Information Theory, Inference, and Learning ... - MAELabs UCSD

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Copyright Cambridge University Press 2003. On-screen viewing permitted. Printing not permitted. http://www.cambridge.org/0521642981 You can buy this book for 30 pounds or $50. See http://www.inference.phy.cam.ac.uk/mackay/itila/ for links. 318 23 — Useful Probability Distributions pR = p1 + p2 and pB = p3 + p4 + p5 + p6. Assuming that the second Bayesian assigns a Dirichlet distribution to (p1, p2, p3, p4, p5, p6) with hyperparameters (u1, u2, u3, u4, u5, u6), show that, in order for the first Bayesian’s inferences to be consistent with those of the second Bayesian, the first Bayesian’s prior should be a Dirichlet distribution with hyperparameters ((u1 + u2), (u3 + u4 + u5 + u6)). Hint: a brute-force approach is to compute the integral P (pR, pB) = d 6 p P (p | u) δ(pR − (p1 + p2)) δ(pB − (p3 + p4 + p5 + p6)). A cheaper approach is to compute the predictive distributions, given arbitrary data (F1, F2, F3, F4, F5, F6), and find the condition for the two predictive distributions to match for all data. The entropic distribution for a probability vector p is sometimes used in the ‘maximum entropy’ image reconstruction community. P (p | α, m) = 1 Z(α, m) exp[−αDKL(p||m)] δ( i pi − 1) , (23.35) where m, the measure, is a positive vector, and DKL(p||m) = i pi log pi/mi. Further reading See (MacKay and Peto, 1995) for fun with Dirichlets. 23.6 Further exercises Exercise 23.4. [2 ] N datapoints {xn} are drawn from a gamma distribution P (x | s, c) = Γ(x; s, c) with unknown parameters s and c. What are the maximum likelihood parameters s and c?
Copyright Cambridge University Press 2003. On-screen viewing permitted. Printing not permitted. http://www.cambridge.org/0521642981 You can buy this book for 30 pounds or $50. See http://www.inference.phy.cam.ac.uk/mackay/itila/ for links. 24 Exact Marginalization How can we avoid the exponentially large cost of complete enumeration of all hypotheses? Before we stoop to approximate methods, we explore two approaches to exact marginalization: first, marginalization over continuous variables (sometimes known as nuisance parameters) by doing integrals; and second, summation over discrete variables by message-passing. Exact marginalization over continuous parameters is a macho activity enjoyed by those who are fluent in definite integration. This chapter uses gamma distributions; as was explained in the previous chapter, gamma distributions are a lot like Gaussian distributions, except that whereas the Gaussian goes from −∞ to ∞, gamma distributions go from 0 to ∞. 24.1 Inferring the mean and variance of a Gaussian distribution We discuss again the one-dimensional Gaussian distribution, parameterized by a mean µ and a standard deviation σ: P (x | µ, σ) = 1 (x − µ)2 √ exp − 2πσ 2σ2 ≡ Normal(x; µ, σ 2 ). (24.1) When inferring these parameters, we must specify their prior distribution. The prior gives us the opportunity to include specific knowledge that we have about µ and σ (from independent experiments, or on theoretical grounds, for example). If we have no such knowledge, then we can construct an appropriate prior that embodies our supposed ignorance. In section 21.2, we assumed a uniform prior over the range of parameters plotted. If we wish to be able to perform exact marginalizations, it may be useful to consider conjugate priors; these are priors whose functional form combines naturally with the likelihood such that the inferences have a convenient form. Conjugate priors for µ and σ The conjugate prior for a mean µ is a Gaussian: we introduce two ‘hyperparameters’, µ0 and σµ, which parameterize the prior on µ, and write P (µ | µ0, σµ) = Normal(µ; µ0, σ 2 µ). In the limit µ0 = 0, σµ → ∞, we obtain the noninformative prior for a location parameter, the flat prior. This is noninformative because it is invariant under the natural reparameterization µ ′ = µ + c. The prior P (µ) = const. is also an improper prior, that is, it is not normalizable. The conjugate prior for a standard deviation σ is a gamma distribution, which has two parameters bβ and cβ. It is most convenient to define the prior 319
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