Estimating information and correlation from finite data

 

 
• Inferring unknown parameters from data
• Inference increases information about the true distribution on average
• Estimating functions of probability distributions from finite samples.
  • Statement of the problem solved here
  • Estimating from finite data is ubiquitous
  • The distribution of data
  • The Bayes' estimator
  • The Bayes' estimator minimizes posterior mean square error
  • Form of the prior
  • Form of the integrals giving the estimator
  • Integrating
  • Estimators for the first and second moments of the entropy
  • Entropy estimator comparison
  • Estimators for moments, correlations, cumulants.
  • Extended notation for more complicated functions of probability distributions
  • More integration techniques
  • Estimators for functions involving up-to-pairwise overlap integrals
  • Multiple overlap integration
• Appendix A. Hypergeometric functions
• Appendix B. Hypergeometric function identities
• Appendix C. Transforms
  • Appendix C.1. The T transform
  • Appendix C.2. The Z transform
  • Appendix C.3. The Laplace transform and inverse
• Appendix D. Commuting linear operators
  • Appendix D.1. Commuting two integrals
  • Appendix D.2. Commuting integrals and derivatives
• Appendix E. Analytic continuation: Expanding 's domain
• Appendix F. Existence conditions
• Appendix G. Derivatives of overlap convolutions: Poles