Articles de revues : « Exact and approximate inferences »

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Wu, Lang. « Exact and Approximate Inferences for Nonlinear Mixed-Effects Models With Missing Covariates ». Journal of the American Statistical Association 99, n^o 467 (septembre 2004) : 700–709. http://dx.doi.org/10.1198/016214504000001006.

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Mekhnacha, Kamel, Juan-Manuel Ahuactzin, Pierre Bessière, Emmanuel Mazer et Linda Smail. « Exact and approximate inference in ProBT ». Revue d'intelligence artificielle 21, n^o 3 (12 juin 2007) : 295–332. http://dx.doi.org/10.3166/ria.21.295-332.

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Akagi, Yasunori, Takuya Nishimura, Yusuke Tanaka, Takeshi Kurashima et Hiroyuki Toda. « Exact and Efficient Inference for Collective Flow Diffusion Model via Minimum Convex Cost Flow Algorithm ». Proceedings of the AAAI Conference on Artificial Intelligence 34, n^o 04 (3 avril 2020) : 3163–70. http://dx.doi.org/10.1609/aaai.v34i04.5713.

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Collective Flow Diffusion Model (CFDM) is a general framework to find the hidden movements underlying aggregated population data. The key procedure in CFDM analysis is MAP inference of hidden variables. Unfortunately, existing approaches fail to offer exact MAP inferences, only approximate versions, and take a lot of computation time when applied to large scale problems. In this paper, we propose an exact and efficient method for MAP inference in CFDM. Our key idea is formulating the MAP inference problem as a combinatorial optimization problem called Minimum Convex Cost Flow Problem (C-MCFP) with no approximation or continuous relaxation. On the basis of this formulation, we propose an efficient inference method that employs the C-MCFP algorithm as a subroutine. Our experiments on synthetic and real datasets show that the proposed method is effective both in single MAP inference and people flow estimation with EM algorithm.

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Abe, Takayuki, et Manabu Iwasaki. « EXACT AND APPROXIMATE INFERENCES FOR AN EXPONENTIAL MEAN FROM TYPE I CENSORED DATA ». Bulletin of informatics and cybernetics 37 (décembre 2005) : 31–39. http://dx.doi.org/10.5109/12589.

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YANG, HANN-PYI JAMES, et WEI-KEI SHIUE. « COMPARISON OF FAILURE INTENSITIES FROM TWO POISSON PROCESSES ». International Journal of Reliability, Quality and Safety Engineering 02, n^o 03 (septembre 1995) : 235–43. http://dx.doi.org/10.1142/s0218539395000186.

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Statistical inferences for ratio and difference of intensities from two Poisson processes are reviewed. Both fixed time periods case and fixed number of failures case are considered. Exact results whenever available are described with emphasis on hypothesis testing procedures and confidence intervals construction. Some approximate confidence intervals are studied and simulation results indicate that these intervals are adequate even with small sample size.

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Karami, Md Jamil Hasan. « Assessing Goodness of Approximate Distributions for Inferences about Parameters in Nonlinear Regression Model ». Dhaka University Journal of Science 71, n^o 1 (29 mai 2023) : 13–16. http://dx.doi.org/10.3329/dujs.v71i1.65267.

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It is often crucial to make inferences about parameters of a nonlinear regression model due to a dependency of Fisher information on the parameter being estimated. Here, the distribution of the relevant test statistic is not exact, but approximate. Therefore, similar conclusion, based on the values of different test statistics, may not be reached. This study shows, in this circumstance, how to come up with a nonlinear regression model that can be used for forecasting and other related purposes. The goodness of the approximate distributions, F and χ 2 , has been assessed to reach a correct decision. The simulation results show that the simulated probability of committing a type I error is very close to its true value in case of F distribution corresponding to F statistic. However, the χ 2 distribution does not do a similar job for the LRT statistic since the simulated type I error is quite larger. Dhaka Univ. J. Sci. 71(1): 13-16, 2023 (Jan)

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El-Sagheer, Rashad M., Taghreed M. Jawa et Neveen Sayed-Ahmed. « Inferences for Generalized Pareto Distribution Based on Progressive First-Failure Censoring Scheme ». Complexity 2021 (7 décembre 2021) : 1–11. http://dx.doi.org/10.1155/2021/9325928.

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In this article, we consider estimation of the parameters of a generalized Pareto distribution and some lifetime indices such as those relating to reliability and hazard rate functions when the failure data are progressive first-failure censored. Both classical and Bayesian techniques are obtained. In the Bayesian framework, the point estimations of unknown parameters under both symmetric and asymmetric loss functions are discussed, after having been estimated using the conjugate gamma and discrete priors for the shape and scale parameters, respectively. In addition, both exact and approximate confidence intervals as well as the exact confidence region for the estimators are constructed. A practical example using a simulated data set is analyzed. Finally, the performance of Bayes estimates is compared with that of maximum likelihood estimates through a Monte Carlo simulation study.

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Lintusaari, Jarno, Paul Blomstedt, Tuomas Sivula, Michael U. Gutmann, Samuel Kaski et Jukka Corander. « Resolving outbreak dynamics using approximate Bayesian computation for stochastic birth-death models ». Wellcome Open Research 4 (25 janvier 2019) : 14. http://dx.doi.org/10.12688/wellcomeopenres.15048.1.

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Earlier research has suggested that approximate Bayesian computation (ABC) makes it possible to fit simulator-based intractable birth-death models to investigate communicable disease outbreak dynamics with accuracy comparable to that of exact Bayesian methods. However, recent findings have indicated that key parameters such as the reproductive number R may remain poorly identifiable with these models. Here we show that the identifiability issue can be resolved by taking into account disease-specific characteristics of the transmission process in closer detail. Using tuberculosis (TB) in the San Francisco Bay area as a case-study, we consider a model that generates genotype data from a mixture of three stochastic processes, each with their distinct dynamics and clear epidemiological interpretation. We show that our model allows for accurate posterior inferences about outbreak dynamics from aggregated annual case data with genotype information. As a by-product of the inference, the model provides an estimate of the infectious population size at the time the data was collected. The acquired estimate is approximately two orders of magnitude smaller compared to the assumptions made in the earlier related studies, and much better aligned with epidemiological knowledge about active TB prevalence. Similarly, the reproductive number R related to the primary underlying transmission process is estimated to be nearly three-fold compared with the previous estimates, which has a substantial impact on the interpretation of the fitted outbreak model.

