Posterior Quantiles Research Articles

Monotone Function Intervals: Theory and Applications

A monotone function interval is the set of monotone functions that lie pointwise between two fixed, -monotone functions. We characterize the set of extreme points of monotone function intervals and apply this to a number of economic settings. First, we leverage the main result to characterize the set of distributions of posterior quantiles that can be induced by a signal, with applications to political economy, Bayesian persuasion, and the psychology of judgment. Second, we combine our characterization with properties of convex optimization problems to unify and generalize seminal results in the literature on security design under adverse selection and moral hazard. (JEL C61, C65, D72, D82, D91, G12)

Marginal Bayesian Posterior Inference using Recurrent Neural Networks with Application to Sequential Models.

In Bayesian data analysis, it is often important to evaluate quantiles of the posterior distribution of a parameter of interest (e.g., to form posterior intervals). In multi-dimensional problems, when non-conjugate priors are used, this is often difficult generally requiring either an analytic or sampling-based approximation, such as Markov chain Monte-Carlo (MCMC), Approximate Bayesian computation (ABC) or variational inference. We discuss a general approach that reframes this as a multi-task learning problem and uses recurrent deep neural networks (RNNs) to approximately evaluate posterior quantiles. As RNNs carry information along a sequence, this application is particularly useful in time-series. An advantage of this risk-minimization approach is that we do not need to sample from the posterior or calculate the likelihood. We illustrate the proposed approach in several examples.

Estimating the effects of demographics on interest rates: A robust Bayesian perspective

We highlight a reason for the vast range of estimates for the effect of demographics on interest rates: the magnitudes are not well-identified without often omitted data on capital and life-cycle consumption. Using nonparametric prior sensitivity analysis for an overlapping generations model estimated through Bayesian methods, we show small changes in the prior for the discount rate, intertemporal elasticity of substitution, and depreciation rate can shift posterior quantiles for the effects of demographics by up to 1.5 percentage points. Capital-output ratio data substantially tighten estimates of the depreciation rate but not the discount rate. Life-cycle consumption is especially informative about the intertemporal elasticity of substitution.

Distributions of Posterior Quantiles and Economic Applications

Distributions of Posterior Quantiles and Economic Applications

Calibration of a bumble bee foraging model using Approximate Bayesian Computation

1. Challenging calibration of complex models can be approached by using prior knowledge on the parameters. However, the natural choice of Bayesian inference can be computationally heavy when relying on Markov Chain Monte Carlo (MCMC) sampling. When the likelihood of the data is intractable, alternative Bayesian methods have been proposed. Approximate Bayesian Computation (ABC) only requires sampling from the data generative model, but may be problematic when the dimension of the data is high.2. We studied alternative strategies to handle high dimensional data in ABC applied to the calibration of a spatially explicit foraging model for Bombus terrestris. The first step consisted in building a set of summary statistics carrying enough biological meaning, i.e. as much as the original data, and then applying ABC on this set. Two ABC strategies, the use of regression adjustment leading to the production of ABC posterior samples, and the use of machine learning approaches to approximate ABC posterior quantiles, were compared with respect to coverage of model estimates and true parameter values. The comparison was made on simulated data as well as on data from two field studies.3. Results from simulated data showed that some model parameters were easier to calibrate than others. Approaches based on random forests in general performed better on simulated data. They also performed well on field data, even though the posterior predictive distribution exhibited a higher variance. Nonlinear regression adjustment performed better than linear ones, and the classical ABC rejection algorithm performed badly.4. ABC is an interesting and appealing approach for the calibration of complex models in biology, such as spatially explicit foraging models. However, while ABC methods are easy to implement, they often require considerable tuning.

