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A tutorial on cue combination and Signal Detection Theory: Using changes in sensitivity to evaluate how observers integrate sensory information
Many sensory inputs contain multiple sources of information (‘cues’), such as two sounds of different frequencies, or a voice heard in unison with moving lips. Often, each cue provides a separate estimate of the same physical attribute, such as the size or location of an object. An ideal observer can exploit such redundant sensory information to improve the accuracy of their perceptual judgments. For example, if each cue is modeled as an independent, Gaussian, random variable, then combining Ncues should provide up to a √N improvement in detection/discrimination sensitivity. Alternatively, a less efficient observer may base their decision on only a subset of the available information, and so gain little or no benefit from having access to multiple sources of information. Here we use Signal Detection Theory to formulate and compare various models of cue-combination, many of which are commonly used to explain empirical data. We alert the reader to the key assumptions inherent in each model, and provide formulas for deriving quantitative predictions. Code is also provided for simulating each model, allowing expected levels of measurement error to be quantified. Based on these results, it is shown that predicted sensitivity often differs surprisingly little between qualitatively distinct models of combination. This means that sensitivity alone is not sufficient for understanding decision efficiency, and the implications of this are discussed
The promising future of a robust cosmological neutrino mass measurement
We forecast the sensitivity of thirty-five different combinations of future
Cosmic Microwave Background and Large Scale Structure data sets to cosmological
parameters and to the total neutrino mass. We work under conservative
assumptions accounting for uncertainties in the modelling of systematics. In
particular, for galaxy redshift surveys, we remove the information coming from
non-linear scales. We use Bayesian parameter extraction from mock likelihoods
to avoid Fisher matrix uncertainties. Our grid of results allows for a direct
comparison between the sensitivity of different data sets. We find that future
surveys will measure the neutrino mass with high significance and will not be
substantially affected by potential parameter degeneracies between neutrino
masses, the density of relativistic relics, and a possible time-varying
equation of state of Dark Energy.Comment: 27 pages, 4 figures, 8 tables. v2: updated Euclid sensitivity
settings, matches published versio
Sensitivity And Out-Of-Sample Error in Continuous Time Data Assimilation
Data assimilation refers to the problem of finding trajectories of a
prescribed dynamical model in such a way that the output of the model (usually
some function of the model states) follows a given time series of observations.
Typically though, these two requirements cannot both be met at the same
time--tracking the observations is not possible without the trajectory
deviating from the proposed model equations, while adherence to the model
requires deviations from the observations. Thus, data assimilation faces a
trade-off. In this contribution, the sensitivity of the data assimilation with
respect to perturbations in the observations is identified as the parameter
which controls the trade-off. A relation between the sensitivity and the
out-of-sample error is established which allows to calculate the latter under
operational conditions. A minimum out-of-sample error is proposed as a
criterion to set an appropriate sensitivity and to settle the discussed
trade-off. Two approaches to data assimilation are considered, namely
variational data assimilation and Newtonian nudging, aka synchronisation.
Numerical examples demonstrate the feasibility of the approach.Comment: submitted to Quarterly Journal of the Royal Meteorological Societ
Nonparametric Covariate Adjustment for Receiver Operating Characteristic Curves
The accuracy of a diagnostic test is typically characterised using the
receiver operating characteristic (ROC) curve. Summarising indexes such as the
area under the ROC curve (AUC) are used to compare different tests as well as
to measure the difference between two populations. Often additional information
is available on some of the covariates which are known to influence the
accuracy of such measures. We propose nonparametric methods for covariate
adjustment of the AUC. Models with normal errors and non-normal errors are
discussed and analysed separately. Nonparametric regression is used for
estimating mean and variance functions in both scenarios. In the general noise
case we propose a covariate-adjusted Mann-Whitney estimator for AUC estimation
which effectively uses available data to construct working samples at any
covariate value of interest and is computationally efficient for
implementation. This provides a generalisation of the Mann-Whitney approach for
comparing two populations by taking covariate effects into account. We derive
asymptotic properties for the AUC estimators in both settings, including
asymptotic normality, optimal strong uniform convergence rates and MSE
consistency. The usefulness of the proposed methods is demonstrated through
simulated and real data examples
Differentially Private Model Selection with Penalized and Constrained Likelihood
In statistical disclosure control, the goal of data analysis is twofold: The
released information must provide accurate and useful statistics about the
underlying population of interest, while minimizing the potential for an
individual record to be identified. In recent years, the notion of differential
privacy has received much attention in theoretical computer science, machine
learning, and statistics. It provides a rigorous and strong notion of
protection for individuals' sensitive information. A fundamental question is
how to incorporate differential privacy into traditional statistical inference
procedures. In this paper we study model selection in multivariate linear
regression under the constraint of differential privacy. We show that model
selection procedures based on penalized least squares or likelihood can be made
differentially private by a combination of regularization and randomization,
and propose two algorithms to do so. We show that our private procedures are
consistent under essentially the same conditions as the corresponding
non-private procedures. We also find that under differential privacy, the
procedure becomes more sensitive to the tuning parameters. We illustrate and
evaluate our method using simulation studies and two real data examples
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