1,707 research outputs found
Rank-based model selection for multiple ions quantum tomography
The statistical analysis of measurement data has become a key component of
many quantum engineering experiments. As standard full state tomography becomes
unfeasible for large dimensional quantum systems, one needs to exploit prior
information and the "sparsity" properties of the experimental state in order to
reduce the dimensionality of the estimation problem. In this paper we propose
model selection as a general principle for finding the simplest, or most
parsimonious explanation of the data, by fitting different models and choosing
the estimator with the best trade-off between likelihood fit and model
complexity. We apply two well established model selection methods -- the Akaike
information criterion (AIC) and the Bayesian information criterion (BIC) -- to
models consising of states of fixed rank and datasets such as are currently
produced in multiple ions experiments. We test the performance of AIC and BIC
on randomly chosen low rank states of 4 ions, and study the dependence of the
selected rank with the number of measurement repetitions for one ion states. We
then apply the methods to real data from a 4 ions experiment aimed at creating
a Smolin state of rank 4. The two methods indicate that the optimal model for
describing the data lies between ranks 6 and 9, and the Pearson test
is applied to validate this conclusion. Additionally we find that the mean
square error of the maximum likelihood estimator for pure states is close to
that of the optimal over all possible measurements.Comment: 24 pages, 6 figures, 3 table
Exact and approximate stepdown methods for multiple hypothesis testing
Consider the problem of testing k hypotheses simultaneously. In this paper, we discuss finite and large sample theory of stepdown methods that provide control of the familywise error rate (FWE). In order to improve upon the Bonferroni method or Holm's (1979) stepdown method, Westfall and Young (1993) make eective use of resampling to construct stepdown methods that implicitly estimate the dependence structure of the test statistics. However, their methods depend on an assumption called subset pivotality. The goal of this paper is to construct general stepdown methods that do not require such an assumption. In order to accomplish this, we take a close look at what makes stepdown procedures work, and a key component is a monotonicity requirement of critical values. By imposing such monotonicity on estimated critical values (which is not an assumption on the model but an assumption on the method), it is demonstrated that the problem of constructing a valid multiple test procedure which controls the FWE can be reduced to the problem of contructing a single test which controls the usual probability of a Type 1 error. This reduction allows us to draw upon an enormous resampling literature as a general means of test contruction.Bootstrap, familywise error rate, multiple testing, permutation test, randomization test, stepdown procedure, subsampling
Fast Two-Sample Testing with Analytic Representations of Probability Measures
We propose a class of nonparametric two-sample tests with a cost linear in
the sample size. Two tests are given, both based on an ensemble of distances
between analytic functions representing each of the distributions. The first
test uses smoothed empirical characteristic functions to represent the
distributions, the second uses distribution embeddings in a reproducing kernel
Hilbert space. Analyticity implies that differences in the distributions may be
detected almost surely at a finite number of randomly chosen
locations/frequencies. The new tests are consistent against a larger class of
alternatives than the previous linear-time tests based on the (non-smoothed)
empirical characteristic functions, while being much faster than the current
state-of-the-art quadratic-time kernel-based or energy distance-based tests.
Experiments on artificial benchmarks and on challenging real-world testing
problems demonstrate that our tests give a better power/time tradeoff than
competing approaches, and in some cases, better outright power than even the
most expensive quadratic-time tests. This performance advantage is retained
even in high dimensions, and in cases where the difference in distributions is
not observable with low order statistics
Contributions to Mediation Analysis and First Principles Modeling for Mechanistic Statistical Analysis
This thesis contains three projects that propose novel methods for studying mechanisms that explain statistical relationships. The ultimate goal of each of these methods is to help researchers describe how or why complex relationships between observed variables exist.
The first project proposes and studies a method for recovering mediation structure in high dimensions. We take a dimension reduction approach that generalizes the ``product of coefficients'' concept for univariate mediation analysis through the optimization of a loss function. We devise an efficient algorithm for optimizing the product-of-coefficients inspired loss function. Through extensive simulation studies, we show that the method is capable of consistently identifying mediation structure. Finally, two case studies are presented that demonstrate how the method can be used to conduct multivariate mediation analysis.
The second project uses tools from conditional inference to improve the calibration of tests of univariate mediation hypotheses. The key insight of the project is that the non-Euclidean geometry of the null parameter space causes the test statistic’s sampling distribution to depend on a nuisance parameter. After identifying a statistic that is both sufficient for the nuisance parameter and approximately ancillary for the parameter of interest, we derive the test statistic’s limiting conditional sampling distribution. We additionally develop a non-standard bootstrap procedure for calibration in finite samples. We demonstrate through simulation studies that improved evidence calibration leads to substantial power increases over existing methods. This project suggests that conditional inference might be a useful tool in evidence calibration for other non-standard or otherwise challenging problems.
In the last project, we present a methodological contribution to a pharmaceutical science study of {em in vivo} ibuprofen pharmacokinetics. We demonstrate how model misspecification in a first-principles analysis can be addressed by augmenting the model to include a term corresponding to an omitted source of variation. In previously used first-principles models, gastric emptying, which is pulsatile and stochastic, is modeled as first-order diffusion for simplicity. However, analyses suggest that the actual gastric emptying process is expected to be a unimodal smooth function, with phase and amplitude varying by subject. Therefore, we adopt a flexible approach in which a highly idealized parametric version of gastric emptying is combined with a Gaussian process to capture deviations from the idealized form. These functions are characterized by their distributions, which allows us to learn their common and unique features across subjects despite that these features are not directly observed. Through simulation studies, we show that the proposed approach is able to identify certain features of latent function distributions.PHDStatisticsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/163026/1/josephdi_1.pd
Cluster-Robust Bootstrap Inference in Quantile Regression Models
In this paper I develop a wild bootstrap procedure for cluster-robust
inference in linear quantile regression models. I show that the bootstrap leads
to asymptotically valid inference on the entire quantile regression process in
a setting with a large number of small, heterogeneous clusters and provides
consistent estimates of the asymptotic covariance function of that process. The
proposed bootstrap procedure is easy to implement and performs well even when
the number of clusters is much smaller than the sample size. An application to
Project STAR data is provided.Comment: 46 pages, 4 figure
No imminent quantum supremacy by boson sampling
It is predicted that quantum computers will dramatically outperform their
conventional counterparts. However, large-scale universal quantum computers are
yet to be built. Boson sampling is a rudimentary quantum algorithm tailored to
the platform of photons in linear optics, which has sparked interest as a rapid
way to demonstrate this quantum supremacy. Photon statistics are governed by
intractable matrix functions known as permanents, which suggests that sampling
from the distribution obtained by injecting photons into a linear-optical
network could be solved more quickly by a photonic experiment than by a
classical computer. The contrast between the apparently awesome challenge faced
by any classical sampling algorithm and the apparently near-term experimental
resources required for a large boson sampling experiment has raised
expectations that quantum supremacy by boson sampling is on the horizon. Here
we present classical boson sampling algorithms and theoretical analyses of
prospects for scaling boson sampling experiments, showing that near-term
quantum supremacy via boson sampling is unlikely. While the largest boson
sampling experiments reported so far are with 5 photons, our classical
algorithm, based on Metropolised independence sampling (MIS), allowed the boson
sampling problem to be solved for 30 photons with standard computing hardware.
We argue that the impact of experimental photon losses means that demonstrating
quantum supremacy by boson sampling would require a step change in technology.Comment: 25 pages, 9 figures. Comments welcom
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