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Lintusaari, Jarno, Paul Blomstedt, Brittany Rose, Tuomas Sivula, Michael U. Gutmann, Samuel Kaski et Jukka Corander. « Resolving outbreak dynamics using approximate Bayesian computation for stochastic birth–death models ». Wellcome Open Research 4 (30 août 2019) : 14. http://dx.doi.org/10.12688/wellcomeopenres.15048.2.

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Earlier research has suggested that approximate Bayesian computation (ABC) makes it possible to fit simulator-based intractable birth–death models to investigate communicable disease outbreak dynamics with accuracy comparable to that of exact Bayesian methods. However, recent findings have indicated that key parameters, such as the reproductive number R, may remain poorly identifiable with these models. Here we show that this identifiability issue can be resolved by taking into account disease-specific characteristics of the transmission process in closer detail. Using tuberculosis (TB) in the San Francisco Bay area as a case study, we consider a model that generates genotype data from a mixture of three stochastic processes, each with its own distinct dynamics and clear epidemiological interpretation. We show that our model allows for accurate posterior inferences about outbreak dynamics from aggregated annual case data with genotype information. As a byproduct of the inference, the model provides an estimate of the infectious population size at the time the data were collected. The acquired estimate is approximately two orders of magnitude smaller than assumed in earlier related studies, and it is much better aligned with epidemiological knowledge about active TB prevalence. Similarly, the reproductive number R related to the primary underlying transmission process is estimated to be nearly three times larger than previous estimates, which has a substantial impact on the interpretation of the fitted outbreak model.

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Shapovalova, Yuliya. « “Exact” and Approximate Methods for Bayesian Inference : Stochastic Volatility Case Study ». Entropy 23, n^o 4 (15 avril 2021) : 466. http://dx.doi.org/10.3390/e23040466.

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We conduct a case study in which we empirically illustrate the performance of different classes of Bayesian inference methods to estimate stochastic volatility models. In particular, we consider how different particle filtering methods affect the variance of the estimated likelihood. We review and compare particle Markov Chain Monte Carlo (MCMC), RMHMC, fixed-form variational Bayes, and integrated nested Laplace approximation to estimate the posterior distribution of the parameters. Additionally, we conduct the review from the point of view of whether these methods are (1) easily adaptable to different model specifications; (2) adaptable to higher dimensions of the model in a straightforward way; (3) feasible in the multivariate case. We show that when using the stochastic volatility model for methods comparison, various data-generating processes have to be considered to make a fair assessment of the methods. Finally, we present a challenging specification of the multivariate stochastic volatility model, which is rarely used to illustrate the methods but constitutes an important practical application.

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Fioretto, Ferdinando, Enrico Pontelli, William Yeoh et Rina Dechter. « Accelerating exact and approximate inference for (distributed) discrete optimization with GPUs ». Constraints 23, n^o 1 (18 août 2017) : 1–43. http://dx.doi.org/10.1007/s10601-017-9274-1.

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Tarvirdizade, Bahman, et Hossein Kazemzadeh Garehchobogh. « Interval Estimation of Stress-Strength Reliability Based on Lower Record Values from Inverse Rayleigh Distribution ». Journal of Quality and Reliability Engineering 2014 (16 novembre 2014) : 1–8. http://dx.doi.org/10.1155/2014/192072.

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We consider the estimation of stress-strength reliability based on lower record values when X and Y are independently but not identically inverse Rayleigh distributed random variables. The maximum likelihood, Bayes, and empirical Bayes estimators of R are obtained and their properties are studied. Confidence intervals, exact and approximate, as well as the Bayesian credible sets for R are obtained. A real example is presented in order to illustrate the inferences discussed in the previous sections. A simulation study is conducted to investigate and compare the performance of the intervals presented in this paper and some bootstrap intervals.

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Guo, Yuanzhen, Hao Xiong et Nicholas Ruozzi. « Marginal Inference in Continuous Markov Random Fields Using Mixtures ». Proceedings of the AAAI Conference on Artificial Intelligence 33 (17 juillet 2019) : 7834–41. http://dx.doi.org/10.1609/aaai.v33i01.33017834.

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Exact marginal inference in continuous graphical models is computationally challenging outside of a few special cases. Existing work on approximate inference has focused on approximately computing the messages as part of the loopy belief propagation algorithm either via sampling methods or moment matching relaxations. In this work, we present an alternative family of approximations that, instead of approximating the messages, approximates the beliefs in the continuous Bethe free energy using mixture distributions. We show that these types of approximations can be combined with numerical quadrature to yield algorithms with both theoretical guarantees on the quality of the approximation and significantly better practical performance in a variety of applications that are challenging for current state-of-the-art methods.

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Kenig, Batya, et Benny Kimelfeld. « Approximate Inference of Outcomes in Probabilistic Elections ». Proceedings of the AAAI Conference on Artificial Intelligence 33 (17 juillet 2019) : 2061–68. http://dx.doi.org/10.1609/aaai.v33i01.33012061.

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We study the complexity of estimating the probability of an outcome in an election over probabilistic votes. The focus is on voting rules expressed as positional scoring rules, and two models of probabilistic voters: the uniform distribution over the completions of a partial voting profile (consisting of a partial ordering of the candidates by each voter), and the Repeated Insertion Model (RIM) over the candidates, including the special case of the Mallows distribution. Past research has established that, while exact inference of the probability of winning is computationally hard (#P-hard), an additive polynomial-time approximation (additive FPRAS) is attained by sampling and averaging. There is often, though, a need for multiplicative approximation guarantees that are crucial for important measures such as conditional probabilities. Unfortunately, a multiplicative approximation of the probability of winning cannot be efficient (under conventional complexity assumptions) since it is already NP-complete to determine whether this probability is nonzero. Contrastingly, we devise multiplicative polynomial-time approximations (multiplicative FPRAS) for the probability of the complement event, namely, losing the election.

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Seridi, Hamid, Herman Akdag, Rachid Mansouri et Mohamed Nemissi. « Approximate Reasoning in Supervised Classification Systems ». Journal of Advanced Computational Intelligence and Intelligent Informatics 10, n^o 4 (20 juillet 2006) : 586–93. http://dx.doi.org/10.20965/jaciii.2006.p0586.

