what is going to be the shortest, most concise restatement of information?

Formal data theory restatement of Occam's Razor

Minimum message length (MML) is a Bayesian data-theoretic method for statistical model comparison and choice.^[1] It provides a formal data theory restatement of Occam'due south Razor: fifty-fifty when models are equal in their measure of fit-accurateness to the observed information, the one generating the nigh concise explanation of data is more likely to exist correct (where the explanation consists of the statement of the model, followed past the lossless encoding of the data using the stated model). MML was invented by Chris Wallace, first appearing in the seminal newspaper "An information measure for classification".^[2] MML is intended non just as a theoretical construct, only as a technique that may exist deployed in practice.^[3] Information technology differs from the related concept of Kolmogorov complication in that it does non crave utilise of a Turing-complete language to model data.^[4]

Definition [edit]

Shannon's A Mathematical Theory of Communication (1948) states that in an optimal lawmaking, the bulletin length (in binary) of an event ${\displaystyle Eastward}$ , ${\displaystyle \operatorname {length} (E)}$ , where ${\displaystyle Eastward}$ has probability ${\displaystyle P(Eastward)}$ , is given past ${\displaystyle \operatorname {length} (E)=-\log _{2}(P(Due east))}$ .

Bayes's theorem states that the probability of a (variable) hypothesis ${\displaystyle H}$ given fixed prove ${\displaystyle E}$ is proportional to ${\displaystyle P(E|H)P(H)}$ , which, by the definition of conditional probability, is equal to ${\displaystyle P(H\country E)}$ . We want the model (hypothesis) with the highest such posterior probability. Suppose nosotros encode a message which represents (describes) both model and data jointly. Since ${\displaystyle \operatorname {length} (H\land E)=-\log _{two}(P(H\state East))}$ , the almost probable model will take the shortest such message. The message breaks into two parts: ${\displaystyle -\log _{ii}(P(H\state Due east))=-\log _{2}(P(H))+-\log _{2}(P(E|H))}$ . The kickoff role encodes the model itself. The second part contains information (e.g., values of parameters, or initial conditions, etc.) that, when processed by the model, outputs the observed data.

MML naturally and precisely trades model complexity for goodness of fit. A more complicated model takes longer to state (longer starting time office) just probably fits the information better (shorter second part). And so, an MML metric won't choose a complicated model unless that model pays for itself.

Continuous-valued parameters [edit]

One reason why a model might exist longer would be simply because its diverse parameters are stated to greater precision, thus requiring transmission of more digits. Much of the power of MML derives from its handling of how accurately to country parameters in a model, and a variety of approximations that make this feasible in practice. This allows it to usefully compare, say, a model with many parameters imprecisely stated against a model with fewer parameters more than accurately stated.

Cardinal features of MML [edit]

• MML tin can be used to compare models of different structure. For case, its earliest application was in finding mixture models with the optimal number of classes. Adding extra classes to a mixture model will always permit the data to exist fitted to greater accurateness, but according to MML this must exist weighed against the extra $.25 required to encode the parameters defining those classes.
• MML is a method of Bayesian model comparison. It gives every model a score.
• MML is scale-invariant and statistically invariant. Unlike many Bayesian option methods, MML doesn't care if you change from measuring length to volume or from Cartesian co-ordinates to polar co-ordinates.
• MML is statistically consistent. For problems like the Neyman-Scott (1948) problem or factor analysis where the amount of data per parameter is divisional above, MML can estimate all parameters with statistical consistency.
• MML accounts for the precision of measurement. It uses the Fisher information (in the Wallace-Freeman 1987 approximation, or other hyper-volumes in other approximations) to optimally discretize continuous parameters. Therefore the posterior is always a probability, not a probability density.
• MML has been in utilise since 1968. MML coding schemes have been developed for several distributions, and many kinds of car learners including unsupervised nomenclature, decision trees and graphs, DNA sequences, Bayesian networks, neural networks (1-layer only so far), image compression, image and function segmentation, etc.

See likewise [edit]

• Algorithmic probability
• Algorithmic information theory
• Grammar induction
• Inductive inference
• Inductive probability
• Kolmogorov complexity – absolute complexity (within a constant, depending on the detail option of Universal Turing Machine); MML is typically a computable approximation (encounter ^[5])
• Minimum clarification length – an alternative with a mayhap dissimilar (non-Bayesian) motivation, developed 10 years after MML.
• Occam's razor

References [edit]

