109 research outputs found
Fluctuating Currents in Stochastic Thermodynamics I. Gauge Invariance of Asymptotic Statistics
Stochastic Thermodynamics uses Markovian jump processes to model random
transitions between observable mesoscopic states. Physical currents are
obtained from anti-symmetric jump observables defined on the edges of the graph
representing the network of states. The asymptotic statistics of such currents
are characterized by scaled cumulants. In the present work, we use the
algebraic and topological structure of Markovian models to prove a gauge
invariance of the scaled cumulant-generating function. Exploiting this
invariance yields an efficient algorithm for practical calculations of
asymptotic averages and correlation integrals. We discuss how our approach
generalizes the Schnakenberg decomposition of the average entropy-production
rate, and how it unifies previous work. The application of our results to
concrete models is presented in an accompanying publication.Comment: PACS numbers: 05.40.-a, 05.70.Ln, 02.50.Ga, 02.10.Ox. An accompanying
pre-print "Fluctuating Currents in Stochastic Thermodynamics II. Energy
Conversion and Nonequilibrium Response in Kinesin Models" by the same authors
is available as arXiv:1504.0364
Dynamic and stochastic systems as a framework for metaphysics and the philosophy of science
Scientists often think of the world (or some part of it) as a dynamical
system, a stochastic process, or a generalization of such a system. Prominent
examples of systems are (i) the system of planets orbiting the sun or any other
classical mechanical system, (ii) a hydrogen atom or any other
quantum-mechanical system, and (iii) the earth's atmosphere or any other
statistical mechanical system. We introduce a simple and general framework for
describing such systems and show how it can be used to examine some familiar
philosophical questions, including the following: how can we define nomological
possibility, necessity, determinism, and indeterminism; what are symmetries and
laws; what regularities must a system display to make scientific inference
possible; is there any metaphysical basis for invoking principles of parsimony
such as Occam's Razor when we make such inferences; and what is the role of
space and time in a system? Our framework is intended to serve as a toolbox for
the formal analysis of systems that is applicable in several areas of
philosophy.Comment: 57 pages, including three appendices; five figure
Dynamic and stochastic systems as a framework for metaphysics and the philosophy of science
Scientists often think of the world (or some part of it) as a dynamical system, a stochastic process, or a generalization of such a system. Prominent examples of systems are (i) the system of planets orbiting the sun or any other classical mechanical system, (ii) a hydrogen atom or any other quantum-mechanical system, and (iii) the earth’s atmosphere or any other statistical mechanical system. We introduce a general and unified framework for describing such systems and show how it can be used to examine some familiar philosophical questions, including the following: how can we define nomological possibility, necessity, determinism, and indeterminism; what are symmetries and laws; what regularities must a system display to make scientific inference possible; how might principles of parsimony such as Occam’s Razor help when we make such inferences; what is the role of space and time in a system; and might they be emergent features? Our framework is intended to serve as a toolbox for the formal analysis of systems that is applicable in several areas of philosophy
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Algebraic Statistics
Algebraic Statistics is concerned with the interplay of techniques from commutative algebra, combinatorics, (real) algebraic geometry, and related fields with problems arising in statistics and data science. This workshop was the first at Oberwolfach dedicated to this emerging subject area. The participants highlighted recent achievements in this field, explored exciting new applications, and mapped out future directions for research
Mathematical foundations for a compositional account of the Bayesian brain
This dissertation reports some first steps towards a compositional account of active inference and the Bayesian brain.
Specifically, we use the tools of contemporary applied category theory to supply functorial semantics for approximate inference.
To do so, we define on the 'syntactic' side the new notion of Bayesian lens and show that Bayesian updating composes according to the compositional lens pattern.
Using Bayesian lenses, and inspired by compositional game theory, we define fibrations of statistical games and classify various problems of statistical inference as corresponding sections: the chain rule of the relative entropy is formalized as a strict section, while maximum likelihood estimation and the free energy give lax sections.
In the process, we introduce a new notion of 'copy-composition'.