Global robust Bayesian analysis in large models

This paper develops a tool for global prior sensitivity analysis in large Bayesian models. Without imposing parametric restrictions, the methodology provides bounds for posterior means or quantiles given any prior close to the original in relative entropy and reveals features of the prior that are important for the posterior statistics of interest. We develop a sequential Monte Carlo algorithm and use approximations to the likelihood and statistic of interest to implement the calculations. The methodology finds that the prior tightness hyperparameters in the hierarchical vector autoregression model from Giannone et al. (2015) are relatively insensitive to their hyperpriors. However, in the New Keynesian model of Smets and Wouters (2007), the error bands for the impulse response of output to a monetary policy shock depend heavily on the prior. The upper bound is especially sensitive, and the prior on wage rigidity plays a particularly important role.

Consistent intensity-duration-frequency curves by post-processing of estimated Bayesian posterior quantiles

As a warming climate leads to more frequent heavy rainfall, the importance of accurate rainfall statistics is increasing. Rainfall statistics are often presented as intensity-duration-frequency (IDF) curves showing the rainfall intensity (return level) that can be expected at a location for a duration, and the frequency of this intensity (return period). IDF curves are commonly constructed by fitting generalized extreme value (GEV) distributions to observed annual maximum rainfall for several target durations. As the estimation is performed independently across durations, the resulting IDF curves may be inconsistent across durations and return periods. This paper proposes to ensure consistency by post-processing the estimated IDF curves. Two post-processing approaches are considered, a quantile selection algorithm that searches for consistent return levels within the posterior quantiles of a Bayesian inference approach, and adjustments based on isotonic regression. The methods are evaluated for simulated data and for Norwegian rainfall data from 83 locations, for hourly and sub-hourly durations. The post-processing yields consistent estimates that are at least as accurate as the unadjusted, inconsistent estimates. We also demonstrate how our approach differs from d-GEV, a method that performs simultaneous estimation across durations. An R implementation for the post-processing methods is available athttps://github.com/ClimDesign/fixIDF.

Posterior moments and quantiles for the normal location model with Laplace prior

We derive explicit expressions for arbitrary moments and quantiles of the posterior distribution of the location parameter η in the normal location model with Laplace prior, and use the results to approximate the posterior distribution of sums of independent copies of η.

Global Robust Bayesian Analysis in Large Models

This paper develops a tool for global prior sensitivity analysis in large Bayesian models. Without imposing parametric restrictions, the methodology provides bounds for posterior means or quantiles given any prior close to the original in relative entropy, and reveals features of the prior that are important for the posterior statistics of interest. We develop a sequential Monte Carlo algorithm and use approximations to the likelihood and statistic of interest to implement the calculations. Applying the methodology to study the error bands for the impulse response of output to a monetary policy shock in the New Keynesian model of Smets and Wouters (2007), we find a high degree of prior sensitivity. The error bands depend asymmetrically on the prior through features of the likelihood such as correlations or modes in the tail of the posterior, which are hard to account for without the systematic and global analysis here.

A default prior for regression coefficients.

When the sample size is not too small, M-estimators of regression coefficients are approximately normal and unbiased. This leads to the familiar frequentist inference in terms of normality-based confidence intervals and p-values. From a Bayesian perspective, use of the (improper) uniform prior yields matching results in the sense that posterior quantiles agree with one-sided confidence bounds. For this, and various other reasons, the uniform prior is often considered objective or non-informative. In spite of this, we argue that the uniform prior is not suitable as a default prior for inference about a regression coefficient in the context of the bio-medical and social sciences. We propose that a more suitable default choice is the normal distribution with mean zero and standard deviation equal to the standard error of the M-estimator. We base this recommendation on two arguments. First, we show that this prior is non-informative for inference about the sign of the regression coefficient. Second, we show that this prior agrees well with a meta-analysis of 50 articles from the MEDLINE database.

