239,140 research outputs found
Explainable subgraphs with surprising densities : a subgroup discovery approach
The connectivity structure of graphs is typically related to the attributes of the nodes. In social networks for example, the probability of a friendship between any pair of people depends on a range of attributes, such as their age, residence location, workplace, and hobbies. The high-level structure of a graph can thus possibly be described well by means of patterns of the form `the subgroup of all individuals with a certain properties X are often (or rarely) friends with individuals in another subgroup defined by properties Y', in comparison to what is expected. Such rules present potentially actionable and generalizable insight into the graph.
We present a method that finds node subgroup pairs between which the edge density is interestingly high or low, using an information-theoretic definition of interestingness. Additionally, the interestingness is quantified subjectively, to contrast with prior information an analyst may have about the connectivity. This view immediatly enables iterative mining of such patterns. This is the first method aimed at graph connectivity relations between different subgroups. Our method generalizes prior work on dense subgraphs induced by a subgroup description. Although this setting has been studied already, we demonstrate for this special case considerable practical advantages of our subjective interestingness measure with respect to a wide range of (objective) interestingness measures
Probabilities from Entanglement, Born's Rule from Envariance
I show how probabilities arise in quantum physics by exploring implications
of {\it environment - assisted invariance} or {\it envariance}, a recently
discovered symmetry exhibited by entangled quantum systems. Envariance of
perfectly entangled ``Bell-like'' states can be used to rigorously justify
complete ignorance of the observer about the outcome of any measurement on
either of the members of the entangled pair. For more general states,
envariance leads to Born's rule, for the outcomes
associated with Schmidt states. Probabilities derived in this manner are an
objective reflection of the underlying state of the system -- they represent
experimentally verifiable symmetries, and not just a subjective ``state of
knowledge'' of the observer. Envariance - based approach is compared with and
found superior to pre-quantum definitions of probability including the {\it
standard definition} based on the `principle of indifference' due to Laplace,
and the {\it relative frequency approach} advocated by von Mises. Implications
of envariance for the interpretation of quantum theory go beyond the derivation
of Born's rule: Envariance is enough to establish dynamical independence of
preferred branches of the evolving state vector of the composite system, and,
thus, to arrive at the {\it environment - induced superselection (einselection)
of pointer states}, that was usually derived by an appeal to decoherence.
Envariant origin of Born's rule for probabilities sheds a new light on the
relation between ignorance (and hence, information) and the nature of quantum
states.Comment: Figure and an appendix (Born's rule for continuous spectra) added.
Presentation improved. (Comments still welcome...
Unknown Quantum States and Operations, a Bayesian View
The classical de Finetti theorem provides an operational definition of the
concept of an unknown probability in Bayesian probability theory, where
probabilities are taken to be degrees of belief instead of objective states of
nature. In this paper, we motivate and review two results that generalize de
Finetti's theorem to the quantum mechanical setting: Namely a de Finetti
theorem for quantum states and a de Finetti theorem for quantum operations. The
quantum-state theorem, in a closely analogous fashion to the original de
Finetti theorem, deals with exchangeable density-operator assignments and
provides an operational definition of the concept of an "unknown quantum state"
in quantum-state tomography. Similarly, the quantum-operation theorem gives an
operational definition of an "unknown quantum operation" in quantum-process
tomography. These results are especially important for a Bayesian
interpretation of quantum mechanics, where quantum states and (at least some)
quantum operations are taken to be states of belief rather than states of
nature.Comment: 37 pages, 3 figures, to appear in "Quantum Estimation Theory," edited
by M.G.A. Paris and J. Rehacek (Springer-Verlag, Berlin, 2004
Beyond subjective and objective in statistics
We argue that the words "objectivity" and "subjectivity" in statistics
discourse are used in a mostly unhelpful way, and we propose to replace each of
them with broader collections of attributes, with objectivity replaced by
transparency, consensus, impartiality, and correspondence to observable
reality, and subjectivity replaced by awareness of multiple perspectives and
context dependence. The advantage of these reformulations is that the
replacement terms do not oppose each other. Instead of debating over whether a
given statistical method is subjective or objective (or normatively debating
the relative merits of subjectivity and objectivity in statistical practice),
we can recognize desirable attributes such as transparency and acknowledgment
of multiple perspectives as complementary goals. We demonstrate the
implications of our proposal with recent applied examples from pharmacology,
election polling, and socioeconomic stratification.Comment: 35 page
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