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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.
Signature Generation Algorithms for Polymorphic Worms
Published in Mohssen Mohammed, Al-Sakib Khan Pathan, Automatic Defense Against Zero-day Polymorphic Worms in Communication Networks, 2016
Mohssen Mohammed, Al-Sakib Khan Pathan
The marginal likelihood has an interesting interpretation. It is the probability of generating dataset D from parameters that are randomly sampled from under the prior for mi. This should be contrasted with the maximum likelihood for mi, which is the probability of the data under the single setting of the parameters θ^i that maximizes P (D | θi, mi). A more complicated model will have a higher maximum likelihood, which is the reason why maximizing the likelihood results in overfitting (i.e., a preference for more complicated models than necessary). In contrast, the marginal likelihood can decrease as the model becomes more complicated. In a more complicated model, sampling random parameter values can generate a wider range of possible datasets, but since the probability over datasets has to integrate to 1 (assuming a fixed number of data points) spreading the density to allow for more complicated datasets necessarily results in some simpler datasets having lower density under the model. The decrease in the marginal likelihood as additional parameters are added has been called the automatic Occam’s razor [93-95].
Model Selection and Evaluation
Published in Richard M. Golden, Statistical Machine Learning, 2020
A major computational challenge associated with applying the Bayesian Model Selection Criterion in (16.21) is the computation of the marginal likelihood of the observed data in (16.20). For example, since p(Dn|θ,M)=exp(−nℓ^n(θ)),p(Dn|M)=∫ΘMexp(−nℓ^n(θ))pΘ(θ|M)dθ. Monte Carlo simulation methods such as the bootstrap simulation methods described in Chapter 14 are often used to numerically evaluate (16.20).
Determinants of renewable energy production in WAEMU countries: New empirical insights and policy implications
Published in International Journal of Green Energy, 2021
Nimonka Bayale, Essossinam Ali, Abdou-Fataou Tchagnao, Amandine Nakumuryango
where, denotes the integrated likelihood which is constant over all models and is thus simply a multiplicative term (Okafor and Piesse 2017; Zeugner and Feldkircher 2015). Therefore, the posterior model probability (PMP) is proportional to the integrated likelihood , which reflects the probability of the data given model . The marginal likelihood of model is multiplied by its prior model probability indicating how probable the researcher thinks model is before looking at the data. The difference between and is that integration is once over the model space ( and once for a given model over the parameter space . By re-normalization of the product from above one can infer the PMPs and thus the model weighted posterior distribution for any statistic (Equation (3)).
A Bayesian Allocation Model Based Approach to Mixed Membership Stochastic Blockmodels
Published in Applied Artificial Intelligence, 2022
For a given latent variable model, the model selection problem corresponds to choosing the optimal number of blocks that explains the latent structure in the observed data best. Bayesian statistics provides a principled likelihood-based approach for this task. The aim is to choose the model, which produces the largest marginal likelihood of the observed data . However, the marginal of is often intractable. Therefore, we choose to approximate the marginal by its mean-field variational approximation similar to the work of Latouche et al. (Latouche, Birmele, and Ambroise 2012).
Multi-Output Gaussian Processes for Inverse Uncertainty Quantification in Neutron Noise Analysis
Published in Nuclear Science and Engineering, 2023
Paul Lartaud, Philippe Humbert, and Josselin Garnier
The common practice for selecting the hyperparameters is to find the values that maximize the marginal likelihood . The marginal likelihood refers to the probability of the observations integrated over all the possible function values drawn from the Gaussian process. It is defined by