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In knowledge-based systems, uncertainty in propositions can be represented by various degrees of belief encoded by numerical or symbolic values. The use of symbolic values is necessary in areas where the exact numerical values associated with a fact are unknown by experts. In this paper we present an expert system of supervised automatic classification based on a symbolic approach. This last is composed of two sub-systems. The first sub-system automatically generates the production rules using training set; the generated rules are accompanied by a symbolic degree of belief which characterizes their classes of memberships. The second is the inference system, which receives in entry the base of rules and the object to classify. Using classical reasoning (Modus Ponens), the inference system provides the membership class of this object with a certain symbolic degree of belief. Methods to evaluate the degree of belief are numerous but they are often tarnished with uncertainty. To appreciate the performances of our symbolic approach, tests are made on the Iris data basis.

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Tucci, Beatriz, et Fabian Schmidt. « EFTofLSS meets simulation-based inference : σ ₈ from biased tracers ». Journal of Cosmology and Astroparticle Physics 2024, n^o 05 (1 mai 2024) : 063. http://dx.doi.org/10.1088/1475-7516/2024/05/063.

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Abstract Cosmological inferences typically rely on explicit expressions for the likelihood and covariance of the data vector, which normally consists of a set of summary statistics. However, in the case of nonlinear large-scale structure, exact expressions for either likelihood or covariance are unknown, and even approximate expressions can become very cumbersome, depending on the scales and summary statistics considered. Simulation-based inference (SBI), in contrast, does not require an explicit form for the likelihood but only a prior and a simulator, thereby naturally circumventing these issues. In this paper, we explore how this technique can be used to infer σ 8 from a Lagrangian effective field theory (EFT) based forward model for biased tracers. The power spectrum and bispectrum are used as summary statistics to obtain the posterior of the cosmological, bias and noise parameters via neural density estimation. We compare full simulation-based inference with cases where the data vector is drawn from a Gaussian likelihood with sample and analytical covariances. We conclude that, for k max = 0.1hMpc-1 and 0.2hMpc-1, the form of the covariance is more important than the non-Gaussianity of the likelihood, although this conclusion is expected to depend on the cosmological parameter inferred, the summary statistics considered and range of scales probed.

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Domínguez, E., et H. J. Kappen. « Efficient inference in the transverse field Ising model ». Journal of Statistical Mechanics : Theory and Experiment 2023, n^o 3 (1 mars 2023) : 033301. http://dx.doi.org/10.1088/1742-5468/acba02.

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Abstract In this paper we introduce an approximate method to solve the quantum cavity equations for transverse field Ising models. The method relies on a projective approximation of the exact cavity distributions of imaginary time trajectories (paths). A key feature, novel in the context of similar algorithms, is the explicit separation of the classical and quantum parts of the distributions. Numerical simulations show accurate results in comparison with the sampled solution of the cavity equations, the exact diagonalization of the Hamiltonian (when possible) and other approximate inference methods in the literature. The computational complexity of this new algorithm scales linearly with the connectivity of the underlying lattice, enabling the study of highly connected networks, as the ones often encountered in quantum machine learning problems.

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Atkinson, Eric, Charles Yuan, Guillaume Baudart, Louis Mandel et Michael Carbin. « Semi-symbolic inference for efficient streaming probabilistic programming ». Proceedings of the ACM on Programming Languages 6, OOPSLA2 (31 octobre 2022) : 1668–96. http://dx.doi.org/10.1145/3563347.

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A streaming probabilistic program receives a stream of observations and produces a stream of distributions that are conditioned on these observations. Efficient inference is often possible in a streaming context using Rao-Blackwellized particle filters (RBPFs), which exactly solve inference problems when possible and fall back on sampling approximations when necessary. While RBPFs can be implemented by hand to provide efficient inference, the goal of streaming probabilistic programming is to automatically generate such efficient inference implementations given input probabilistic programs. In this work, we propose semi-symbolic inference, a technique for executing probabilistic programs using a runtime inference system that automatically implements Rao-Blackwellized particle filtering. To perform exact and approximate inference together, the semi-symbolic inference system manipulates symbolic distributions to perform exact inference when possible and falls back on approximate sampling when necessary. This approach enables the system to implement the same RBPF a developer would write by hand. To ensure this, we identify closed families of distributions – such as linear-Gaussian and finite discrete models – on which the inference system guarantees exact inference. We have implemented the runtime inference system in the ProbZelus streaming probabilistic programming language. Despite an average 1.6× slowdown compared to the state of the art on existing benchmarks, our evaluation shows that speedups of 3×-87× are obtainable on a new set of challenging benchmarks we have designed to exploit closed families.

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Demidenko, Eugene. « Exact and Approximate Statistical Inference for Nonlinear Regression and the Estimating Equation Approach ». Scandinavian Journal of Statistics 44, n^o 3 (29 mars 2017) : 636–65. http://dx.doi.org/10.1111/sjos.12269.

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Çakmak, Burak, Yue M. Lu et Manfred Opper. « Analysis of random sequential message passing algorithms for approximate inference ». Journal of Statistical Mechanics : Theory and Experiment 2022, n^o 7 (1 juillet 2022) : 073401. http://dx.doi.org/10.1088/1742-5468/ac764a.

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Abstract We analyze the dynamics of a random sequential message passing algorithm for approximate inference with large Gaussian latent variable models in a student–teacher scenario. To model nontrivial dependencies between the latent variables, we assume random covariance matrices drawn from rotation invariant ensembles. Moreover, we consider a model mismatching setting, where the teacher model and the one used by the student may be different. By means of dynamical functional approach, we obtain exact dynamical mean-field equations characterizing the dynamics of the inference algorithm. We also derive a range of model parameters for which the sequential algorithm does not converge. The boundary of this parameter range coincides with the de Almeida Thouless (AT) stability condition of the replica-symmetric ansatz for the static probabilistic model.

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Randone, Francesca, Luca Bortolussi, Emilio Incerto et Mirco Tribastone. « Inference of Probabilistic Programs with Moment-Matching Gaussian Mixtures ». Proceedings of the ACM on Programming Languages 8, POPL (5 janvier 2024) : 1882–912. http://dx.doi.org/10.1145/3632905.

