35,034 research outputs found
Bayesian Approximate Kernel Regression with Variable Selection
Nonlinear kernel regression models are often used in statistics and machine
learning because they are more accurate than linear models. Variable selection
for kernel regression models is a challenge partly because, unlike the linear
regression setting, there is no clear concept of an effect size for regression
coefficients. In this paper, we propose a novel framework that provides an
effect size analog of each explanatory variable for Bayesian kernel regression
models when the kernel is shift-invariant --- for example, the Gaussian kernel.
We use function analytic properties of shift-invariant reproducing kernel
Hilbert spaces (RKHS) to define a linear vector space that: (i) captures
nonlinear structure, and (ii) can be projected onto the original explanatory
variables. The projection onto the original explanatory variables serves as an
analog of effect sizes. The specific function analytic property we use is that
shift-invariant kernel functions can be approximated via random Fourier bases.
Based on the random Fourier expansion we propose a computationally efficient
class of Bayesian approximate kernel regression (BAKR) models for both
nonlinear regression and binary classification for which one can compute an
analog of effect sizes. We illustrate the utility of BAKR by examining two
important problems in statistical genetics: genomic selection (i.e. phenotypic
prediction) and association mapping (i.e. inference of significant variants or
loci). State-of-the-art methods for genomic selection and association mapping
are based on kernel regression and linear models, respectively. BAKR is the
first method that is competitive in both settings.Comment: 22 pages, 3 figures, 3 tables; theory added; new simulations
presented; references adde
Discrete versus continuous domain models for disease mapping
The main goal of disease mapping is to estimate disease risk and identify
high-risk areas. Such analyses are hampered by the limited geographical
resolution of the available data. Typically the available data are counts per
spatial unit and the common approach is the Besag--York--Molli{\'e} (BYM)
model. When precise geocodes are available, it is more natural to use
Log-Gaussian Cox processes (LGCPs). In a simulation study mimicking childhood
leukaemia incidence using actual residential locations of all children in the
canton of Z\"urich, Switzerland, we compare the ability of these models to
recover risk surfaces and identify high-risk areas. We then apply both
approaches to actual data on childhood leukaemia incidence in the canton of
Z\"urich during 1985-2015. We found that LGCPs outperform BYM models in almost
all scenarios considered. Our findings suggest that there are important gains
to be made from the use of LGCPs in spatial epidemiology.Comment: 28 pages, 4 figures, 2 Table
Active inference, evidence accumulation, and the urn task
Deciding how much evidence to accumulate before making a decision is a problem we and other animals often face, but one that is not completely understood. This issue is particularly important because a tendency to sample less information (often known as reflection impulsivity) is a feature in several psychopathologies, such as psychosis. A formal understanding of information sampling may therefore clarify the computational anatomy of psychopathology. In this theoretical letter, we consider evidence accumulation in terms of active (Bayesian) inference using a generic model of Markov decision processes. Here, agents are equipped with beliefs about their own behavior--in this case, that they will make informed decisions. Normative decision making is then modeled using variational Bayes to minimize surprise about choice outcomes. Under this scheme, different facets of belief updating map naturally onto the functional anatomy of the brain (at least at a heuristic level). Of particular interest is the key role played by the expected precision of beliefs about control, which we have previously suggested may be encoded by dopaminergic neurons in the midbrain. We show that manipulating expected precision strongly affects how much information an agent characteristically samples, and thus provides a possible link between impulsivity and dopaminergic dysfunction. Our study therefore represents a step toward understanding evidence accumulation in terms of neurobiologically plausible Bayesian inference and may cast light on why this process is disordered in psychopathology
Approximating Cross-validatory Predictive P-values with Integrated IS for Disease Mapping Models
An important statistical task in disease mapping problems is to identify out-
lier/divergent regions with unusually high or low residual risk of disease.
Leave-one-out cross-validatory (LOOCV) model assessment is a gold standard for
computing predictive p-value that can flag such outliers. However, actual LOOCV
is time-consuming because one needs to re-simulate a Markov chain for each
posterior distribution in which an observation is held out as a test case. This
paper introduces a new method, called iIS, for approximating LOOCV with only
Markov chain samples simulated from a posterior based on a full data set. iIS
is based on importance sampling (IS). iIS integrates the p-value and the
likelihood of the test observation with respect to the distribution of the
latent variable without reference to the actual observation. The predictive
p-values computed with iIS can be proved to be equivalent to the LOOCV
predictive p-values, following the general theory for IS. We com- pare iIS and
other three existing methods in the literature with a lip cancer dataset
collected in Scotland. Our empirical results show that iIS provides predictive
p-values that are al- most identical to the actual LOOCV predictive p-values
and outperforms the existing three methods, including the recently proposed
ghosting method by Marshall and Spiegelhalter (2007).Comment: 21 page
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