APPROXIMATE MULTIVARIATE POSTERIOR PROBABILITY DISTRIBUTIONS FROM SIMULATED SAMPLES

• US 20140278239A1
• Filed: 03/18/2014
• Published: 09/18/2014
• Est. Priority Date: 03/15/2013
• Status: Active Grant

• Current Assignee: SAS Institute Incorporated
• Original Assignee: SAS Institute Incorporated
• Inventors: Little, Mark Roland, Macaro, Christian, Chvosta, Jan
• Granted Patent: US 10,146,741 B2
• Time in Patent Office: Days
• US Class Current: 702/181
• 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
• CPC Class Codes: G06N 7/01 Probabilistic graphical models, e.g. probabilistic networks

Abstract

Various embodiments are directed to techniques for deriving a sample representation from a random sample. A computer-program product includes instructions to cause a first computing device to fit an empirical distribution function to a marginal probability distribution of a variable within a first sample portion of a random sample to derive a partial marginal probability distribution approximation, wherein the random sample is divided into multiple sample portions distributed among multiple computing devices; fit a first portion of a copula function to a multivariate probability distribution of the first sample portion, wherein the copula function is divided into multiple portions; and transmit an indication of a first likelihood contribution of the first sample portion to a coordinating device to cause a second computing device to fit a second portion of the copula function to a multivariate probability distribution of a second sample portion. Other embodiments are described and claimed.

Claims

1. A computer-program product tangibly embodied in a non-transitory machine-readable storage medium, the computer-program product including instructions operable to cause a first computing device to perform operations including:

fit an empirical distribution function to a marginal probability distribution of a variable within a first sample portion of a random sample to derive a first partial marginal probability distribution approximation, wherein the random sample is divided into a plurality of sample portions comprising the first sample portion and a second sample portion, and wherein the plurality of sample portions is distributed among a plurality of computing devices;

fit a first portion of a copula function to a multivariate probability distribution of the first sample portion, wherein the copula function is divided into a plurality of portions comprising the first portion of the copula function and a second portion of the copula function; and

transmit an indication of a first likelihood contribution of the first sample portion to a coordinating device to cause a second computing device to fit the second portion of the copula function to a multivariate probability distribution of the second sample portion.

2. The computer-program product of claim 1, the first computing device caused to perform operations including:

perform a first portion of a Bayesian analysis to derive a first portion of a posterior probability distribution;

generate the first sample portion from the first portion of the posterior probability distribution; and

exchange communications with another computing device to coordinate at least one of performance of a second portion of the Bayesian analysis by the second computing device to derive a second portion of the posterior probability distribution, or generation of the second sample portion by the second computing device from the second portion of the posterior probability distribution.

3. The computer-program product of claim 1, the first computing device caused to perform operations including:

receive copula coefficients from the coordinating device;

fit the first portion of the copula function to the multivariate probability distribution of the first sample portion again using the copula coefficients; and

transmit another indication of the first likelihood contribution of the first sample portion to the coordinating device to enable the coordinating device to determine to repeat fitting the copula function based on a degree of fit to the multivariate probability distribution of at least the first sample portion.

4. The computer-program product of claim 1, the first computing device caused to perform operations including:

compare a quantity of draws of parameters within at least the first sample portion to a threshold quantity of draws;

select the empirical distribution function as the first partial marginal probability distribution approximation associated with the variable based on the quantity of variables within at least the first sample portion exceeding the threshold; and

select quantiles taken of the empirical distribution function as the first partial marginal probability distribution approximation based on the quantity of variables within at least the first sample portion not exceeding the threshold.

5. The computer-program product of claim 1, the first computing device caused to perform operations including transmit the first portion of the copula and the first partial marginal probability distribution approximation to another computing device as a representation of the first sample portion.

6. The computer-program product of claim 1, the first computing device caused to perform operations including:

combine the first partial marginal probability distribution approximation with at least a second partial marginal probability distribution approximation associated with the variable to generate a marginal probability distribution approximation; and

combine the first portion of the copula function with at least the second portion of the copula function to generate the copula function.