1. ^ Wallace, C. S. (Christopher South.), -2004. (2005). Statistical and inductive inference past minimum message length. New York: Springer. ISBN9780387237954. OCLC 62889003. {{cite book}}: CS1 maint: multiple names: authors list (link)
2. ^ Wallace, C. South.; Boulton, D. M. (1968-08-01). "An Information Mensurate for Classification". The Figurer Journal. 11 (ii): 185–194. doi:10.1093/comjnl/11.2.185. ISSN 0010-4620.
3. ^ Allison, Lloyd. (2019). Coding Ockham's Razor. Springer. ISBN978-3030094881. OCLC 1083131091.
4. ^ Wallace, C. S.; Dowe, D. L. (1999-01-01). "Minimum Message Length and Kolmogorov Complexity". The Computer Journal. 42 (4): 270–283. doi:10.1093/comjnl/42.four.270. ISSN 0010-4620.
5. ^ Wallace, C. Due south.; Dowe, D. L. (1999-01-01). "Minimum Message Length and Kolmogorov Complexity". The Estimator Periodical. 42 (4): 270–283. doi:10.1093/comjnl/42.4.270. ISSN 0010-4620.

External links [edit]

Original Publication:

• Wallace; Boulton (Baronial 1968). "An information measure for classification". Computer Periodical. 11 (two): 185–194. doi:ten.1093/comjnl/eleven.2.185.

Books:

• Wallace, C.S. (May 2005). Statistical and Inductive Inference past Minimum Message Length. Informatics and Statistics. Springer-Verlag. ISBN978-0-387-23795-iv.
• Allison, L. (2018). Coding Ockham's Razor. Springer. doi:10.1007/978-3-319-76433-seven. ISBN978-3319764320. S2CID 19136282. , on implementing MML, and source-code.

Related Links:

• Links to all Chris Wallace'southward known publications.
• A searchable database of Chris Wallace'southward publications.
• Wallace, C.S.; Dowe, D.L. (1999). "Minimum Bulletin Length and Kolmogorov Complexity". Reckoner Journal. 42 (4): 270–283. CiteSeerX10.1.1.17.321. doi:10.1093/comjnl/42.4.270.
• "Special Issue on Kolmogorov Complication". Computer Journal. 42 (four). 1999. ^{[ dead link ]}
• Dowe, D.L.; Wallace, C.South. (1997). Resolving the Neyman-Scott Trouble past Minimum Message Length. 28th Symposium on the interface, Sydney, Australia. Computing Science and Statistics. Vol. 28. pp. 614–618.
• History of MML, CSW's terminal talk.
• Needham, S.; Dowe, D. (2001). Bulletin Length equally an Effective Ockham'south Razor in Decision Tree Consecration (PDF). Proc. 8th International Workshop on AI and Statistics. pp. 253–260. (Shows how Occam'due south razor works fine when interpreted as MML.)
• Allison, L. (Jan 2005). "Models for car learning and data mining in functional programming". Journal of Functional Programming. 15 (1): 15–32. doi:10.1017/S0956796804005301. S2CID 5218889. (MML, FP, and Haskell lawmaking).
• Comley, J.W.; Dowe, D.L. (April 2005). "Chapter 11: Minimum Message Length, MDL and Generalised Bayesian Networks with Asymmetric Languages". In Grunwald, P.; Pitt, M. A.; Myung, I. J. (eds.). Advances in Minimum Description Length: Theory and Applications. Yard.I.T. Press. pp. 265–294. ISBN978-0-262-07262-five.
• Comley, Joshua Due west.; Dowe, D.L. (5–eight June 2003). General Bayesian Networks and Asymmetric Languages. Proc. 2nd Hawaii International Conference on Statistics and Related Fields. , .pdf. Comley & Dowe (2003, 2005) are the commencement ii papers on MML Bayesian nets using both discrete and continuous valued parameters.

• Dowe, David Fifty. (2010). "MML, hybrid Bayesian network graphical models, statistical consistency, invariance and uniqueness" (PDF). Handbook of Philosophy of Science (Book 7: Handbook of Philosophy of Statistics). Elsevier. pp. 901–982. ISBN978-0-444-51862-0.
• Minimum Bulletin Length (MML), LA's MML introduction, (MML alt.).
• Minimum Bulletin Length (MML), researchers and links.
• "Another MML enquiry website". Archived from the original on 12 April 2017.
• Snob folio for MML mixture modelling.
• MITECS: Chris Wallace wrote an entry on MML for MITECS. (Requires account)
• mikko.ps: Curt introductory slides by Mikko Koivisto in Helsinki
• Akaike information benchmark (AIC) method of model pick, and a comparison with MML: Dowe, D.L.; Gardner, South.; Oppy, Thou. (Dec 2007). "Bayes not Bust! Why Simplicity is no Problem for Bayesians". Br. J. Philos. Sci. 58 (4): 709–754. doi:10.1093/bjps/axm033.

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Source: https://en.wikipedia.org/wiki/Minimum_message_length

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