18,451 research outputs found
Realms: A Structure for Consolidating Knowledge about Mathematical Theories
Since there are different ways of axiomatizing and developing a mathematical
theory, knowledge about a such a theory may reside in many places and in many
forms within a library of formalized mathematics. We introduce the notion of a
realm as a structure for consolidating knowledge about a mathematical theory. A
realm contains several axiomatizations of a theory that are separately
developed. Views interconnect these developments and establish that the
axiomatizations are equivalent in the sense of being mutually interpretable. A
realm also contains an external interface that is convenient for users of the
library who want to apply the concepts and facts of the theory without delving
into the details of how the concepts and facts were developed. We illustrate
the utility of realms through a series of examples. We also give an outline of
the mechanisms that are needed to create and maintain realms.Comment: As accepted for CICM 201
Unifying Cubical Models of Univalent Type Theory
We present a new constructive model of univalent type theory based on cubical sets. Unlike prior work on cubical models, ours depends neither on diagonal cofibrations nor connections. This is made possible by weakening the notion of fibration from the cartesian cubical set model, so that it is not necessary to assume that the diagonal on the interval is a cofibration. We have formally verified in Agda that these fibrations are closed under the type formers of cubical type theory and that the model satisfies the univalence axiom. By applying the construction in the presence of diagonal cofibrations or connections and reversals, we recover the existing cartesian and De Morgan cubical set models as special cases. Generalizing earlier work of Sattler for cubical sets with connections, we also obtain a Quillen model structure
Learning about a Categorical Latent Variable under Prior Near-Ignorance
It is well known that complete prior ignorance is not compatible with
learning, at least in a coherent theory of (epistemic) uncertainty. What is
less widely known, is that there is a state similar to full ignorance, that
Walley calls near-ignorance, that permits learning to take place. In this paper
we provide new and substantial evidence that also near-ignorance cannot be
really regarded as a way out of the problem of starting statistical inference
in conditions of very weak beliefs. The key to this result is focusing on a
setting characterized by a variable of interest that is latent. We argue that
such a setting is by far the most common case in practice, and we show, for the
case of categorical latent variables (and general manifest variables) that
there is a sufficient condition that, if satisfied, prevents learning to take
place under prior near-ignorance. This condition is shown to be easily
satisfied in the most common statistical problems.Comment: 15 LaTeX page
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Semantic memory redux: an experimental test of hierarchical category representation
Four experiments investigated the classic issue in semantic memory of whether people organize categorical information in hierarchies and use inference to retrieve information from them, as proposed by Collins & Quillian (1969). Past evidence has focused on RT to confirm sentences such as “All birds are animals” or “Canaries breathe.” However, confounding variables such as familiarity and associations between the terms have led to contradictory results. Our experiments avoided such problems by teaching subjects novel materials. Experiment 1 tested an implicit hierarchical structure in the features of a set of studied objects (e.g., all brown objects were large). Experiment 2 taught subjects nested categories of artificial bugs. In Experiment 3, subjects learned a tree structure of novel category hierarchies. In all three, the results differed from the predictions of the hierarchical inference model. In Experiment 4, subjects learned a hierarchy by means of paired associates of novel category names. Here we finally found the RT signature of hierarchical inference. We conclude that it is possible to store information in a hierarchy and retrieve it via inference, but it is difficult and avoided whenever possible. The results are more consistent with feature comparison models than hierarchical models of semantic memory
Belief propagation in monoidal categories
We discuss a categorical version of the celebrated belief propagation
algorithm. This provides a way to prove that some algorithms which are known or
suspected to be analogous, are actually identical when formulated generically.
It also highlights the computational point of view in monoidal categories.Comment: In Proceedings QPL 2014, arXiv:1412.810
Limits of Learning about a Categorical Latent Variable under Prior Near-Ignorance
In this paper, we consider the coherent theory of (epistemic) uncertainty of
Walley, in which beliefs are represented through sets of probability
distributions, and we focus on the problem of modeling prior ignorance about a
categorical random variable. In this setting, it is a known result that a state
of prior ignorance is not compatible with learning. To overcome this problem,
another state of beliefs, called \emph{near-ignorance}, has been proposed.
Near-ignorance resembles ignorance very closely, by satisfying some principles
that can arguably be regarded as necessary in a state of ignorance, and allows
learning to take place. What this paper does, is to provide new and substantial
evidence that also near-ignorance cannot be really regarded as a way out of the
problem of starting statistical inference in conditions of very weak beliefs.
The key to this result is focusing on a setting characterized by a variable of
interest that is \emph{latent}. We argue that such a setting is by far the most
common case in practice, and we provide, for the case of categorical latent
variables (and general \emph{manifest} variables) a condition that, if
satisfied, prevents learning to take place under prior near-ignorance. This
condition is shown to be easily satisfied even in the most common statistical
problems. We regard these results as a strong form of evidence against the
possibility to adopt a condition of prior near-ignorance in real statistical
problems.Comment: 27 LaTeX page
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