# Quantum statistical inference for density estimation

## Abstract

A new penalized likelihood method for non-parametric density estimation is proposed, which is based on a mathematical analogy to quantum statistical physics. The mathematical procedure for density estimation is related to maximum entropy methods for inverse problems; the penalty function is a convex information divergence enforcing global smoothing toward default models, positivity, extensivity and normalization. The novel feature is the replacement of classical entropy by quantum entropy, so that local smoothing may be enforced by constraints on the expectation values of differential operators. Although the hyperparameters, covariance, and linear response to perturbations can be estimated by a variety of statistical methods, we develop the Bayesian interpretation. The linear response of the MAP estimate is proportional to the covariance. The hyperparameters are estimated by type-II maximum likelihood. The method is demonstrated on standard data sets.

- Authors:

- Publication Date:

- Research Org.:
- Los Alamos National Lab., NM (United States)

- Sponsoring Org.:
- USDOE, Washington, DC (United States)

- OSTI Identifier:
- 10193654

- Report Number(s):
- LA-UR-93-3552; CONF-9308107-4

ON: DE94002625

- DOE Contract Number:
- W-7405-ENG-36

- Resource Type:
- Conference

- Resource Relation:
- Conference: Joint American Statistical Association (ASA), Institute of Mathematics Statistics and Biometric Society conference,San Francisco, CA (United States),8-12 Aug 1993; Other Information: PBD: [1993]

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 99 GENERAL AND MISCELLANEOUS//MATHEMATICS, COMPUTING, AND INFORMATION SCIENCE; 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; DENSITY; STATISTICAL MODELS; STATISTICS; ENTROPY; STATISTICAL MECHANICS; RESPONSE FUNCTIONS; PARAMETRIC ANALYSIS; 990200; 661300; MATHEMATICS AND COMPUTERS; OTHER ASPECTS OF PHYSICAL SCIENCE

### Citation Formats

```
Silver, R N, Martz, H F, and Wallstrom, T.
```*Quantum statistical inference for density estimation*. United States: N. p., 1993.
Web.

```
Silver, R N, Martz, H F, & Wallstrom, T.
```*Quantum statistical inference for density estimation*. United States.

```
Silver, R N, Martz, H F, and Wallstrom, T. Mon .
"Quantum statistical inference for density estimation". United States. https://www.osti.gov/servlets/purl/10193654.
```

```
@article{osti_10193654,
```

title = {Quantum statistical inference for density estimation},

author = {Silver, R N and Martz, H F and Wallstrom, T},

abstractNote = {A new penalized likelihood method for non-parametric density estimation is proposed, which is based on a mathematical analogy to quantum statistical physics. The mathematical procedure for density estimation is related to maximum entropy methods for inverse problems; the penalty function is a convex information divergence enforcing global smoothing toward default models, positivity, extensivity and normalization. The novel feature is the replacement of classical entropy by quantum entropy, so that local smoothing may be enforced by constraints on the expectation values of differential operators. Although the hyperparameters, covariance, and linear response to perturbations can be estimated by a variety of statistical methods, we develop the Bayesian interpretation. The linear response of the MAP estimate is proportional to the covariance. The hyperparameters are estimated by type-II maximum likelihood. The method is demonstrated on standard data sets.},

doi = {},

journal = {},

number = ,

volume = ,

place = {United States},

year = {1993},

month = {11}

}