Data Assimilation: A Mathematical Introduction

These notes provide a systematic mathematical treatment of the subject of data assimilation

Maximum Fidelity

The most fundamental problem in statistics is the inference of an unknown probability distribution from a finite number of samples. For a specific observed data set, answers to the following questions would be desirable: (1) Estimation: Which candidate distribution provides the best fit to the observed data?, (2) Goodness-of-fit: How concordant is this distribution with the observed data?, and (3) Uncertainty: How concordant are other candidate distributions with the observed data? A simple unified approach for univariate data that addresses these traditionally distinct statistical notions is presented called "maximum fidelity". Maximum fidelity is a strict frequentist approach that is fundamentally based on model concordance with the observed data. The fidelity statistic is a general information measure based on the coordinate-independent cumulative distribution and critical yet previously neglected symmetry considerations. An approximation for the null distribution of the fidelity allows its direct conversion to absolute model concordance (p value). Fidelity maximization allows identification of the most concordant model distribution, generating a method for parameter estimation, with neighboring, less concordant distributions providing the "uncertainty" in this estimate. Maximum fidelity provides an optimal approach for parameter estimation (superior to maximum likelihood) and a generally optimal approach for goodness-of-fit assessment of arbitrary models applied to univariate data. Extensions to binary data, binned data, multidimensional data, and classical parametric and nonparametric statistical tests are described. Maximum fidelity provides a philosophically consistent, robust, and seemingly optimal foundation for statistical inference. All findings are presented in an elementary way to be immediately accessible to all researchers utilizing statistical analysis.Comment: 66 pages, 32 figures, 7 tables, submitte

RAINIER: A Simulation Tool for Distributions of Excited Nuclear States and Cascade Fluctuations

A new code has been developed named RAINIER that simulates the $\gamma$-ray decay of discrete and quasi-continuum nuclear levels for a user-specified range of energy, angular momentum, and parity including a realistic treatment of level spacing and transition width fluctuations. A similar program, DICEBOX, uses the Monte Carlo method to simulate level and width fluctuations but is restricted to $\gamma$-ray decay from no more than two initial states such as de-excitation following thermal neutron capture. On the other hand, modern reaction codes such as TALYS and EMPIRE populate a wide range of states in the residual nucleus prior to $\gamma$-ray decay, but do not go beyond the use of deterministic functions and therefore neglect cascade fluctuations. This combination of capabilities allows RAINIER to be used to determine quasi-continuum properties through comparison with experimental data. Several examples are given that demonstrate how cascade fluctuations influence experimental high-resolution $\gamma$-ray spectra from reactions that populate a wide range of initial states.Comment: 14 pages, 13 figures, Nuclear Instrumentation and Methods A, 201

How unitary cosmology generalizes thermodynamics and solves the inflationary entropy problem

We analyze cosmology assuming unitary quantum mechanics, using a tripartite partition into system, observer and environment degrees of freedom. This generalizes the second law of thermodynamics to "The system's entropy can't decrease unless it interacts with the observer, and it can't increase unless it interacts with the environment." The former follows from the quantum Bayes Theorem we derive. We show that because of the long-range entanglement created by cosmological inflation, the cosmic entropy decreases exponentially rather than linearly with the number of bits of information observed, so that a given observer can reduce entropy by much more than the amount of information her brain can store. Indeed, we argue that as long as inflation has occurred in a non-negligible fraction of the volume, almost all sentient observers will find themselves in a post-inflationary low-entropy Hubble volume, and we humans have no reason to be surprised that we do so as well, which solves the so-called inflationary entropy problem. An arguably worse problem for unitary cosmology involves gamma-ray-burst constraints on the "Big Snap", a fourth cosmic doomsday scenario alongside the "Big Crunch", "Big Chill" and "Big Rip", where an increasingly granular nature of expanding space modifies our life-supporting laws of physics. Our tripartite framework also clarifies when it is valid to make the popular quantum gravity approximation that the Einstein tensor equals the quantum expectation value of the stress-energy tensor, and how problems with recent attempts to explain dark energy as gravitational backreaction from super-horizon scale fluctuations can be understood as a failure of this approximation.Comment: Updated to match accepted PRD version, including Quantum Bayes Theorem derivation and rigorous proof that decoherence increases von Neumann entropy. 20 pages, 5 fig