9,690 research outputs found
Visualization of Conjugate Distributions in Latent Dirichlet Allocation Model
In Bayesian probability theory, if the posterior distributions p(θx) are in the same family as the prior probability distribution p(θ), the prior and posterior are then called conjugate distributions, and the prior is called a conjugate prior for the likelihood function. In the Latent Dirichlet Allocation model, the likelihood function is Multinomial and the prior function is Dirichlet. There the Dirichlet distribution is a conjugate prior and then the posterior function becomes also Dirichlet. The posterior function is a parameter mixture distribution where the parameter of the likelihood function is distributed according to the given Dirichlet distribution. The compound probability distribution is, however, complicated to understand and have the image. To make many persons understand the image intuitively, the paper visualizes the parameter mixture distribution
A conjugate prior for discrete hierarchical log-linear models
In Bayesian analysis of multi-way contingency tables, the selection of a
prior distribution for either the log-linear parameters or the cell
probabilities parameters is a major challenge. In this paper, we define a
flexible family of conjugate priors for the wide class of discrete hierarchical
log-linear models, which includes the class of graphical models. These priors
are defined as the Diaconis--Ylvisaker conjugate priors on the log-linear
parameters subject to "baseline constraints" under multinomial sampling. We
also derive the induced prior on the cell probabilities and show that the
induced prior is a generalization of the hyper Dirichlet prior. We show that
this prior has several desirable properties and illustrate its usefulness by
identifying the most probable decomposable, graphical and hierarchical
log-linear models for a six-way contingency table.Comment: Published in at http://dx.doi.org/10.1214/08-AOS669 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Bayesian nonparametric modeling of latent partitions via Stirling-gamma priors
Dirichlet process mixtures are particularly sensitive to the value of the
so-called precision parameter, which controls the behavior of the underlying
latent partition. Randomization of the precision through a prior distribution
is a common solution, which leads to more robust inferential procedures.
However, existing prior choices do not allow for transparent elicitation, due
to the lack of analytical results. We introduce and investigate a novel prior
for the Dirichlet process precision, the Stirling-gamma distribution. We study
the distributional properties of the induced random partition, with an emphasis
on the number of clusters. Our theoretical investigation clarifies the reasons
of the improved robustness properties of the proposed prior. Moreover, we show
that, under specific choices of its hyperparameters, the Stirling-gamma
distribution is conjugate to the random partition of a Dirichlet process. We
illustrate with an ecological application the usefulness of our approach for
the detection of communities of ant workers
Fast Predictive Uncertainty for Classification with Bayesian Deep Networks
In Bayesian Deep Learning, distributions over the output of classification
neural networks are approximated by first constructing a Gaussian distribution
over the weights, then sampling from it to receive a distribution over the
categorical output distribution. This is costly. We reconsider old work to
construct a Dirichlet approximation of this output distribution, which yields
an analytic map between Gaussian distributions in logit space and Dirichlet
distributions (the conjugate prior to the categorical) in the output space. We
argue that the resulting Dirichlet distribution has theoretical and practical
advantages, in particular more efficient computation of the uncertainty
estimate, scaling to large datasets and networks like ImageNet and DenseNet. We
demonstrate the use of this Dirichlet approximation by using it to construct a
lightweight uncertainty-aware output ranking for the ImageNet setup
Thompson sampling based Monte-Carlo planning in POMDPs
Monte-Carlo tree search (MCTS) has been drawinggreat interest in recent years for planning under uncertainty. One of the key challenges is the tradeoffbetween exploration and exploitation. To addressthis, we introduce a novel online planning algorithmfor large POMDPs using Thompson sampling basedMCTS that balances between cumulative and simple regrets.The proposed algorithm — Dirichlet-Dirichlet-NormalGamma based Partially Observable Monte-Carlo Planning (D2NG-POMCP) — treats the accumulatedreward of performing an action from a beliefstate in the MCTS search tree as a random variable followingan unknown distribution with hidden parameters.Bayesian method is used to model and infer theposterior distribution of these parameters by choosingthe conjugate prior in the form of a combination of twoDirichlet and one NormalGamma distributions. Thompsonsampling is exploited to guide the action selection inthe search tree. Experimental results confirmed that ouralgorithm outperforms the state-of-the-art approacheson several common benchmark problems
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