Differentially private posterior summaries for linear regression coefficients

In Bayesian regression modeling, often analysts summarize inferences using posterior probabilities and quantiles, such as the posterior probability that a coefficient exceeds zero or the posterior median of that coefficient. However, with potentially unbounded outcomes and explanatory variables, regression inferences based on typical prior distributions can be sensitive to values of individual data points. Thus, releasing posterior summaries of regression coefficients can result in disclosure risks. In this article, we propose some differentially private algorithms for reporting posterior probabilities and posterior quantiles of linear regression coefficients. The algorithms use the general strategy of subsample and aggregate, a technique that requires randomly partitioning the data into disjoint subsets, estimating the regression within each subset, and combining results in ways that satisfy differential privacy. We illustrate the performance of some of the algorithms using repeated sampling studies. The non-private versions also can be used for Bayesian inference with big data in non-private settings.

Second-order matching prior family parametrized by sample size and matching probability

We propose a family of priors that satisfies the second-order probability matching property. The posterior quantile of a probability matching prior is exactly or approximately equal to the frequentist one. Most models lack an exact matching prior. If all quantiles of a prior’s posterior converge to the frequentist ones up to $$o(n^{-1/2})$$ or $$o(n^{-1})$$ as the sample size n increases, the prior is called a first-order probability matching prior and a second-order probability matching prior, respectively. Although a second-order matching prior does not necessarily exist, a first-order matching prior always exists. We introduce a class of priors that depend on the sample size and matching probability. We derive the condition under which the family satisfies the second-order probability matching property even when a second-order probability matching prior does not exist. The superiority of the proposed priors is illustrated in several numerical experiments.

On Monte Carlo computation of posterior expectations with uncertainty

ABSTRACTIn this paper, we study computation of the range of posterior expectations that arise from robust Bayesian statistics. We compute supremum and infimum of the posterior expectations, when allowing uncertainty for the choice of the likelihood function, or uncertainty for the choice of the prior distribution. In the standard approach of sensitivity analysis, posterior statistics is computed a multiple number of times for each choice of the uncertainty scenarios, which might involve heavy computation due to running Monte Carlo sampling many times. Our paper proposes a more efficient computational method that only requires one Monte Carlo sample for all possible choices of the uncertainty scenarios. The proposed computational method involves three steps (with the mnemonic PSI):(Prior step.) Introduce an auxiliary hyperprior distribution on a parameter λ that indexes the uncertainty.(Sampling step.) For any parameter of interest h, we derive a sample of from the joint posterior distribution given the observed data, using a Monte Carlo method. 1(Inference step.) Based on this posterior sample of , we estimate the range of posterior expectations (and similarly the range for any posterior quantile).

Exact Bayesian Inference Comparing Binomial Proportions, With Application to Proof-of-Concept Clinical Trials

The authors revisit the problem of exact Bayesian inference comparing two independent binomial proportions. Numerical integration in R is used to compute exact posterior distribution functions, probability densities, and quantiles of the risk difference, relative risk, and odds ratio. An application of the methodology is given in the context of randomized comparative proof-of-concept clinical trials that are driven by evaluation of quantitative criteria combining statistical significance and clinical relevance. A two-stage adaptive design based on predictive probability of success is proposed and its operating characteristics are studied via Monte Carlo simulation. The authors conclude that exact Bayesian methods provide an elegant and efficient way to facilitate design and analysis of proof-of-concept studies.

A Computational Implementation of GMM

In this paper we study a statistical method of implementing quasi-Bayes estimators for nonlinear and nonseparable GMM models, that is motivated by the ideas proposed in Chernozhukov and Hong (2003) and Creel and Kristensen (2011) and that combines simulation with nonparametric regression in the computation of GMM models. We provide formal conditions under which frequentist inference is asymptotically valid and demonstrate the validity of the use of posterior quantiles. We also show that in this setting, local linear regression methods have theoretical advantages over kernel methods that are also reected in nite sample simulation results. Our results also apply to both exactly and over identied models. These estimators do not need to rely on numerical optimization or Markov Chain Monte Carlo simulations. They provide an eective compliment to the classical M-estimators and to MCMC methods, and can be applied to both likelihood based models and method of moment based models.