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Computing the posterior distribution of a probabilistic program is a hard task for which no one-fit-for-all solution exists. We propose Gaussian Semantics, which approximates the exact probabilistic semantics of a bounded program by means of Gaussian mixtures. It is parametrized by a map that associates each program location with the moment order to be matched in the approximation. We provide two main contributions. The first is a universal approximation theorem stating that, under mild conditions, Gaussian Semantics can approximate the exact semantics arbitrarily closely. The second is an approximation that matches up to second-order moments analytically in face of the generally difficult problem of matching moments of Gaussian mixtures with arbitrary moment order. We test our second-order Gaussian approximation (SOGA) on a number of case studies from the literature. We show that it can provide accurate estimates in models not supported by other approximation methods or when exact symbolic techniques fail because of complex expressions or non-simplified integrals. On two notable classes of problems, namely collaborative filtering and programs involving mixtures of continuous and discrete distributions, we show that SOGA significantly outperforms alternative techniques in terms of accuracy and computational time.

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De Santis, Fulvio, et Stefania Gubbiotti. « Sample Size Requirements for Calibrated Approximate Credible Intervals for Proportions in Clinical Trials ». International Journal of Environmental Research and Public Health 18, n^o 2 (12 janvier 2021) : 595. http://dx.doi.org/10.3390/ijerph18020595.

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In Bayesian analysis of clinical trials data, credible intervals are widely used for inference on unknown parameters of interest, such as treatment effects or differences in treatments effects. Highest Posterior Density (HPD) sets are often used because they guarantee the shortest length. In most of standard problems, closed-form expressions for exact HPD intervals do not exist, but they are available for intervals based on the normal approximation of the posterior distribution. For small sample sizes, approximate intervals may be not calibrated in terms of posterior probability, but for increasing sample sizes their posterior probability tends to the correct credible level and they become closer and closer to exact sets. The article proposes a predictive analysis to select appropriate sample sizes needed to have approximate intervals calibrated at a pre-specified level. Examples are given for interval estimation of proportions and log-odds.

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CANO, ANDRÉS, MANUEL GÓMEZ-OLMEDO, CORA B. PÉREZ-ARIZA et ANTONIO SALMERÓN. « FAST FACTORISATION OF PROBABILISTIC POTENTIALS AND ITS APPLICATION TO APPROXIMATE INFERENCE IN BAYESIAN NETWORKS ». International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 20, n^o 02 (avril 2012) : 223–43. http://dx.doi.org/10.1142/s0218488512500110.

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We present an efficient procedure for factorising probabilistic potentials represented as probability trees. This new procedure is able to detect some regularities that cannot be captured by existing methods. In cases where an exact decomposition is not achievable, we propose a heuristic way to carry out approximate factorisations guided by a parameter called factorisation degree, which is fast to compute. We show how this parameter can be used to control the tradeoff between complexity and accuracy in approximate inference algorithms for Bayesian networks.

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Schälte, Yannik, et Jan Hasenauer. « Efficient exact inference for dynamical systems with noisy measurements using sequential approximate Bayesian computation ». Bioinformatics 36, Supplement_1 (1 juillet 2020) : i551—i559. http://dx.doi.org/10.1093/bioinformatics/btaa397.

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Abstract Motivation Approximate Bayesian computation (ABC) is an increasingly popular method for likelihood-free parameter inference in systems biology and other fields of research, as it allows analyzing complex stochastic models. However, the introduced approximation error is often not clear. It has been shown that ABC actually gives exact inference under the implicit assumption of a measurement noise model. Noise being common in biological systems, it is intriguing to exploit this insight. But this is difficult in practice, as ABC is in general highly computationally demanding. Thus, the question we want to answer here is how to efficiently account for measurement noise in ABC. Results We illustrate exemplarily how ABC yields erroneous parameter estimates when neglecting measurement noise. Then, we discuss practical ways of correctly including the measurement noise in the analysis. We present an efficient adaptive sequential importance sampling-based algorithm applicable to various model types and noise models. We test and compare it on several models, including ordinary and stochastic differential equations, Markov jump processes and stochastically interacting agents, and noise models including normal, Laplace and Poisson noise. We conclude that the proposed algorithm could improve the accuracy of parameter estimates for a broad spectrum of applications. Availability and implementation The developed algorithms are made publicly available as part of the open-source python toolbox pyABC (https://github.com/icb-dcm/pyabc). Supplementary information Supplementary data are available at Bioinformatics online.

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Demidenko, Eugene, Benjamin B. Williams, Ann Barry Flood et Harold M. Swartz. « Standard error of inverse prediction for dose-response relationship : approximate and exact statistical inference ». Statistics in Medicine 32, n^o 12 (5 novembre 2012) : 2048–61. http://dx.doi.org/10.1002/sim.5668.

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Van den Broek, B., W. Wiegerinck et B. Kappen. « Graphical Model Inference in Optimal Control of Stochastic Multi-Agent Systems ». Journal of Artificial Intelligence Research 32 (16 mai 2008) : 95–122. http://dx.doi.org/10.1613/jair.2473.

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In this article we consider the issue of optimal control in collaborative multi-agent systems with stochastic dynamics. The agents have a joint task in which they have to reach a number of target states. The dynamics of the agents contains additive control and additive noise, and the autonomous part factorizes over the agents. Full observation of the global state is assumed. The goal is to minimize the accumulated joint cost, which consists of integrated instantaneous costs and a joint end cost. The joint end cost expresses the joint task of the agents. The instantaneous costs are quadratic in the control and factorize over the agents. The optimal control is given as a weighted linear combination of single-agent to single-target controls. The single-agent to single-target controls are expressed in terms of diffusion processes. These controls, when not closed form expressions, are formulated in terms of path integrals, which are calculated approximately by Metropolis-Hastings sampling. The weights in the control are interpreted as marginals of a joint distribution over agent to target assignments. The structure of the latter is represented by a graphical model, and the marginals are obtained by graphical model inference. Exact inference of the graphical model will break down in large systems, and so approximate inference methods are needed. We use naive mean field approximation and belief propagation to approximate the optimal control in systems with linear dynamics. We compare the approximate inference methods with the exact solution, and we show that they can accurately compute the optimal control. Finally, we demonstrate the control method in multi-agent systems with nonlinear dynamics consisting of up to 80 agents that have to reach an equal number of target states.

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Daly, Aidan C., Jonathan Cooper, David J. Gavaghan et Chris Holmes. « Comparing two sequential Monte Carlo samplers for exact and approximate Bayesian inference on biological models ». Journal of The Royal Society Interface 14, n^o 134 (septembre 2017) : 20170340. http://dx.doi.org/10.1098/rsif.2017.0340.

