56,077 research outputs found
Interval-based uncertain reasoning
This thesis examines three interval based uncertain reasoning approaches: reasoning
under interval constraints, reasoning using necessity and possibility functions, and
reasoning with rough set theory. In all these approaches, intervals are used to characterize
the uncertainty involved in a reasoning process when the available information
is insufficient for single-valued truth evaluation functions. Approaches using interval
constraints can be applied to both interval fuzzy sets and interval probabilities. The
notion of interval triangular norms, or interval t-norms for short, is introduced and
studied in both numeric and non-numeric settings. Algorithms for computing interval
t-norms are proposed. Basic issues on the use of t-norms for approximate reasoning
with interval fuzzy sets are studied. Inference rules for reasoning under interval constraints
are investigated. In the second approach, a pair of necessity and possibility
functions is used to bound the fuzzy truth values of propositions. Inference in this
case is to narrow the gap between the pair of the functions. Inference rules are derived
from the properties of necessity and possibility functions. The theory of rough sets
is used to approximate truth values of propositions and to explore modal structures
in many-valued logic. It offers an uncertain reasoning method complementary to the
other two
The impact of temporal sampling resolution on parameter inference for biological transport models
Imaging data has become widely available to study biological systems at
various scales, for example the motile behaviour of bacteria or the transport
of mRNA, and it has the potential to transform our understanding of key
transport mechanisms. Often these imaging studies require us to compare
biological species or mutants, and to do this we need to quantitatively
characterise their behaviour. Mathematical models offer a quantitative
description of a system that enables us to perform this comparison, but to
relate these mechanistic mathematical models to imaging data, we need to
estimate the parameters of the models. In this work, we study the impact of
collecting data at different temporal resolutions on parameter inference for
biological transport models by performing exact inference for simple velocity
jump process models in a Bayesian framework. This issue is prominent in a host
of studies because the majority of imaging technologies place constraints on
the frequency with which images can be collected, and the discrete nature of
observations can introduce errors into parameter estimates. In this work, we
avoid such errors by formulating the velocity jump process model within a
hidden states framework. This allows us to obtain estimates of the
reorientation rate and noise amplitude for noisy observations of a simple
velocity jump process. We demonstrate the sensitivity of these estimates to
temporal variations in the sampling resolution and extent of measurement noise.
We use our methodology to provide experimental guidelines for researchers
aiming to characterise motile behaviour that can be described by a velocity
jump process. In particular, we consider how experimental constraints resulting
in a trade-off between temporal sampling resolution and observation noise may
affect parameter estimates.Comment: Published in PLOS Computational Biolog
Birth/birth-death processes and their computable transition probabilities with biological applications
Birth-death processes track the size of a univariate population, but many
biological systems involve interaction between populations, necessitating
models for two or more populations simultaneously. A lack of efficient methods
for evaluating finite-time transition probabilities of bivariate processes,
however, has restricted statistical inference in these models. Researchers rely
on computationally expensive methods such as matrix exponentiation or Monte
Carlo approximation, restricting likelihood-based inference to small systems,
or indirect methods such as approximate Bayesian computation. In this paper, we
introduce the birth(death)/birth-death process, a tractable bivariate extension
of the birth-death process. We develop an efficient and robust algorithm to
calculate the transition probabilities of birth(death)/birth-death processes
using a continued fraction representation of their Laplace transforms. Next, we
identify several exemplary models arising in molecular epidemiology,
macro-parasite evolution, and infectious disease modeling that fall within this
class, and demonstrate advantages of our proposed method over existing
approaches to inference in these models. Notably, the ubiquitous stochastic
susceptible-infectious-removed (SIR) model falls within this class, and we
emphasize that computable transition probabilities newly enable direct
inference of parameters in the SIR model. We also propose a very fast method
for approximating the transition probabilities under the SIR model via a novel
branching process simplification, and compare it to the continued fraction
representation method with application to the 17th century plague in Eyam.
Although the two methods produce similar maximum a posteriori estimates, the
branching process approximation fails to capture the correlation structure in
the joint posterior distribution
Coherent frequentism
By representing the range of fair betting odds according to a pair of
confidence set estimators, dual probability measures on parameter space called
frequentist posteriors secure the coherence of subjective inference without any
prior distribution. The closure of the set of expected losses corresponding to
the dual frequentist posteriors constrains decisions without arbitrarily
forcing optimization under all circumstances. This decision theory reduces to
those that maximize expected utility when the pair of frequentist posteriors is
induced by an exact or approximate confidence set estimator or when an
automatic reduction rule is applied to the pair. In such cases, the resulting
frequentist posterior is coherent in the sense that, as a probability
distribution of the parameter of interest, it satisfies the axioms of the
decision-theoretic and logic-theoretic systems typically cited in support of
the Bayesian posterior. Unlike the p-value, the confidence level of an interval
hypothesis derived from such a measure is suitable as an estimator of the
indicator of hypothesis truth since it converges in sample-space probability to
1 if the hypothesis is true or to 0 otherwise under general conditions.Comment: The confidence-measure theory of inference and decision is explicitly
extended to vector parameters of interest. The derivation of upper and lower
confidence levels from valid and nonconservative set estimators is formalize
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