Download The Variational Bayes Method in Signal Processing (Signals by Vaclav Smidl, Anthony Quinn PDF

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By Vaclav Smidl, Anthony Quinn

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Extra resources for The Variational Bayes Method in Signal Processing (Signals and Communication Technology)

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Our state of knowledge of θ after observing D is quantified by the posterior distribution, f (θ|D). 2) where Θ∗ is the space of θ. We will refer to f (θ, D) as the joint distribution of parameters and data, or, more concisely, as the joint distribution. We will refer to f (D|θ) as the observation model. If this is viewed as a (non-measure) function of θ, it is known as the likelihood function [3, 43–45]: l(θ|D) ≡ f (D|θ) . 3) ζ = f (D) is the normalizing constant, sometimes known as the partition function in the physics literature [46]: f (θ, D) dθ = ζ = f (D) = Θ∗ f (D|θ) f (θ) dθ.

7), and will be encountered again. 7), being a family of tractable parametric distributions, with members f˘ (θ|D) ≡ f0 (θ|β). Here, the approximating family members are indexed by an unknown (shaping) parameter, β, but their distributional form, f0 (·), is set a priori. The optimal approximation f˜ (θ|D) = f0 θ|βˆ , is then determined via βˆ = arg min KL (f (θ|D) ||f0 (θ|β)) . 39) are used for specific problems. Examples include the Levy, chi-squared and L2 norms. These are reviewed in [32]. g.

4) can be computationally expensive, or even intractable. 4) does not converge, the distribution is called improper [47]. 5) will be called the normalized distribution. In Fig. 2) as an operator, B, transforming the prior into the posterior, via the observation model, f (D|θ) . f (D|θ) f (θ) B f (θ|D) Fig. 1. Bayes’ rule as an operator. 2). e. e. the prior measure f (θ). In this sense, Bayesian methods are born from a subjective philosophy, which conditions all inference on the prior knowledge of the observer [2,36].

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