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Bayesian methods are advantageous for biological modelling studies due to their ability to quantify and characterize posterior variability in model parameters. When Bayesian methods cannot be applied, due either to non-determinism in the model or limitations on system observability, approximate Bayesian computation (ABC) methods can be used to similar effect, despite producing inflated estimates of the true posterior variance. Owing to generally differing application domains, there are few studies comparing Bayesian and ABC methods, and thus there is little understanding of the properties and magnitude of this uncertainty inflation. To address this problem, we present two popular strategies for ABC sampling that we have adapted to perform exact Bayesian inference, and compare them on several model problems. We find that one sampler was impractical for exact inference due to its sensitivity to a key normalizing constant, and additionally highlight sensitivities of both samplers to various algorithmic parameters and model conditions. We conclude with a study of the O'Hara–Rudy cardiac action potential model to quantify the uncertainty amplification resulting from employing ABC using a set of clinically relevant biomarkers. We hope that this work serves to guide the implementation and comparative assessment of Bayesian and ABC sampling techniques in biological models.

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Cabañas, Rafael, Manuel Gómez-Olmedo et Andrés Cano. « Using Binary Trees for the Evaluation of Influence Diagrams ». International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 24, n^o 01 (février 2016) : 59–89. http://dx.doi.org/10.1142/s0218488516500045.

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This paper proposes the use of binary trees for representing and managing the potentials involved in Influence Diagrams. This kind of tree allows representing context-specific independencies that are finer-grained compared to those encoded using other representations. This enhanced capability can be used to improve the efficiency of the inference algorithms used for Influence Diagrams. Moreover, binary trees allow computing approximate solutions when exact inference is not feasible. In this work we describe how binary trees can be used to perform this approximate evaluation and we compare them with other structures present in the literature.

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NAMPALLY, ARUN, TIMOTHY ZHANG et C. R. RAMAKRISHNAN. « Constraint-Based Inference in Probabilistic Logic Programs ». Theory and Practice of Logic Programming 18, n^o 3-4 (juillet 2018) : 638–55. http://dx.doi.org/10.1017/s1471068418000273.

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AbstractProbabilistic Logic Programs (PLPs) generalize traditional logic programs and allow the encoding of models combining logical structure and uncertainty. In PLP, inference is performed by summarizing the possible worlds which entail the query in a suitable data structure, and using this data structure to compute the answer probability. Systems such as ProbLog, PITA, etc., use propositional data structures like explanation graphs, BDDs, SDDs, etc., to represent the possible worlds. While this approach saves inference time due to substructure sharing, there are a number of problems where a more compact data structure is possible. We propose a data structure called Ordered Symbolic Derivation Diagram (OSDD) which captures the possible worlds by means of constraint formulas. We describe a program transformation technique to construct OSDDs via query evaluation, and give procedures to perform exact and approximate inference over OSDDs. Our approach has two key properties. Firstly, the exact inference procedure is a generalization of traditional inference, and results in speedup over the latter in certain settings. Secondly, the approximate technique is a generalization of likelihood weighting in Bayesian Networks, and allows us to perform sampling-based inference with lower rejection rate and variance. We evaluate the effectiveness of the proposed techniques through experiments on several problems.

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Ensinger, Katharina, Nicholas Tagliapietra, Sebastian Ziesche et Sebastian Trimpe. « Exact Inference for Continuous-Time Gaussian Process Dynamics ». Proceedings of the AAAI Conference on Artificial Intelligence 38, n^o 11 (24 mars 2024) : 11883–91. http://dx.doi.org/10.1609/aaai.v38i11.29074.

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Many physical systems can be described as a continuous-time dynamical system. In practice, the true system is often unknown and has to be learned from measurement data. Since data is typically collected in discrete time, e.g. by sensors, most methods in Gaussian process (GP) dynamics model learning are trained on one-step ahead predictions. While this scheme is mathematically tempting, it can become problematic in several scenarios, e.g. if measurements are provided at irregularly-sampled time steps or physical system properties have to be conserved. Thus, we aim for a GP model of the true continuous-time dynamics. We tackle this task by leveraging higher-order numerical integrators. These integrators provide the necessary tools to discretize dynamical systems with arbitrary accuracy. However, most higher-order integrators require dynamics evaluations at intermediate time steps, making exact GP inference intractable. In previous work, this problem is often addressed by approximate inference techniques. However, exact GP inference is preferable in many scenarios, e.g. due to its mathematical guarantees. In order to enable direct inference, we propose to leverage multistep and Taylor integrators. We demonstrate how exact inference schemes can be derived for these types of integrators. Further, we derive tailored sampling schemes that allow one to draw consistent dynamics functions from the posterior. The learned model can thus be integrated with arbitrary integrators, just like a standard dynamical system. We show empirically and theoretically that our approach yields an accurate representation of the continuous-time system.

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Cano, Andrés, Manuel Gómez, Serafín Moral et Joaquín Abellán. « Hill-climbing and branch-and-bound algorithms for exact and approximate inference in credal networks ». International Journal of Approximate Reasoning 44, n^o 3 (mars 2007) : 261–80. http://dx.doi.org/10.1016/j.ijar.2006.07.020.

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Drovandi, Christopher C., Anthony N. Pettitt et Roy A. McCutchan. « Exact and Approximate Bayesian Inference for Low Integer-Valued Time Series Models with Intractable Likelihoods ». Bayesian Analysis 11, n^o 2 (juin 2016) : 325–52. http://dx.doi.org/10.1214/15-ba950.

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Feldman, A., G. Provan et A. Van Gemund. « Approximate Model-Based Diagnosis Using Greedy Stochastic Search ». Journal of Artificial Intelligence Research 38 (27 juillet 2010) : 371–413. http://dx.doi.org/10.1613/jair.3025.

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We propose a StochAstic Fault diagnosis AlgoRIthm, called SAFARI, which trades off guarantees of computing minimal diagnoses for computational efficiency. We empirically demonstrate, using the 74XXX and ISCAS-85 suites of benchmark combinatorial circuits, that SAFARI achieves several orders-of-magnitude speedup over two well-known deterministic algorithms, CDA* and HA*, for multiple-fault diagnoses; further, SAFARI can compute a range of multiple-fault diagnoses that CDA* and HA* cannot. We also prove that SAFARI is optimal for a range of propositional fault models, such as the widely-used weak-fault models (models with ignorance of abnormal behavior). We discuss the optimality of SAFARI in a class of strong-fault circuit models with stuck-at failure modes. By modeling the algorithm itself as a Markov chain, we provide exact bounds on the minimality of the diagnosis computed. SAFARI also displays strong anytime behavior, and will return a diagnosis after any non-trivial inference time.