On the 'semantic' side, we present a new formalization of general open dynamical systems (particularly: deterministic, stochastic, and random; and discrete- and continuous-time) as certain coalgebras of polynomial functors, which we show collect into monoidal opindexed categories (or, alternatively, into algebras for multicategories of generalized polynomial functors).
We use these opindexed categories to define monoidal bicategories of 'cilia': dynamical systems which control lenses, and which supply the target for our functorial semantics.
Accordingly, we construct functors which explain the bidirectional compositional structure of predictive coding neural circuits under the free energy principle, thereby giving a formal mathematical underpinning to the bidirectionality observed in the cortex.
Along the way, we explain how to compose rate-coded neural circuits using an algebra for a multicategory of linear circuit diagrams, showing subsequently that this is subsumed by lenses and polynomial functors.
Because category theory is unfamiliar to many computational neuroscientists and cognitive scientists, we have made a particular effort to give clear, detailed, and approachable expositions of all the category-theoretic structures and results of which we make use.
We hope that this dissertation will prove helpful in establishing a new "well-typed'' science of life and mind, and in facilitating interdisciplinary communication
Mathematical Foundations for a Compositional Account of the Bayesian Brain
This dissertation reports some first steps towards a compositional account of
active inference and the Bayesian brain. Specifically, we use the tools of
contemporary applied category theory to supply functorial semantics for
approximate inference. To do so, we define on the `syntactic' side the new
notion of Bayesian lens and show that Bayesian updating composes according to
the compositional lens pattern. Using Bayesian lenses, and inspired by
compositional game theory, we define fibrations of statistical games and
classify various problems of statistical inference as corresponding sections:
the chain rule of the relative entropy is formalized as a strict section, while
maximum likelihood estimation and the free energy give lax sections. In the
process, we introduce a new notion of `copy-composition'.
On the `semantic' side, we present a new formalization of general open
dynamical systems (particularly: deterministic, stochastic, and random; and
discrete- and continuous-time) as certain coalgebras of polynomial functors,
which we show collect into monoidal opindexed categories (or, alternatively,
into algebras for multicategories of generalized polynomial functors). We use
these opindexed categories to define monoidal bicategories of cilia: dynamical
systems which control lenses, and which supply the target for our functorial
semantics. Accordingly, we construct functors which explain the bidirectional
compositional structure of predictive coding neural circuits under the free
energy principle, thereby giving a formal mathematical underpinning to the
bidirectionality observed in the cortex. Along the way, we explain how to
compose rate-coded neural circuits using an algebra for a multicategory of
linear circuit diagrams, showing subsequently that this is subsumed by lenses
and polynomial functors.Comment: DPhil thesis; as submitted. Main change from v1: improved treatment
of statistical games. A number of errors also fixed, and some presentation
improved. Comments most welcom
The Algebra of Open and Interconnected Systems
Herein we develop category-theoretic tools for understanding network-style
diagrammatic languages. The archetypal network-style diagrammatic language is
that of electric circuits; other examples include signal flow graphs, Markov
processes, automata, Petri nets, chemical reaction networks, and so on. The key
feature is that the language is comprised of a number of components with
multiple (input/output) terminals, each possibly labelled with some type, that
may then be connected together along these terminals to form a larger network.
The components form hyperedges between labelled vertices, and so a diagram in
this language forms a hypergraph. We formalise the compositional structure by
introducing the notion of a hypergraph category. Network-style diagrammatic
languages and their semantics thus form hypergraph categories, and semantic
interpretation gives a hypergraph functor.
The first part of this thesis develops the theory of hypergraph categories.
In particular, we introduce the tools of decorated cospans and corelations.
Decorated cospans allow straightforward construction of hypergraph categories
from diagrammatic languages: the inputs, outputs, and their composition are
modelled by the cospans, while the 'decorations' specify the components
themselves. Not all hypergraph categories can be constructed, however, through
decorated cospans. Decorated corelations are a more powerful version that
permits construction of all hypergraph categories and hypergraph functors.
These are often useful for constructing the semantic categories of diagrammatic
languages and functors from diagrams to the semantics. To illustrate these
principles, the second part of this thesis details applications to linear
time-invariant dynamical systems and passive linear networks.Comment: 230 pages. University of Oxford DPhil Thesi
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