7,550 research outputs found
Bayesian Nonparametric Hidden Semi-Markov Models
There is much interest in the Hierarchical Dirichlet Process Hidden Markov
Model (HDP-HMM) as a natural Bayesian nonparametric extension of the ubiquitous
Hidden Markov Model for learning from sequential and time-series data. However,
in many settings the HDP-HMM's strict Markovian constraints are undesirable,
particularly if we wish to learn or encode non-geometric state durations. We
can extend the HDP-HMM to capture such structure by drawing upon
explicit-duration semi-Markovianity, which has been developed mainly in the
parametric frequentist setting, to allow construction of highly interpretable
models that admit natural prior information on state durations.
In this paper we introduce the explicit-duration Hierarchical Dirichlet
Process Hidden semi-Markov Model (HDP-HSMM) and develop sampling algorithms for
efficient posterior inference. The methods we introduce also provide new
methods for sampling inference in the finite Bayesian HSMM. Our modular Gibbs
sampling methods can be embedded in samplers for larger hierarchical Bayesian
models, adding semi-Markov chain modeling as another tool in the Bayesian
inference toolbox. We demonstrate the utility of the HDP-HSMM and our inference
methods on both synthetic and real experiments
Stochastic Block Transition Models for Dynamic Networks
There has been great interest in recent years on statistical models for
dynamic networks. In this paper, I propose a stochastic block transition model
(SBTM) for dynamic networks that is inspired by the well-known stochastic block
model (SBM) for static networks and previous dynamic extensions of the SBM.
Unlike most existing dynamic network models, it does not make a hidden Markov
assumption on the edge-level dynamics, allowing the presence or absence of
edges to directly influence future edge probabilities while retaining the
interpretability of the SBM. I derive an approximate inference procedure for
the SBTM and demonstrate that it is significantly better at reproducing
durations of edges in real social network data.Comment: To appear in proceedings of AISTATS 201
Beyond the Spectral Theorem: Spectrally Decomposing Arbitrary Functions of Nondiagonalizable Operators
Nonlinearities in finite dimensions can be linearized by projecting them into
infinite dimensions. Unfortunately, often the linear operator techniques that
one would then use simply fail since the operators cannot be diagonalized. This
curse is well known. It also occurs for finite-dimensional linear operators. We
circumvent it by developing a meromorphic functional calculus that can
decompose arbitrary functions of nondiagonalizable linear operators in terms of
their eigenvalues and projection operators. It extends the spectral theorem of
normal operators to a much wider class, including circumstances in which poles
and zeros of the function coincide with the operator spectrum. By allowing the
direct manipulation of individual eigenspaces of nonnormal and
nondiagonalizable operators, the new theory avoids spurious divergences. As
such, it yields novel insights and closed-form expressions across several areas
of physics in which nondiagonalizable dynamics are relevant, including
memoryful stochastic processes, open non unitary quantum systems, and
far-from-equilibrium thermodynamics.
The technical contributions include the first full treatment of arbitrary
powers of an operator. In particular, we show that the Drazin inverse,
previously only defined axiomatically, can be derived as the negative-one power
of singular operators within the meromorphic functional calculus and we give a
general method to construct it. We provide new formulae for constructing
projection operators and delineate the relations between projection operators,
eigenvectors, and generalized eigenvectors.
By way of illustrating its application, we explore several, rather distinct
examples.Comment: 29 pages, 4 figures, expanded historical citations;
http://csc.ucdavis.edu/~cmg/compmech/pubs/bst.ht
Explicit computations for some Markov modulated counting processes
In this paper we present elementary computations for some Markov modulated
counting processes, also called counting processes with regime switching.
Regime switching has become an increasingly popular concept in many branches of
science. In finance, for instance, one could identify the background process
with the `state of the economy', to which asset prices react, or as an
identification of the varying default rate of an obligor. The key feature of
the counting processes in this paper is that their intensity processes are
functions of a finite state Markov chain. This kind of processes can be used to
model default events of some companies.
Many quantities of interest in this paper, like conditional characteristic
functions, can all be derived from conditional probabilities, which can, in
principle, be analytically computed. We will also study limit results for
models with rapid switching, which occur when inflating the intensity matrix of
the Markov chain by a factor tending to infinity. The paper is largely
expository in nature, with a didactic flavor
Criticality in Formal Languages and Statistical Physics
We show that the mutual information between two symbols, as a function of the
number of symbols between the two, decays exponentially in any probabilistic
regular grammar, but can decay like a power law for a context-free grammar.
This result about formal languages is closely related to a well-known result in
classical statistical mechanics that there are no phase transitions in
dimensions fewer than two. It is also related to the emergence of power-law
correlations in turbulence and cosmological inflation through recursive
generative processes. We elucidate these physics connections and comment on
potential applications of our results to machine learning tasks like training
artificial recurrent neural networks. Along the way, we introduce a useful
quantity which we dub the rational mutual information and discuss
generalizations of our claims involving more complicated Bayesian networks.Comment: Replaced to match final published version. Discussion improved,
references adde
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