7. The computer-program product of claim 6, the first computing device caused to perform operations including:

derive an inverse cumulative distribution function (ICDF) of the marginal probability distribution approximation; and

employ at least the inverse marginal CDF, the copula function and a random number generator to generate a simulated random sample as a representation of the random sample, wherein the simulated random sample is smaller in size than the random sample, and wherein the simulated random sample and the random sample have at least one similar statistical characteristic.

8. The computer-program product of claim 6, the first computing device caused to perform operations including:

derive a prior probability distribution from the marginal probability distribution approximation and the copula function; and

perform another Bayesian analysis using the prior probability distribution.

9. The computer-program product of claim 6, the first computing device caused to perform operations including:

derive an inverse marginal CDF of the marginal probability distribution approximation; and

employ at least the inverse marginal CDF and the copula function as a representation of the random sample in an analysis of a statistical characteristic of the random sample.

10. The computer-program product of claim 6, the first computing device caused to perform operations including transmit at least the marginal probability distribution approximation and the copula function to another computing device as a representation of the random sample.

11. A computer-implemented method comprising:

fitting, on a first computing device, an empirical distribution function to a marginal probability distribution of a variable within a first sample portion of a random sample to derive a first partial marginal probability distribution approximation, wherein the random sample is divided into a plurality of sample portions comprising the first sample portion and a second sample portion, and wherein the plurality of sample portions is distributed among a plurality of computing devices;

fitting, on the first computing device, a first portion of a copula function to a multivariate probability distribution of the first sample portion, wherein the copula function is divided into a plurality of portions comprising the first portion of the copula function and a second portion of the copula function; and

transmitting, from the first computing device, an indication of a first likelihood contribution of the first sample portion to a coordinating device to cause a second computing device to fit the second portion of the copula function to a multivariate probability distribution of the second sample portion.

12. The computer-implemented method of claim 11, comprising:

performing, on the first computing device, a first portion of a Bayesian analysis to derive a first portion of a posterior probability distribution;

generating, on the first computing device, the first sample portion from the first portion of the posterior probability distribution; and

exchanging communications with another computing device to coordinate at least one of performance of a second portion of the Bayesian analysis by the second computing device to derive a second portion of the posterior probability distribution, or generation of the second sample portion by the second computing device from the second portion of the posterior probability distribution.

13. The computer-implemented method of claim 11, comprising:

receiving copula coefficients from the coordinating device;

fitting the first portion of the copula function to the multivariate probability distribution of the first sample portion again using the copula coefficients; and

transmitting another indication of the first likelihood contribution of the first sample portion to the coordinating device to enable the coordinating device to determine to repeat fitting the copula function based on a degree of fit to the multivariate probability distribution of at least the first sample portion.

14. The computer-implemented method of claim 11, comprising:

comparing, on the first computing device, a quantity of draws of parameters within at least the first sample portion to a threshold quantity of variables;

selecting, on the first computing device, the empirical distribution function as the first partial marginal probability distribution approximation associated with the variable based on the quantity of variables within at least the first sample portion exceeding the threshold; and

selecting, on the first computing device, quantiles taken of the empirical distribution function as the first partial marginal probability distribution approximation based on the quantity of variables within at least the first sample portion not exceeding the threshold.

15. The computer-implemented method of claim 11, comprising transmitting, from the first computing device, the first portion of the copula and the first partial marginal probability distribution approximation to another computing device as a representation of the first sample portion.

16. The computer-implemented method of claim 11, comprising:

combining, on the first computing device, the first partial marginal probability distribution approximation with at least a second partial marginal probability distribution approximation associated with the variable to generate a marginal probability distribution approximation; and

combining, on the first computing device, the first portion of the copula function with at least the second portion of the copula function to generate the copula function.

17. The computer-implemented method of claim 16, comprising:

deriving, on the first computing device, an inverse cumulative distribution function (ICDF) of the marginal probability distribution approximation; and

employing, on the first computing device, at least the inverse marginal CDF, the copula function and a random number generator to generate a simulated random sample as a representation of the random sample, wherein the simulated random sample is smaller in size than the random sample, and wherein the simulated random sample and the random sample have at least one similar statistical characteristic.

