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Supervised Learning
Published in Peter Wlodarczak, Machine Learning and its Applications, 2019
Bayesian models are based on Bayes theorem. Generally speaking, the Bayes classifier minimizes the probability of misclassification. It is a model that draws its inferences from the posterior distribution. Bayesian models utilize a prior distribution and a likelihood, which are related by Bayes’ theorem. Bayes rule decomposes the computation of a posterior probability into the computation of a likelihood and a prior probability [30]. It calculates the posterior probability P(c|x) from P(c), P(x) and P(x|c), as shown in equation 4.7. () P(c|x)=P(x|c)P(c)P(x)
Parameter Estimation
Published in Alex Martynenko, Andreas Bück, Intelligent Control in Drying, 2018
The prior distribution represents the belief in a parameter vector before observing the data, for example, values from literature can be used as means for a normal prior distribution. If no such information is available, flat (uninformative) priors can be used. In analog to the profile likelihood, the concept of profile posteriors can be employed to analyze identifiability (Hug et al., 2013). The marginal likelihood p(y) (also occasionally referred to as evidence for the data) is a usually high-dimensional integral taken over the whole parameter space and thus usually hard to compute analytically or numerically, representing a major bottleneck in the Bayesian approach.
Search, Tracking, and Surveillance
Published in Yasmina Bestaoui Sebbane, Intelligent Autonomy of Uavs, 2018
Since time is a major factor determining the success of a search and rescue operation and hence the system, the goal of each agent is to minimize t, the number of iterations required to locate the subject. Models and algorithms are developed for the following four steps of the search process [12]:Agent Frame Request: All agents generate frame requests based on their individual PDFs of the subject’s location.UAV Frame Allocation: Collects requests and computes an optimal frame assignment to the UAVs.Sensor Data Extraction: Processes the resulting image data and specifies whether or not the subject was detected.Prior Distribution Update: All agents update their probability distribution functions to incorporate the new data.
Data-driven model-based flow measurement uncertainty quantification for building central cooling systems using a probabilistic approach
Published in Science and Technology for the Built Environment, 2023
Shaobo Sun, Kui Shan, Shengwei Wang
Bayesian inference is a statistical inference method, and its core is Bayes’ theorem expressed by Eq. (5). Where, θ is the unknown parameter that needs to be quantified, D is the observation data, is the posterior distribution/probability, is the likelihood function, is the prior distribution/probability, and is the marginal likelihood. This equation can be simplified as Eq. (6) by dropping which does not rely on θ. The posterior probability () is proportional to the product of prior probability () and likelihood function (). In Bayesian analysis, the posterior distribution is obtained by updating the prior distribution with the information in observation data (van de Schoot et al. 2021).
Understanding regional streamflow trend magnitudes in the Southern Murray-Darling Basin, Australia
Published in Australasian Journal of Water Resources, 2022
Zitian Gao, Danlu Guo, Murray C. Peel, Michael J. Stewardson
The calibration process in each BHM compares the model-simulated with the corresponding observations. To initiate a BHM, the prior distributions of variables and parameters need to be specified (see equations (9), (11) and equation annotations). The prior distribution represents the expected parameter range before they are inferred from any observed data, and the posterior distribution is the final parameter estimated based on observations (calibration). The calibration process involves drawing independent samples from the prior distribution using a Markov chain Monte Carlo (MCMC) technique, and then using maximum likelihood to encourage convergence towards the posterior distribution. We extracted the mean of the posterior distributions for as the region-level trends. The posterior distributions of , , … were also assessed for the influence of each predictor (i.e. catchment characteristics) on regional trends.
Bayesian Methods for Planning Accelerated Repeated Measures Degradation Tests
Published in Technometrics, 2021
Brian P. Weaver, William Q. Meeker
Case C is different than Cases A and B in that it uses the informative prior distribution for inference. With this additional prior information for inference, one cannot factor out the sample size in (9), because the sample size n affects only but not the prior distribution. Thus (unlike Cases A and B) the optimum test plans for the different sample sizes will result in different allocations. Thus for Case C we will again use sample sizes n = 29 and n = 2900. When the sample size is large, the posterior distribution will be less dependent on the prior distribution for inference. When the sample size is small, however, the posterior and the design will be more strongly influenced by the prior distribution used for inference.