BAYESIAN THEORY

By Daniel Restrepo-Montoya. 2008

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Citations in the corpus

Variational Inference: A Review for Statisticians (2016, methodology): “Next, we take the prior on the global variables to be the corresponding conjugate prior (Diaconis et al. 1979; Bernardo and Smith 1994), This prior has natural (hyper)parameter α = [ α 1 , α 2 ] , a column vector, and sufficient statistics that concatenate the global variable and its log normalizer…”
Recasting Gradient-Based Meta-Learning as Hierarchical Bayes (2018, methodology): “…, the marginal likelihood of the observed data is given by Maximizing (2) as a function of θ gives a point estimate for θ , an instance of a method known as empirical Bayes (Bernardo & Smith, 2006; Gelman et al., 2014) due to its use of the data to estimate the parameters of the prior distribution.”
Continual Learning with Adaptive Weights (CLAW) (2019, positive): “…Eq(w|γ) logp(wi,j) (9) =−0.5 ( 1+log(2π)+log(b2i,jpi,j(1− pi,j)) ) − Eq(w|γ) log 1 | | (10) =−logbi,j −0.5 logpi,j −0.5 log(1− pi,j) +c+ Eq(w|γ) log | |, (11) where the switch from (9) to (10) is due to the entropy computation (Bernardo & Smith, 2000) of the Gaussian q(wi,j|γi,j) defined in (8).”
Automatic Differentiation Variational Inference (2016, methodology): “If we place a multivariate Gaussian prior on µ then the posterior is also a Gaussian that we can compute analytically (Bernardo and Smith, 2009).”
A General Method for Robust Bayesian Modeling (2015, methodology): “As we mentioned, it is only the right procedure when the data comes from the model (Bernardo and Smith, 1994).”
The Boxer, the Wrestler, and the Coin Flip (2006, background): “(2)As reviewed by Bernardo and Smith (1994) and Kass and Wasserman (1996), a uniform distribution on the probability scale is only one of the many ways to define “ignorance” in this sort of problem.”
Dependent Credit Migrations (2006, methodology): “…is a conjugate prior of a covariance matrix: [Λ | · ] ∼ Wishart (ν + T,Λ0 + C3(b)) , where C3(b) is the following H ×H-matrix: C3(b) = T∑ t=1 btb ′ t + α 2 T−1∑ t=2 btb ′ t − α T∑ t=2 ( btb ′ t−1 + bt−1b ′ t ) , see Appendix A.2 of Bernardo and Smith (1994) or McCulloch and Rossi (1994).”
Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC (2015, methodology): “Here we take the log score as the default for evaluating the predictive density (Geisser and Eddy, 1979, Bernardo and Smith, 1994, Gneiting and Raftery, 2007).”
A Comparison of Three Methods of Estimation in the Context of Spatial Modeling (2009, background): “The aim is to determine whether the foundational differences between the two Schools, expertly discussed by Bernardo and Smith (1994), Jeffreys (1961) and others, manifest in significantly different outcomes – parameter and prediction estimates – within the context of property market analysis.”
A Bayesian Analysis of Directional Data Using the von Mises–Fisher Distribution (2005, methodology): “It is also desirable that the proposal density g · resemble f · as closely as possible, and that g · have heavier tails than f · (see, for example, Bernardo and Smith, 1994).”
Build, Compute, Critique, Repeat: Data Analysis with Latent Variable Models (2014, background): “The variables form a conjugate pair if the conditional distribution of the latent variable is in the same family as its prior, a property that depends on both the prior and the data generating distribution (Box and Tiao, 1973; Bernardo and Smith, 1994).”
Spatial filtering, model uncertainty and the speed of income convergence in Europe (2013, methodology): “The proposal of Ley and Steel (2009) is to impose that the model size follows a Binomial-Beta(a, b) distribution (Bernardo and Smith (1994)) with a = 1, so that P (k = kj) = Γ(1 + b) Γ(1) + Γ(b) + Γ(1 + b+K) ( K kj ) Γ(1 + kj)Γ(b+K − kj) kj = 0, . . . , K. (9) The prior can be elicited by anchoring…”
Multivariate Bayesian analysis of atmosphere–ocean general circulation models (2007, methodology): “The posterior density can be derived via Bayes’ theorem (e.g. Bernardo and Smith 1994), synthesized as [ process | data, parameters ] ∝ [ data | process, parameters ] · [ process | parameters ] · [ parameters ] .”
Comparing Dynamic Causal Models using AIC, BIC and Free Energy (2012, methodology): “Mathematical models of scienti ﬁ c data can be formally compared using the Bayesian model evidence (Bernardo and Smith, 2000; Gelman et al., 1995; Mackay, 2003), an approach that is now widely used in statistics (Hoeting et al., 1999), signal processing (Penny and Roberts, 2002), machine learning…”
Beyond subjective and objective in statistics (2015, background): “…or objectivist philosophy, because the “true parameter values” can often be interpreted as expected large sample functions given the prior model (Bernardo and Smith, 1994), but the way in which classical subjectivist or objectivist statistical data analysis is determined by the untestable prior…”
Bayesian Compressive Sensing (2008, methodology): “The use of a Laplace sparseness prior makes this difficult, due to the complexity of evaluating the denominator of the posterior; this is because the Laplace prior is not conjugate [14] to the likelihood in (6).”