature to determine a modified estimator, the modified estimator providing a second distribution of the data.

2. The method of claim 1, wherein selectively modifying includes:

applying an epsilon function to the initial estimator to yield the modified estimator.

3. The method of claim 2 further including:

selecting a value of a parameter (ε

) by maximizing or minimizing or increasing or decreasing one or more empirical criteria, wherein the epsilon function is a function of a stretching parameter ε

.

4. The method of claim 2, wherein the epsilon function includes:

a function of a stretching parameter ε

such that the initial estimator fluctuates or changes as a function of the parameter ε

when the epsilon function is applied to the initial estimator.

5. The method of claim 4, wherein when the parameter (ε

) is set to zero, the initial estimator is not fluctuated.

6. The method of claim 5, wherein a linear span of a derivative of the one or more empirical criteria as a function of the stretching parameter ε

at ε

=0 includes components of a predetermined function.

7. The method of claim 1, wherein selectively modifying includes:

iteratively updating the modified estimator by using a previously determined modified estimator to provide the initial estimator for a subsequent iteration.

8. The method of claim 7, wherein iteratively updating includes:

reducing asymptotic bias for the target feature in each iteration of the updating of the modified estimator.

9. The method of claim 7, wherein iteratively updating further includes:

creating a sub-model of stretched versions of the initial estimator, wherein the stretched versions have been stretched by an epsilon function; and

using the sub-model to select a value for a parameter (ε

) of the epsilon function based on a criterion; and

repeating the above steps with each updating of said step of iteratively updating.

10. The method of claim 1, further including:

using the modified estimator and a stretching function to map a prior probability distribution of a target feature to a posterior distribution of the target feature.

11. The method of claim 1, wherein the modified estimator solves or approximately solves an equation by:

0=F(P)(O(1), . . . , O(n)),

where F(P) is a function of the probability distribution P;

where P represents the modified estimator, and where O(n) represents an n-th observation, where n is an integer, and where F(P)(O(1), . . . , O(n)) represents F(P) applied to observations O(1), . . . , O(n).

12. The method of claim 1, wherein the modified estimator solves or approximately solves an equation given by:

O=D(P)(O(1))+ . . . +D(P)(O(n)), [3]

where D(P) is an efficient influence curve equation evaluated P;

where P represents the modified estimator; and

where O(n) represents an n-th observation, where n is an integer; and

where D(P)(O(n)) represents D(P) applied to an observation O(n).

13. The method of claim 1, further including:

determining multiple modified estimators from multiple initial estimators.

14. The method of claim 13, further including:

comparing one or more criteria using likelihood values for the multiple modified estimators to determine a desired modified estimator.

15. A system configured to obtain an estimator of a distribution pertaining to data comprising:

an initial estimator module configured to receive data and a target feature pertaining to the data, wherein the initial estimator module is configured to determine an initial estimator descriptive of a first distribution of the data; and

a targeted estimator module coupled to the initial estimator module and configured to selectively modify the initial estimator based on the target feature to determine a modified estimator, the modified estimator providing a second distribution of the data.

16. The system of claim 15, wherein the targeted estimator module is configured to provide a targeted estimator usable to yield an improved estimator of a particular feature relative to an estimate of the feature that would be provided via an initial estimator provided by the initial estimator module.

17. The system of claim 16, wherein the targeted estimator module is configured to remove asymptotic bias for the target feature from an initial estimator output by the initial estimator module and provide a targeted estimator as output in response thereto.

18. The system of claim 16, further including a targeted feature calculation module coupled to the targeted estimator module, wherein the targeted feature calculation module is adapted to apply a mapping function based on the target information sought to a targeted estimator output by the targeted estimator module.

19. Software encoded in one or more tangible media for execution by the one or more processors and when executed operable to:

determine data;

determine a target feature pertaining to the data;

determine an initial estimator descriptive of a first distribution of the data; and

selectively modify the initial estimator based on the target feature to determine a modified estimator, the modified estimator providing a second distribution of the data.

20. A method comprising:

obtaining a dataset; and

mapping a prior probability distribution of a target feature to a posterior distribution of the target feature using an estimator of a probability distribution of the dataset and a stretching function.

21. The method of claim 20, wherein mapping includes:

calculating the posterior distribution of the target feature using a sub-model of one or more stretched probability distributions obtained by stretching one or more probability distributions of the dataset with the stretching function.

22. The method of claim 21, further including:

determining the stretched targeted estimators by treating a targeted estimator, which represents the estimator of the probability distribution of the dataset, as fixed or non-random and letting a stretching parameter (ε

) of the stretching function vary to determine the posterior distribution of the target feature given the dataset.

23. The method of claim 20, wherein mapping further includes:

calculating the posterior distribution of the target feature by first calculating a posterior distribution of a parameter (ε

) of the stretching function, given the data set, from a prior distribution of the parameter ε

.

24. The method of claim 20, wherein mapping further includes:

determining the estimator of a probability distribution of the dataset, wherein the estimator of a probability distribution of the dataset includes a targeted estimator that is targeted based on the target feature.

25. The method of claim 24, wherein the targeted estimator solves or approximately solves an efficient influence curve equation or other estimating equation.

26. The method of claim 24, wherein mapping further includes:

determining the stretching function, wherein the stretching function is a function of a variable parameter (ε

) such that when ε

=0, the targeted estimator to which the stretching function is applied is not stretched.

27. The method of claim 26, wherein mapping further includes:

determining a sub-model that includes targeted probability distributions of the dataset indexed by different values of the variable parameter (ε

), wherein the targeted probability distributions are treated as non-random.

28. The method of claim 27, wherein determining a sub-model further includes:

treating the stretched targeted probability distributions as conditional probability distributions of the dataset given choices of the parameter ε

.

29. The method of claim 28, wherein mapping further includes:

determining a prior distribution of the parameter ε

that implies, for the sub-model, the prior probability distribution of the target feature.

30. The method of claim 29, wherein mapping further includes:

using the sub-model and the prior probability distribution of the parameter ε

to determine the posterior distribution of the target feature given the dataset.