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Taghipour, N., D. Fierens, J. Davis et H. Blockeel. « Lifted Variable Elimination : Decoupling the Operators from the Constraint Language ». Journal of Artificial Intelligence Research 47 (8 juillet 2013) : 393–439. http://dx.doi.org/10.1613/jair.3793.

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Lifted probabilistic inference algorithms exploit regularities in the structure of graphical models to perform inference more efficiently. More specifically, they identify groups of interchangeable variables and perform inference once per group, as opposed to once per variable. The groups are defined by means of constraints, so the flexibility of the grouping is determined by the expressivity of the constraint language. Existing approaches for exact lifted inference use specific languages for (in)equality constraints, which often have limited expressivity. In this article, we decouple lifted inference from the constraint language. We define operators for lifted inference in terms of relational algebra operators, so that they operate on the semantic level (the constraints' extension) rather than on the syntactic level, making them language-independent. As a result, lifted inference can be performed using more powerful constraint languages, which provide more opportunities for lifting. We empirically demonstrate that this can improve inference efficiency by orders of magnitude, allowing exact inference where until now only approximate inference was feasible.

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van Lieshout, M. N. M., et E. W. van Zwet. « Exact sampling from conditional Boolean models with applications to maximum likelihood inference ». Advances in Applied Probability 33, n^o 2 (juin 2001) : 339–53. http://dx.doi.org/10.1017/s000186780001082x.

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We are interested in estimating the intensity parameter of a Boolean model of discs (the bombing model) from a single realization. To do so, we derive the conditional distribution of the points (germs) of the underlying Poisson process. We demonstrate how to apply coupling from the past to generate samples from this distribution, and use the samples thus obtained to approximate the maximum likelihood estimator of the intensity. We discuss and compare two methods: one based on a Monte Carlo approximation of the likelihood function, the other a stochastic version of the EM algorithm.

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Alnosaier, Waseem. « Comparisons of the Satterthwaite Approaches for Fixed Effects in Linear Mixed Models ». International Journal of Statistics and Probability 13, n^o 1 (28 février 2024) : 22. http://dx.doi.org/10.5539/ijsp.v13n1p22.

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Four approximate F- tests derived by Fai and Cornelious in 1996 to make inference for fixed effects in mixed linear models of rank greater than one. Two of these approaches derived by introducing a Wald-type statistic distributed approximately as an F distribution, and the denominator degrees of freedom computed by matching the approximated one moment of the Wald-type statistic with the exact one moment of the F distribution. The other two approaches were derived by introducing a scaled Wald-type statistic to be distributed approximately as an F distribution, and the denominator degrees of freedom and the scale factor computed by matching the two moments of the statistic with the moments of the F distribution. This paper proposes two more approximate F-tests analogous to the four approaches where an adjusted estimator of the variance of the estimate of fixed effects used. In addition, the paper evaluates and compares the performance of the six approaches analytically, and some useful results are presented. Also, a simulation study for block designs was run to assess and compare the performance of the approaches based on their observed test levels. The simulation study shows that the approaches usually perform reasonably based on their test levels, and in some cases some approaches found to more adequately than other approaches.

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Masegosa, Andrés R., Rafael Cabañas, Helge Langseth, Thomas D. Nielsen et Antonio Salmerón. « Probabilistic Models with Deep Neural Networks ». Entropy 23, n^o 1 (18 janvier 2021) : 117. http://dx.doi.org/10.3390/e23010117.

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Recent advances in statistical inference have significantly expanded the toolbox of probabilistic modeling. Historically, probabilistic modeling has been constrained to very restricted model classes, where exact or approximate probabilistic inference is feasible. However, developments in variational inference, a general form of approximate probabilistic inference that originated in statistical physics, have enabled probabilistic modeling to overcome these limitations: (i) Approximate probabilistic inference is now possible over a broad class of probabilistic models containing a large number of parameters, and (ii) scalable inference methods based on stochastic gradient descent and distributed computing engines allow probabilistic modeling to be applied to massive data sets. One important practical consequence of these advances is the possibility to include deep neural networks within probabilistic models, thereby capturing complex non-linear stochastic relationships between the random variables. These advances, in conjunction with the release of novel probabilistic modeling toolboxes, have greatly expanded the scope of applications of probabilistic models, and allowed the models to take advantage of the recent strides made by the deep learning community. In this paper, we provide an overview of the main concepts, methods, and tools needed to use deep neural networks within a probabilistic modeling framework.

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Alahmadi, Amani A., Jennifer A. Flegg, Davis G. Cochrane, Christopher C. Drovandi et Jonathan M. Keith. « A comparison of approximate versus exact techniques for Bayesian parameter inference in nonlinear ordinary differential equation models ». Royal Society Open Science 7, n^o 3 (mars 2020) : 191315. http://dx.doi.org/10.1098/rsos.191315.

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The behaviour of many processes in science and engineering can be accurately described by dynamical system models consisting of a set of ordinary differential equations (ODEs). Often these models have several unknown parameters that are difficult to estimate from experimental data, in which case Bayesian inference can be a useful tool. In principle, exact Bayesian inference using Markov chain Monte Carlo (MCMC) techniques is possible; however, in practice, such methods may suffer from slow convergence and poor mixing. To address this problem, several approaches based on approximate Bayesian computation (ABC) have been introduced, including Markov chain Monte Carlo ABC (MCMC ABC) and sequential Monte Carlo ABC (SMC ABC). While the system of ODEs describes the underlying process that generates the data, the observed measurements invariably include errors. In this paper, we argue that several popular ABC approaches fail to adequately model these errors because the acceptance probability depends on the choice of the discrepancy function and the tolerance without any consideration of the error term. We observe that the so-called posterior distributions derived from such methods do not accurately reflect the epistemic uncertainties in parameter values. Moreover, we demonstrate that these methods provide minimal computational advantages over exact Bayesian methods when applied to two ODE epidemiological models with simulated data and one with real data concerning malaria transmission in Afghanistan.

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Seo, Jung-In, Jae-Woo Jeon et Suk-Bok Kang. « Exact Interval Inference for the Two-Parameter Rayleigh Distribution Based on the Upper Record Values ». Journal of Probability and Statistics 2016 (2016) : 1–5. http://dx.doi.org/10.1155/2016/8246390.

