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MML Mixture Modelling of Rnulti-State, Poisson, Von Mises Circular and Gaussian Distributions

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MML Mixture Modelling of Rnulti-State, Poisson, Von Mises Circular and Gaussian Distributions MML mixture modelling of rnulti-state, Poisson, von Mises circular and Gaussian distributions Chris S. Wallace and David L. Dowe Department of Computer Science, Monash University, Clayton, Victoria 3168, Australia e-mail: {csw, dld} @cs.monash.edu.au Abstract Bayes's Theorem. Since D and Pr(D) are given and we wish to infer H , we can regard the problem of maximis­ Minimum Message Length (MML) is an invariant ing the posterior probability, Pr(H!D), as one of choos­ Bayesian point estimation technique which is also con­ ing H so as to maximise Pr(H). Pr(DIH) . sistent and efficient. We provide a brief overview of An information-theoretic interpretation of MML is MML inductive inference (Wallace and Boulton (1968) , that elementary coding theory tells us that an event of Wallace and Freeman (1987)) , and how it has both probability p can be coded (e.g. by a Huffman code) by an information-theoretic and a Bayesian interpretation. a message of length l = - log2 p bits. (Negligible or no We then outline how MML is used for statistical pa­ harm is done by ignoring effects of rounding up to the rameter estimation, and how the MML mixture mod­ next positive integer.) elling program, Snob (Wallace and Boulton (1968), Wal­ So, since - log2 (Pr(H). Pr(D!H)) = - log2 (Pr(H)) - lace (1986) , Wallace and Dowe(1994)) uses the message log2 (Pr(DjH)), maximising the posterior probability, lengths from various parameter estimates to enable it to Pr(HID), is equivalent to minimising combine parameter estimation with selection of the num­ ber of components. The message length is (to within MessLen = -log2(Pr(H))-log2 (Pr(DIH)) (1) a constant) the logarithm of the posterior probability the length of a two-part message conveying the theory, of the theory. So, the MML theory can also be re­ H , and the data, D, in light of the theory, H . Hence garded as the theory with the highest posterior proba­ the name "minimum message length" (principle) for thus bility. Snob currently assumes that variables are uncor­ choosing a theory, H , to fit observed data, D. The prin­ related, and permits multi-variate data from Gaussian, ciple seems to have first been stated by Solomonoff[31, discrete multi-state, Poisson and von Mises circular dis­ p20] and was re-stated and apparently first applied in a tributions. series of papers by Wallace and Boulton[37, p185][4][5, pp63-64][6, 8, 7, 38] dealing with model selection and parameter estimation (for Normal and multi-state vari­ 1 Introduction - About Minimum ables) for problems of mixture modelling (also known as Message Length (MML) clustering, numerical taxonomy or, e.g. [3], "intrinsic classification"). The Minimum Message Length (MML)[37, pl85][43] An important special case of the Minimum Message (and, e.g., [5, pp63-64][38]) principle of inductive infer­ Length principle is an observation of Chaitin[9] that data ence is based on information theory, and hence lies on the can be regarded as "random" if there is no theory, H, de­ interface on computer science and statistics. A Bayesian scribing the data which results in a shorter total message interpretation of the MML principle is that it variously length than the null theory results in. For a comparison states that the best conclusion to draw from data is the with the related Minimum Description Length (MDL) theory with the highest posterior probability or, equiv­ work of Rissanen[28, 29], see, e.g., [32]. We discuss later alently, that theory which maximises the product of the some (other) applications of MML. prior probability of the theory with the probability of the data occuring in light of that theory. We quantify by this immediately below. 2 Parameter Estimation Letting D be the data and H be an hypothesis (or MML theory) with prior probability Pr(H), we can write the posterior probability Pr(H!D) = Pr(H&D)/Pr(D) Given data x and parameters B, let h(B) be the prior Pr(H). Pr(DjH)/ Pr(D), by repeated application of probability distribution on B, let p(xlB) be the likelihood, 529 let L = - logp(xlB) be the negative log-likelihood and 2.1 Gaussian Variables let (2) For a No rm al distribution (with sample size, N), as­ suming a uniform prior on µ and a scale-invariant, be the Fisher information, the determinant of the (Fisher l/cr prior on er , we get that the Maximum Likelihood information) matrix of expected second partial deriva­ (ML) and MM L estimates of the mean concur, i.e., that 2 2 tives of the negative log-likelihood. Then the MML es­ fl.MML =µ ,'VIL= i:. Letting s = L ;(x; - x) , we get timate of B [43, p245] is that value of Bminimising the that cr 2 ML = s2 /N and[37, p190) that message length, •) 2 <T-Miv!L = s /(N - 1) (4) - log( h(B)p(xlB)/ jF{i)) + l / 2 log(e/12) (3) corrects this minor but well-known bias in the Maximum (If € is the measurement accuracy of the data and N Likelihood estimate. is the number of data things, then we add the constant term N log(l /€) to the length of the message. This is elaborated upon elsewhere[43 , p245][39, ppl-3].) 