409 research outputs found
On Individual Risk
We survey a variety of possible explications of the term "Individual Risk."
These in turn are based on a variety of interpretations of "Probability,"
including Classical, Enumerative, Frequency, Formal, Metaphysical, Personal,
Propensity, Chance and Logical conceptions of Probability, which we review and
compare. We distinguish between "groupist" and "individualist" understandings
of Probability, and explore both "group to individual" (G2i) and "individual to
group" (i2G) approaches to characterising Individual Risk. Although in the end
that concept remains subtle and elusive, some pragmatic suggestions for
progress are made.Comment: 31 page
Bayesian Model Selection Based on Proper Scoring Rules
Bayesian model selection with improper priors is not well-defined because of
the dependence of the marginal likelihood on the arbitrary scaling constants of
the within-model prior densities. We show how this problem can be evaded by
replacing marginal log-likelihood by a homogeneous proper scoring rule, which
is insensitive to the scaling constants. Suitably applied, this will typically
enable consistent selection of the true model.Comment: Published at http://dx.doi.org/10.1214/15-BA942 in the Bayesian
Analysis (http://projecteuclid.org/euclid.ba) by the International Society of
Bayesian Analysis (http://bayesian.org/
Rejoinder to "Bayesian Model Selection Based on Proper Scoring Rules"
We are deeply appreciative of the initiative of the editor, Marina Vanucci,
in commissioning a discussion of our paper, and extremely grateful to all the
discussants for their insightful and thought-provoking comments. We respond to
the discussions in alphabetical order [arXiv:1409.5291].Comment: Published at http://dx.doi.org/10.1214/15-BA942REJ in the Bayesian
Analysis (http://projecteuclid.org/euclid.ba) by the International Society of
Bayesian Analysis (http://bayesian.org/
A Formal Treatment of Sequential Ignorability
Taking a rigorous formal approach, we consider sequential decision problems
involving observable variables, unobservable variables, and action variables.
We can typically assume the property of extended stability, which allows
identification (by means of G-computation) of the consequence of a specified
treatment strategy if the unobserved variables are, in fact, observed - but not
generally otherwise. However, under certain additional special conditions we
can infer simple stability (or sequential ignorability), which supports
G-computation based on the observed variables alone. One such additional
condition is sequential randomization, where the unobserved variables
essentially behave as random noise in their effects on the actions. Another is
sequential irrelevance, where the unobserved variables do not influence future
observed variables. In the latter case, to deduce sequential ignorability in
full generality requires additional positivity conditions. We show here that
these positivity conditions are not required when all variables are discrete.Comment: 25 pages, 5 figures, 1 tabl
Theory and Applications of Proper Scoring Rules
We give an overview of some uses of proper scoring rules in statistical
inference, including frequentist estimation theory and Bayesian model selection
with improper priors.Comment: 13 page
A Note on Prediction Markets
In a prediction market, individuals can sequentially place bets on the
outcome of a future event. This leaves a trail of personal probabilities for
the event, each being conditional on the current individual's private
background knowledge and on the previously announced probabilities of other
individuals, which give partial information about their private knowledge. By
means of theory and examples, we revisit some results in this area. In
particular, we consider the case of two individuals, who start with the same
overall probability distribution but different private information, and then
take turns in updating their probabilities. We note convergence of the
announced probabilities to a limiting value, which may or may not be the same
as that based on pooling their private information.Comment: 12 page
Identifying the consequences of dynamic treatment strategies: A decision-theoretic overview
We consider the problem of learning about and comparing the consequences of
dynamic treatment strategies on the basis of observational data. We formulate
this within a probabilistic decision-theoretic framework. Our approach is
compared with related work by Robins and others: in particular, we show how
Robins's 'G-computation' algorithm arises naturally from this
decision-theoretic perspective. Careful attention is paid to the mathematical
and substantive conditions required to justify the use of this formula. These
conditions revolve around a property we term stability, which relates the
probabilistic behaviours of observational and interventional regimes. We show
how an assumption of 'sequential randomization' (or 'no unmeasured
confounders'), or an alternative assumption of 'sequential irrelevance', can be
used to infer stability. Probabilistic influence diagrams are used to simplify
manipulations, and their power and limitations are discussed. We compare our
approach with alternative formulations based on causal DAGs or potential
response models. We aim to show that formulating the problem of assessing
dynamic treatment strategies as a problem of decision analysis brings clarity,
simplicity and generality.Comment: 49 pages, 15 figure
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