Computational estimation of a characteristic of a posterior distribution

Abstract

An apparatus for implementing a computing system to predict preferences includes at least one processor device operatively coupled to a memory. The at least one processor device is configured to calculate a parameter relating to a density of a prior distribution at each sample of a set of samples associated with the prior distribution. The at least one parameter including a distance from each sample to at least one neighboring sample. The at least one processor device is further configured to estimate, for the plurality of samples, at least one differential entropy of at least one posterior distribution associated with at least one observation based on the parameter relating to the density of the prior distribution at each sample and the likelihood of observation for each sample. The estimation is performed without sampling the at least one posterior distribution to reduce consumption of resources of the computing system.

What is claimed is: 
     
         1 . An apparatus for implementing a computing system to predict preferences, comprising:
 at least one processor device operatively coupled to a memory and configured to:
 calculate a parameter relating to a density of a prior distribution at each sample of a set of samples associated with the prior distribution; 
 estimate, for the plurality of samples, at least one differential entropy of at least one posterior distribution associated with at least one observation based on the parameter relating to the density of the prior distribution at each sample and the likelihood of observation for each sample; 
 select an action from a plurality of candidate actions each causing one or more observations; and 
 transmit, to at least one device associated with at least one person, at least one electronic interaction generated based on the action. 
   
     
     
         2 . The apparatus of  claim 1 , wherein the at least one processor device is further configured to:
 generate a plurality of samples from the prior distribution;   obtain, for each sample among the plurality of samples, a likelihood of an observation as an output of a likelihood function given the sample; and   eliminate samples from the plurality of samples having a likelihood less than a threshold value to generate the set of samples.   
     
     
         3 . The apparatus of  claim 1 , wherein the distance from each sample to at least one neighboring sample is a distance from each sample to a k th -nearest neighbor, k being a natural number. 
     
     
         4 . The apparatus of  claim 1 , wherein the at least one processor device is further configured to estimate the at least one differential entropy of the at least one posterior distribution by approximating a probability density function of the prior distribution at each sample using a volume of a sphere having a radius equal to the distance. 
     
     
         5 . The apparatus of  claim 1 , wherein the at least one processor device is further configured to estimate the at least one differential entropy of the at least one posterior distribution having Euler's constant as a constant term. 
     
     
         6 . The apparatus of  claim 1 , wherein the at least one processor device is further configured to estimate the at least one differential entropy of each of a plurality of posterior distributions based on the at least one parameter relating to the density at each sample and a likelihood of transition for each sample from the prior distribution to each posterior distribution, and wherein each likelihood of transition exceeds a threshold likelihood. 
     
     
         7 . The apparatus of  claim 1 , wherein the at least one processor device is further configured to obtain the at least one observation from a model having an internal state estimated by the prior distribution. 
     
     
         8 . The apparatus of  claim 7 , wherein the model is a behavioral model of at least one person. 
     
     
         9 . The apparatus of  claim 1 , wherein the action from the plurality of candidate actions each causing one or more observations is selected based on expected values of the differential entropies estimated for all observations caused by the action. 
     
     
         10 . A computer-implemented method for implementing a computer system to predict preferences, comprising:
 calculating a parameter relating to a density of a prior distribution at each sample of a set of samples associated with the prior distribution;   estimating, for the plurality of samples, at least one differential entropy of at least one posterior distribution associated with at least one observation based on the parameter relating to the density of the prior distribution at each sample and the likelihood of observation for each sample;   selecting an action from a plurality of candidate actions each causing one or more observations; and   transmitting, to at least one device associated with at least one person, at least one electronic interaction generated based on the action.   
     
     
         11 . The method of  claim 10 , further comprising:
 generating a plurality of samples from the prior distribution;   obtaining, for each sample among the plurality of samples, a likelihood of an observation as an output of a likelihood function given the sample; and   eliminating samples from the plurality of samples having a likelihood less than a threshold value to generate the set of samples.   
     
     
         12 . The method of  claim 10 , wherein the distance from each sample to at least one neighboring sample is a distance from each sample to a k th -nearest neighbor, k being a natural number. 
     
     
         13 . The method of  claim 10 , wherein estimating the at least one differential entropy of the at least one posterior distribution further includes approximating a probability density function of the prior distribution at each sample using a volume of a sphere having a radius equal to the distance. 
     
     
         14 . The method of  claim 10 , wherein the at least one differential entropy of the at least one posterior distribution is estimated having Euler's constant as a constant term. 
     
     
         15 . The method of  claim 10 , wherein the at least one differential entropy of each of a plurality of posterior distributions is estimated based on the at least one parameter relating to the density at each sample and a likelihood of transition for each sample from the prior distribution to each posterior distribution, and wherein each likelihood of transition exceeds a threshold likelihood. 
     
     
         16 . The method of  claim 10 , wherein the at least one processor device is further configured to obtain the at least one observation from a model having an internal state estimated by the prior distribution. 
     
     
         17 . The method of  claim 16 , wherein the model is a behavioral model of at least one person. 
     
     
         18 . The method of  claim 10 , wherein the action from the plurality of candidate actions each causing one or more observations is selected based on expected values of the differential entropies estimated for all observations caused by the action. 
     
     
         19 . A computer program product for implementing a computer system to predict preferences, the computer program product comprising a non-transitory computer readable storage medium having program instructions embodied therewith, the program instructions executable by a computer to cause the computer to perform operations comprising:
 calculating a parameter relating to a density of a prior distribution at each sample of a set of samples associated with the prior distribution;   estimating, for the plurality of samples, at least one differential entropy of at least one posterior distribution associated with at least one observation based on the parameter relating to the density of the prior distribution at each sample and the likelihood of observation for each sample;   selecting an action from a plurality of candidate actions each causing one or more observations; and   transmitting, to at least one device associated with at least one person, at least one electronic interaction generated based on the action.   
     
     
         20 . The computer program product of  claim 19 , wherein the action from the plurality of candidate actions each causing one or more observations is selected based on expected values of the differential entropies estimated for all observations caused by the action.