18. The computer-implemented method of claim 16, comprising:

deriving, on the first computing device, a prior probability distribution from the marginal probability distribution approximation and the copula function; and

performing another Bayesian analysis using the prior probability distribution.

19. The computer-implemented method of claim 16, comprising:

deriving, on the first computing device, an inverse marginal CDF of the marginal probability distribution approximation; and

employing at least the inverse marginal CDF and the copula function as a representation of the random sample in an analysis of a statistical characteristic of the random sample.

20. The computer-implemented method of claim 16, comprising transmitting, from the first computing device, at least the marginal probability distribution approximation and the copula function to another computing device as a representation of the random sample.

21. An apparatus comprising:

a processor component of a first computing device;

an empirical function component for execution by the processor component to fit an empirical distribution function to a marginal probability distribution of a variable within a first sample portion of a random sample to derive a first partial marginal probability distribution approximation, wherein the random sample is divided into a plurality of sample portions comprising the first sample portion and a second sample portion, and wherein the plurality of sample portions is distributed among a plurality of computing devices;

a copula fitting component for execution by the processor component to fit a first portion of a copula function to a multivariate probability distribution of the first sample portion, wherein the copula function is divided into a plurality of portions comprising the first portion of the copula function and a second portion of the copula function; and

a communications component for execution by the processor component to transmit an indication of a first likelihood contribution of the first sample portion to a coordinating device to cause a second computing device to fit the second portion of the copula function to a multivariate probability distribution of the second sample portion.

22. The apparatus of claim 21, comprising:

a Bayesian component for execution by the processor component to perform a first portion of a Bayesian analysis to derive a first portion of a posterior probability distribution; and

a simulation component for execution by the processor component to generate the first sample portion from the first portion of the posterior probability distribution, wherein the communications component exchanges communications with another computing device to coordinate at least one of performance of a second portion of the Bayesian analysis by the second computing device to derive a second portion of the posterior probability distribution, or generation of the second sample portion by the second computing device from the second portion of the posterior probability distribution.

23. The apparatus of claim 21, wherein:

the communications component receives copula coefficients from the coordinating device;

the copula fitting component fits the first portion of the copula function to the multivariate probability distribution of the first sample portion again using the copula coefficients; and

the communications component transmits another indication of the first likelihood contribution of the first sample portion to the coordinating device to enable the coordinating device to determine to repeat fitting the copula function based on a degree of fit to the multivariate probability distribution of at least the first sample portion.

24. The apparatus of claim 21, comprising a quantile component to:

compare a quantity of variables within at least the first sample portion to a threshold quantity of variables;

select the empirical distribution function as a first partial marginal probability distribution approximation associated with the variable based on the quantity of variables within at least the first sample portion exceeding the threshold; and

derive quantiles of the empirical distribution function and select the quantiles as the first partial marginal probability distribution approximation based on the quantity of variables within at least the first sample portion not exceeding the threshold.

25. The apparatus of claim 21, wherein the communications component transmits the first portion of the copula and the first partial marginal probability distribution approximation to another computing device as a representation of the first sample portion.

26. The apparatus of claim 21, comprising:

a marginal combining component for execution by the processor component to combine the first partial marginal probability distribution approximation with at least a second partial marginal probability distribution approximation associated with the variable to generate a marginal probability distribution approximation; and

a copula combining component for execution by the processor component to combine the first portion of the copula function with at least the second portion of the copula function to generate the copula function.

27. The apparatus of claim 26, comprising a simulation component for execution by the processor component to:

derive an inverse cumulative distribution function (ICDF) of the marginal probability distribution approximation; and

employ at least the inverse marginal CDF, the copula function and a random number generator to generate a simulated random sample as a representation of the random sample, wherein the simulated random sample is smaller in size than the random sample, and wherein the simulated random sample and the random sample have at least one similar statistical characteristic.

28. The apparatus of claim 26, wherein the communications component transmits at least the marginal probability distribution approximation and the copula function to another computing device as a representation of the random sample.