2,042 research outputs found
A Recursive Algorithm for Computing Inferences in Imprecise Markov Chains
We present an algorithm that can efficiently compute a broad class of
inferences for discrete-time imprecise Markov chains, a generalised type of
Markov chains that allows one to take into account partially specified
probabilities and other types of model uncertainty. The class of inferences
that we consider contains, as special cases, tight lower and upper bounds on
expected hitting times, on hitting probabilities and on expectations of
functions that are a sum or product of simpler ones. Our algorithm exploits the
specific structure that is inherent in all these inferences: they admit a
general recursive decomposition. This allows us to achieve a computational
complexity that scales linearly in the number of time points on which the
inference depends, instead of the exponential scaling that is typical for a
naive approach
Exchangeability and sets of desirable gambles
Sets of desirable gambles constitute a quite general type of uncertainty
model with an interesting geometrical interpretation. We give a general
discussion of such models and their rationality criteria. We study
exchangeability assessments for them, and prove counterparts of de Finetti's
finite and infinite representation theorems. We show that the finite
representation in terms of count vectors has a very nice geometrical
interpretation, and that the representation in terms of frequency vectors is
tied up with multivariate Bernstein (basis) polynomials. We also lay bare the
relationships between the representations of updated exchangeable models, and
discuss conservative inference (natural extension) under exchangeability and
the extension of exchangeable sequences.Comment: 40 page
State sequence prediction in imprecise hidden Markov models
We present an efficient exact algorithm for estimating state sequences from outputs (or observations) in imprecise hidden Markov models (iHMM), where both the uncertainty linking one state to the next, and that linking a state to its output, are represented using coherent lower previsions. The notion of independence we associate with the credal network representing the iHMM is that of epistemic irrelevance. We consider as best estimates for state sequences the (Walley-Sen) maximal sequences for the posterior joint state model (conditioned on the observed output sequence), associated with a gain function that is the indicator of the state sequence. This corresponds to (and generalises) finding the state sequence with the highest posterior probability in HMMs with precise transition and output probabilities (pHMMs). We argue that the computational complexity is at worst quadratic in the length of the Markov chain, cubic in the number of states, and essentially linear in the number of maximal state sequences. For binary iHMMs, we investigate experimentally how the number of maximal state sequences depends on the model parameters
Interpreting, axiomatising and representing coherent choice functions in terms of desirability
Choice functions constitute a simple, direct and very general mathematical
framework for modelling choice under uncertainty. In particular, they are able
to represent the set-valued choices that appear in imprecise-probabilistic
decision making. We provide these choice functions with a clear interpretation
in terms of desirability, use this interpretation to derive a set of basic
coherence axioms, and show that this notion of coherence leads to a
representation in terms of sets of strict preference orders. By imposing
additional properties such as totality, the mixing property and Archimedeanity,
we obtain representation in terms of sets of strict total orders, lexicographic
probability systems, coherent lower previsions or linear previsions.Comment: arXiv admin note: text overlap with arXiv:1806.0104
Computable Randomness is Inherently Imprecise
We use the martingale-theoretic approach of game-theoretic probability to
incorporate imprecision into the study of randomness. In particular, we define
a notion of computable randomness associated with interval, rather than
precise, forecasting systems, and study its properties. The richer mathematical
structure that thus arises lets us better understand and place existing results
for the precise limit. When we focus on constant interval forecasts, we find
that every infinite sequence of zeroes and ones has an associated filter of
intervals with respect to which it is computably random. It may happen that
none of these intervals is precise, which justifies the title of this paper. We
illustrate this by showing that computable randomness associated with
non-stationary precise forecasting systems can be captured by a stationary
interval forecast, which must then be less precise: a gain in model simplicity
is thus paid for by a loss in precision.Comment: 29 pages, 12 of which constitute the main text, and 17 of which
constitute an appendix with proofs and additional material. 3 figures.
Conference paper (ISIPTA 2017
Accept & Reject Statement-Based Uncertainty Models
We develop a framework for modelling and reasoning with uncertainty based on
accept and reject statements about gambles. It generalises the frameworks found
in the literature based on statements of acceptability, desirability, or
favourability and clarifies their relative position. Next to the
statement-based formulation, we also provide a translation in terms of
preference relations, discuss---as a bridge to existing frameworks---a number
of simplified variants, and show the relationship with prevision-based
uncertainty models. We furthermore provide an application to modelling symmetry
judgements.Comment: 35 pages, 17 figure
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