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The maximum likelihood method is the most widely used estimation method. On the other hand, it can produce substantial bias, and an approximate confidence interval based on the maximum likelihood estimator cannot be valid when the sample size is small. Because the sizes of the record values are considerably smaller than the original sequence observed in the majority of cases, a method appropriate for this situation is required for precise inference. This paper provides the exact confidence intervals for unknown parameters and exact predictive intervals for the future upper record values by providing some pivotal quantities in the two-parameter Rayleigh distribution based on the upper record values. Finally, the validity of the proposed inference methods was examined from Monte Carlo simulations and real data.

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Volaufová, Júlia, et Viktor Witkovský. « On exact inference in linear models with two variance-covariance components ». Tatra Mountains Mathematical Publications 51, n^o 1 (1 novembre 2012) : 173–81. http://dx.doi.org/10.2478/v10127-012-0017-9.

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ABSTRACT Linear models with variance-covariance components are used in a wide variety of applications. In most situations it is possible to partition the response vector into a set of independent subvectors, such as in longitudinal models where the response is observed repeatedly on a set of sampling units (see, e.g., Laird & Ware 1982). Often the objective of inference is either a test of linear hypotheses about the mean or both, the mean and the variance components. Confidence intervals for parameters of interest can be constructed as an alter- native to a test. These questions have kept many statisticians busy for several decades. Even under the assumption that the response can be modeled by a multivariate normal distribution, it is not clear what test to recommend except in a few settings such as balanced or orthogonal designs. Here we investigate statistical properties, such as accuracy of p-values and powers of exact (Crainiceanu & Ruppert 2004) tests and compare with properties of approximate asymptotic tests. Simultaneous exact confidence regions for variance components and mean parameters are constructed as well.

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GHAHRAMANI, ZOUBIN. « AN INTRODUCTION TO HIDDEN MARKOV MODELS AND BAYESIAN NETWORKS ». International Journal of Pattern Recognition and Artificial Intelligence 15, n^o 01 (février 2001) : 9–42. http://dx.doi.org/10.1142/s0218001401000836.

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We provide a tutorial on learning and inference in hidden Markov models in the context of the recent literature on Bayesian networks. This perspective makes it possible to consider novel generalizations of hidden Markov models with multiple hidden state variables, multiscale representations, and mixed discrete and continuous variables. Although exact inference in these generalizations is usually intractable, one can use approximate inference algorithms such as Markov chain sampling and variational methods. We describe how such methods are applied to these generalized hidden Markov models. We conclude this review with a discussion of Bayesian methods for model selection in generalized HMMs.

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Friston, Karl J., Lancelot Da Costa et Thomas Parr. « Some Interesting Observations on the Free Energy Principle ». Entropy 23, n^o 8 (19 août 2021) : 1076. http://dx.doi.org/10.3390/e23081076.

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Biehl et al. (2021) present some interesting observations on an early formulation of the free energy principle. We use these observations to scaffold a discussion of the technical arguments that underwrite the free energy principle. This discussion focuses on solenoidal coupling between various (subsets of) states in sparsely coupled systems that possess a Markov blanket—and the distinction between exact and approximate Bayesian inference, implied by the ensuing Bayesian mechanics.

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Ullah, Insha, Sudhir Paul, Zhenjie Hong et You-Gan Wang. « Significance tests for analyzing gene expression data with small sample sizes ». Bioinformatics 35, n^o 20 (15 mars 2019) : 3996–4003. http://dx.doi.org/10.1093/bioinformatics/btz189.

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Abstract Motivation Under two biologically different conditions, we are often interested in identifying differentially expressed genes. It is usually the case that the assumption of equal variances on the two groups is violated for many genes where a large number of them are required to be filtered or ranked. In these cases, exact tests are unavailable and the Welch’s approximate test is most reliable one. The Welch’s test involves two layers of approximations: approximating the distribution of the statistic by a t-distribution, which in turn depends on approximate degrees of freedom. This study attempts to improve upon Welch’s approximate test by avoiding one layer of approximation. Results We introduce a new distribution that generalizes the t-distribution and propose a Monte Carlo based test that uses only one layer of approximation for statistical inferences. Experimental results based on extensive simulation studies show that the Monte Carol based tests enhance the statistical power and performs better than Welch’s t-approximation, especially when the equal variance assumption is not met and the sample size of the sample with a larger variance is smaller. We analyzed two gene-expression datasets, namely the childhood acute lymphoblastic leukemia gene-expression dataset with 22 283 genes and Golden Spike dataset produced by a controlled experiment with 13 966 genes. The new test identified additional genes of interest in both datasets. Some of these genes have been proven to play important roles in medical literature. Availability and implementation R scripts and the R package mcBFtest is available in CRAN and to reproduce all reported results are available at the GitHub repository, https://github.com/iullah1980/MCTcodes. Supplementary information Supplementary data is available at Bioinformatics online.

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Huang, Kai, et Jie Mi. « Inference about Weibull Distribution Using Upper Record Values ». International Journal of Reliability, Quality and Safety Engineering 22, n^o 04 (août 2015) : 1550016. http://dx.doi.org/10.1142/s0218539315500163.

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This paper studies the frequentist inference about the shape and scale parameters of the two-parameter Weibull distribution using upper record values. The exact sampling distribution of the MLE of the shape parameter is derived. The asymptotic normality of the MLEs of both parameters are obtained. Based on these results this paper proposes various confidence intervals of the two parameters. Assuming one parameter is known certain testing procedures are proposed. Furthermore, approximate prediction interval for the immediately consequent record value is derived too. Conclusions are made based on intensive simulations.

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Jaakkola, T. S., et M. I. Jordan. « Variational Probabilistic Inference and the QMR-DT Network ». Journal of Artificial Intelligence Research 10 (1 mai 1999) : 291–322. http://dx.doi.org/10.1613/jair.583.

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We describe a variational approximation method for efficient inference in large-scale probabilistic models. Variational methods are deterministic procedures that provide approximations to marginal and conditional probabilities of interest. They provide alternatives to approximate inference methods based on stochastic sampling or search. We describe a variational approach to the problem of diagnostic inference in the `Quick Medical Reference' (QMR) network. The QMR network is a large-scale probabilistic graphical model built on statistical and expert knowledge. Exact probabilistic inference is infeasible in this model for all but a small set of cases. We evaluate our variational inference algorithm on a large set of diagnostic test cases, comparing the algorithm to a state-of-the-art stochastic sampling method.

