3,057 research outputs found
Examining how teachers use graphs to teach mathematics during a professional development program
There are urgent calls for more studies examining the impact of Professional Development (PD) programs on teachers’ instructional practices. In this study, we analyzed how grades 5-9 mathematics teachers used graphs to teach mathematics at the start and end of a PD program. This topic is relevant because while many studies have investigated students’ difficulties with graphs, there is limited research on how teachers use graphs in their classrooms and no research on how PD impacts the way teachers use graphs in class to teach mathematics. Participant teachers took three graduate level semester-long courses focused on mathematics and student mathematical thinking. The program provided teachers with multiple opportunities for exploration and discussion, systematic feedback, contexts for collaboration and
collegial sharing, and extended follow-up support. We analyzed all lessons where teachers used graphs in class at the start and end of the program, finding that teachers’ use of graphs was qualitatively more sophisticated in the end lessons. Results suggest that the features of the PD program had a positive effect on teachers’ classroom practices regarding the use of graphs
Half-trek criterion for generic identifiability of linear structural equation models
A linear structural equation model relates random variables of interest and
corresponding Gaussian noise terms via a linear equation system. Each such
model can be represented by a mixed graph in which directed edges encode the
linear equations and bidirected edges indicate possible correlations among
noise terms. We study parameter identifiability in these models, that is, we
ask for conditions that ensure that the edge coefficients and correlations
appearing in a linear structural equation model can be uniquely recovered from
the covariance matrix of the associated distribution. We treat the case of
generic identifiability, where unique recovery is possible for almost every
choice of parameters. We give a new graphical condition that is sufficient for
generic identifiability and can be verified in time that is polynomial in the
size of the graph. It improves criteria from prior work and does not require
the directed part of the graph to be acyclic. We also develop a related
necessary condition and examine the "gap" between sufficient and necessary
conditions through simulations on graphs with 25 or 50 nodes, as well as
exhaustive algebraic computations for graphs with up to five nodes.Comment: Published in at http://dx.doi.org/10.1214/12-AOS1012 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Graphical modeling of stochastic processes driven by correlated errors
We study a class of graphs that represent local independence structures in
stochastic processes allowing for correlated error processes. Several graphs
may encode the same local independencies and we characterize such equivalence
classes of graphs. In the worst case, the number of conditions in our
characterizations grows superpolynomially as a function of the size of the node
set in the graph. We show that deciding Markov equivalence is coNP-complete
which suggests that our characterizations cannot be improved upon
substantially. We prove a global Markov property in the case of a multivariate
Ornstein-Uhlenbeck process which is driven by correlated Brownian motions.Comment: 43 page
Margins of discrete Bayesian networks
Bayesian network models with latent variables are widely used in statistics
and machine learning. In this paper we provide a complete algebraic
characterization of Bayesian network models with latent variables when the
observed variables are discrete and no assumption is made about the state-space
of the latent variables. We show that it is algebraically equivalent to the
so-called nested Markov model, meaning that the two are the same up to
inequality constraints on the joint probabilities. In particular these two
models have the same dimension. The nested Markov model is therefore the best
possible description of the latent variable model that avoids consideration of
inequalities, which are extremely complicated in general. A consequence of this
is that the constraint finding algorithm of Tian and Pearl (UAI 2002,
pp519-527) is complete for finding equality constraints.
Latent variable models suffer from difficulties of unidentifiable parameters
and non-regular asymptotics; in contrast the nested Markov model is fully
identifiable, represents a curved exponential family of known dimension, and
can easily be fitted using an explicit parameterization.Comment: 41 page
Causal inference via algebraic geometry: feasibility tests for functional causal structures with two binary observed variables
We provide a scheme for inferring causal relations from uncontrolled
statistical data based on tools from computational algebraic geometry, in
particular, the computation of Groebner bases. We focus on causal structures
containing just two observed variables, each of which is binary. We consider
the consequences of imposing different restrictions on the number and
cardinality of latent variables and of assuming different functional
dependences of the observed variables on the latent ones (in particular, the
noise need not be additive). We provide an inductive scheme for classifying
functional causal structures into distinct observational equivalence classes.
For each observational equivalence class, we provide a procedure for deriving
constraints on the joint distribution that are necessary and sufficient
conditions for it to arise from a model in that class. We also demonstrate how
this sort of approach provides a means of determining which causal parameters
are identifiable and how to solve for these. Prospects for expanding the scope
of our scheme, in particular to the problem of quantum causal inference, are
also discussed.Comment: Accepted for publication in Journal of Causal Inference. Revised and
updated in response to referee feedback. 16+5 pages, 26+2 figures. Comments
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Graphs for margins of Bayesian networks
Directed acyclic graph (DAG) models, also called Bayesian networks, impose
conditional independence constraints on a multivariate probability
distribution, and are widely used in probabilistic reasoning, machine learning
and causal inference. If latent variables are included in such a model, then
the set of possible marginal distributions over the remaining (observed)
variables is generally complex, and not represented by any DAG. Larger classes
of mixed graphical models, which use multiple edge types, have been introduced
to overcome this; however, these classes do not represent all the models which
can arise as margins of DAGs. In this paper we show that this is because
ordinary mixed graphs are fundamentally insufficiently rich to capture the
variety of marginal models.
We introduce a new class of hyper-graphs, called mDAGs, and a latent
projection operation to obtain an mDAG from the margin of a DAG. We show that
each distinct marginal of a DAG model is represented by at least one mDAG, and
provide graphical results towards characterizing when two such marginal models
are the same. Finally we show that mDAGs correctly capture the marginal
structure of causally-interpreted DAGs under interventions on the observed
variables
Latent tree models
Latent tree models are graphical models defined on trees, in which only a
subset of variables is observed. They were first discussed by Judea Pearl as
tree-decomposable distributions to generalise star-decomposable distributions
such as the latent class model. Latent tree models, or their submodels, are
widely used in: phylogenetic analysis, network tomography, computer vision,
causal modeling, and data clustering. They also contain other well-known
classes of models like hidden Markov models, Brownian motion tree model, the
Ising model on a tree, and many popular models used in phylogenetics. This
article offers a concise introduction to the theory of latent tree models. We
emphasise the role of tree metrics in the structural description of this model
class, in designing learning algorithms, and in understanding fundamental
limits of what and when can be learned
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