2.2 Discrete, Multi-State Variables The two-part message describing the data thus com­ prises first , a theory, which is the MML parameter esti­ Since multi-state attributes are discrete, the above issues mate(s) , and, second, the data given this theory. of measurement accuracy do not arise. It is reasonably clear to see that a finite coding can be For a multi-state distribution with M states, a given when the data is discrete or multi-state. For con­ ("colourless" ) uniform prior, h(f) = (M-1)!, is assumed tinuous data, we also acknowledge that it must only have over the ( M - 1)-dimensional region of hyper-volume been stated to finite precision by virtue of the fact that 1/(M-1)! given byp1+ P2+ ... +pM=l; p;;:::O . it was able to be (finitely) recorded. (In practice[41], as Letting nm be the number of things in state m and below equation (3), we assume that, for a given continu­ N = n 1 + ... +nM , minimising the message length formula ous or circular attribute, all measurements are made to gives that the MML estimate of Pm is given[37, p187(4), some accuracy, t: .) Just as all recorded data is finitely pp191-194][41) by recorded and can be finitely ·represented, by acknowl­ edging an uncertainty region in the MML estimate of Pm= (nm+ 1/2)/(N + M/2) (5) approximately[43, 36, 39] v12/ F(O) , the MML estimate is stated to a (non-zero) finite precision. The MML es­ This nominally gives rise to a (minimum) message timate has a genuine, non-zero, prior probability (not a length[37, p187(4),p194(28)] of density) and can be encoded by a genuine finite code, (Indeed, the object of MML is to choose a finitely stated (M-1) log(N/12+1)/2-log(M-1)!- L(nm+l/2) logftm estimate or hypothesis, H , to make the two-part message m of length - log2(Pr(H)) - log2(Pr(DIH)) stating H fol­ (6) lowed by D given H as short as possible.) The MML for both stating the parameter estimates and then en­ theory is thus different, in general, from the standard coding the things in light of these parameter estimates. Bayesian maximum a posteriori (MAP) theory. In the remainder of this section, we give several ex­ amples of the result of using the MML formula to ob­ 2.3 Poisson Variables tain parameter estimates from "innocuous" priors. For the Gaussian, multi-state and Poisson distributions, the Earlier versions of Snob originally[37, 33, 34] permitted MML estimate can be written in a simple analytic form models of classes whose variables were assumed to come and closely approximates the Maximum Likelihood (ML) from a COlllbination of either (discrete) multi-state or estimate. For the von Mises distributiOn, the estimators (continuous) Normal distributions. Snob has since been take a messier form[30, 18 , 39, 14). augmented by permitting Poisson distributions and von The following two sections are on extending MML pa­ Mises circular distributions[39, 40, 14]. rameter estimation to MML mixture modelling, and on With a the population rate, c the total count and the invariance[43, 38] of MML and the consistency and t the total time, with a prior on the rate, r , of efficiency[43, 36, 1] of MML. Further sections mention al­ h(r) = (l/a).e-rfc.x, we get an MML estimate of ternative mixture modelling programs, and applications of and extensions to the Snob program. TMML = (c + 1/2)/(t + 1/a) (7) 2.4 von Mises Circular Variables to permit (e.g.) in '2 dimensions, the uncertainty re­ gion to be a hexagon rather than a rectangle since (in The von Mises distribution. J\!h(µ , t1: ), with mean direc­ short) both hexagons and rectangles tile the Euclidean tionµ , and concentration parameter, K, is a circular ana­ plane but a hexagon has a smaller (average or) expected logue of the Normal distribution[l8, 21 , 39], - both be­ squared distance from its centre than a rectangle or any ing maximum entropy distributions. Letting Io(K) be other tiling shape. This is quantified elsewhere [43, 39] in the relevant normalisation constant, it has probability terms of lattice constants[l l] for optimally1 tesselating density function (p.d.f.) Voronoi regions.
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