Combining Outputs from the North American Regional Climate Change Assessment Program by Using A Bayesian Hierarchical Model

SummaryWe investigate the 20-year-average boreal winter temperatures generated by an ensemble of six regional climate models (RCMs) in phase I of the North American Regional Climate Change Assessment Program. We use the long-run average (20-year integration) to smooth out variability and to capture the climate properties from the RCM outputs. We find that, although the RCMs capture the large-scale climate variation from coast to coast and from south to north similarly, their outputs can differ substantially in some regions. We propose a Bayesian hierarchical model to synthesize information from the ensemble of RCMs, and we construct a consensus climate signal with each RCM contributing to the consensus according to its own variability parameter. The Bayesian methodology enables us to make posterior inference on all the unknowns, including the large-scale fixed effects and the small-scale random effects in the consensus climate signal and in each RCM. The joint distributions of the consensus climate and the outputs from the RCMs are also investigated through posterior means, posterior variances and posterior spatial quantiles. We use a spatial random-effects model in the Bayesian hierarchical model and, consequently, we can deal with the large data sets of fine resolution outputs from all the RCMs. Additionally, our model allows a flexible spatial covariance structure without assuming stationarity or isotropy.

Non-informative priors in the generalized gamma stress–strength systems

This article deals with non-informative priors for parameters when both stress and strength follow generalized gamma distributions. First, the orthogonal reparameterization is treated and then, using this reparameterization, Jeffreys’ prior, group ordering reference priors, and matching priors are derived. The propriety of posterior distributions is investigated and marginal posterior distributions are provided under those non-informative priors. The question of whether or not the reference priors satisfy the probability matching criterion is addressed. Finally, the reference prior that satisfies the probability matching criterion is shown to be good in the sense of frequentist coverage probability of the posterior quantile. Accepted in 2005 for a special issue on Reliability co-edited by Hoang Pham, Rutgers University; Dong Ho Park, Hallym University, Korea; and Richard Cassady, University of Arkansas.

On the approximate frequentist validity of the posterior quantiles of a parametric function: results based on empirical and related likelihoods

With reference to a wide class of empirical and related likelihoods, we study priors which ensure approximate frequentist validity of the posterior quantiles of a general parametric function. It is seen that no data-free prior entails such frequentist validity but, at least for the usual empirical likelihood, a data-dependent prior serves the purpose. Accounting for the nonlinearity of the parametric function of interest requires special attention in the derivation. A simulation study is seen to provide support, in finite samples, to our asymptotic results.

Neutral noninformative and informative conjugate beta and gamma prior distributions

The conjugate binomial and Poisson models are commonly used for estimating proportions or rates. However, it is not well known that the conventional noninformative conjugate priors tend to shrink the posterior quantiles toward the boundary or toward the middle of the parameter space, making them thus appear excessively informative. The shrinkage is always largest when the number of observed events is small. This behavior persists for all sample sizes and exposures. The effect of the prior is therefore most conspicuous and potentially controversial when analyzing rare events. As alternative default conjugate priors, I introduce Beta(1/3, 1/3) and Gamma(1/3, 0), which I call ‘neutral’ priors because they lead to posterior distributions with approximately 50 per cent probability that the true value is either smaller or larger than the maximum likelihood estimate. This holds for all sample sizes and exposures as long as the point estimate is not at the boundary of the parameter space. I also discuss the construction of informative prior distributions. Under the suggested formulation, the posterior median coincides approximately with the weighted average of the prior median and the sample mean, yielding priors that perform more intuitively than those obtained by matching moments and quantiles.

Data-dependent probability matching priors of the second order

We consider second-order probability matching priors that ensure frequentist validity of posterior quantiles with margin of error o(n −1), where n is the sample size. It is well known that there are many models of interest where data-free second-order probability matching priors do not exist. We explore how this problem can be resolved via consideration of data-dependent priors. This is done both in the absence and presence of nuisance parameters.