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Jiao, Jiajia. « HEAP : A Holistic Error Assessment Framework for Multiple Approximations Using Probabilistic Graphical Models ». Electronics 9, n^o 2 (22 février 2020) : 373. http://dx.doi.org/10.3390/electronics9020373.

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Approximate computing has been a good paradigm of energy-efficient accelerator design. Accurate and fast error estimation is critical for appropriate approximate techniques selection so that power saving (or performance improvement) can be maximized with acceptable output quality in approximate accelerators. In the paper, we propose HEAP, a Holistic Error assessment framework to characterize multiple Approximate techniques with Probabilistic graphical models (PGM) in a joint way. HEAP maps the problem of evaluating errors induced by different approximate techniques into a PGM issue, including: (1) A heterogeneous Bayesian network is represented by converting an application’s data flow graph, where various approximate options are {precise, approximate} two-state X*-type nodes, while input or operating variables are {precise, approximate, unacceptable} three-state X-type nodes. These two different kinds of nodes are separately used to configure the available approximate techniques and track the corresponding error propagation for guaranteed configurability; (2) node learning is accomplished via an approximate library, which consists of probability mass functions of multiple approximate techniques to fast calculate each node’s Conditional Probability Table by mechanistic modeling or empirical modeling; (3) exact inference provides the probability distribution of output quality at three levels of precise, approximate, and unacceptable. We do a complete case study of 3 × 3 Gaussian kernels with different approximate configurations to verify HEAP. The comprehensive results demonstrate that HEAP is helpful to explore design space for power-efficient approximate accelerators, with just 4.18% accuracy loss and 3.34 × 105 speedup on average over Mentor Carlo simulation.

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Miller, David J., et Lian Yan. « Approximate Maximum Entropy Joint Feature Inference Consistent with Arbitrary Lower-Order Probability Constraints : Application to Statistical Classification ». Neural Computation 12, n^o 9 (1 septembre 2000) : 2175–207. http://dx.doi.org/10.1162/089976600300015105.

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We propose a new learning method for discrete space statistical classifiers. Similar to Chow and Liu (1968) and Cheeseman (1983), we cast classification/inference within the more general framework of estimating the joint probability mass function (p.m.f.) for the (feature vector, class label) pair. Cheeseman's proposal to build the maximum entropy (ME) joint p.m.f. consistent with general lower-order probability constraints is in principle powerful, allowing general dependencies between features. However, enormous learning complexity has severely limited the use of this approach. Alternative models such as Bayesian networks (BNs) require explicit determination of conditional independencies. These may be difficult to assess given limited data. Here we propose an approximate ME method, which, like previous methods, incorporates general constraints while retaining quite tractable learning. The new method restricts joint p.m.f. support during learning to a small subset of the full feature space. Classification gains are realized over dependence trees, tree-augmented naive Bayes networks, BNs trained by the Kutato algorithm, and multilayer perceptrons. Extensions to more general inference problems are indicated. We also propose a novel exact inference method when there are several missing features.

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Lin, Peng, Martin Neil et Norman Fenton. « Improved High Dimensional Discrete Bayesian Network Inference using Triplet Region Construction ». Journal of Artificial Intelligence Research 69 (27 septembre 2020) : 231–95. http://dx.doi.org/10.1613/jair.1.12198.

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Performing efficient inference on high dimensional discrete Bayesian Networks (BNs) is challenging. When using exact inference methods the space complexity can grow exponentially with the tree-width, thus making computation intractable. This paper presents a general purpose approximate inference algorithm, based on a new region belief approximation method, called Triplet Region Construction (TRC). TRC reduces the cluster space complexity for factorized models from worst-case exponential to polynomial by performing graph factorization and producing clusters of limited size. Unlike previous generations of region-based algorithms, TRC is guaranteed to converge and effectively addresses the region choice problem that bedevils other region-based algorithms used for BN inference. Our experiments demonstrate that it also achieves significantly more accurate results than competing algorithms.

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Mozer, Reagan, Luke Miratrix, Aaron Russell Kaufman et L. Jason Anastasopoulos. « Matching with Text Data : An Experimental Evaluation of Methods for Matching Documents and of Measuring Match Quality ». Political Analysis 28, n^o 4 (17 mars 2020) : 445–68. http://dx.doi.org/10.1017/pan.2020.1.

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Matching for causal inference is a well-studied problem, but standard methods fail when the units to match are text documents: the high-dimensional and rich nature of the data renders exact matching infeasible, causes propensity scores to produce incomparable matches, and makes assessing match quality difficult. In this paper, we characterize a framework for matching text documents that decomposes existing methods into (1) the choice of text representation and (2) the choice of distance metric. We investigate how different choices within this framework affect both the quantity and quality of matches identified through a systematic multifactor evaluation experiment using human subjects. Altogether, we evaluate over 100 unique text-matching methods along with 5 comparison methods taken from the literature. Our experimental results identify methods that generate matches with higher subjective match quality than current state-of-the-art techniques. We enhance the precision of these results by developing a predictive model to estimate the match quality of pairs of text documents as a function of our various distance scores. This model, which we find successfully mimics human judgment, also allows for approximate and unsupervised evaluation of new procedures in our context. We then employ the identified best method to illustrate the utility of text matching in two applications. First, we engage with a substantive debate in the study of media bias by using text matching to control for topic selection when comparing news articles from thirteen news sources. We then show how conditioning on text data leads to more precise causal inferences in an observational study examining the effects of a medical intervention.

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Zhu, Jianping, Hua Xin, Chenlu Zheng et Tzong-Ru Tsai. « Inference for the Process Performance Index of Products on the Basis of Power-Normal Distribution ». Mathematics 10, n^o 1 (23 décembre 2021) : 35. http://dx.doi.org/10.3390/math10010035.

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The process performance index (PPI) can be a simple metric to connect the conforming rate of products. The properties of the PPI have been well studied for the normal distribution and other widely used lifetime distributions, such as the Weibull, Gamma, and Pareto distributions. Assume that the quality characteristic of product follows power-normal distribution. Statistical inference procedures for the PPI are established. The maximum likelihood estimation method for the model parameters and PPI is investigated and the exact Fisher information matrix is derived. We discuss the drawbacks of using the exact Fisher information matrix to obtain the confidence interval of the model parameters. The parametric bootstrap percentile and bootstrap bias-corrected percentile methods are proposed to obtain approximate confidence intervals for the model parameters and PPI. Monte Carlo simulations are conducted to evaluate the performance of the proposed methods. One example about the flow width of the resist in the hard-bake process is used for illustration.

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