Talk:Statistical model

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Could this be explained for the layman?

statical modelling

b Aswediscussedabovetheproblemofmaininterestforusistoobtainameasureofboththe complexityandthe(useful)informationinadataset.Asinthealgorithmictheorythecomplexity istheprimarynotion,whichthenallowsustode¯nethemoreintricatenotionofinformation.Our planistode¯nethecomplexityintermsoftheshortestcodelengthwhenthedataisencodedwith aclassofmodelsascodes.Intheprevioussectionwesawthatthisleadsintothenoncomputability problemifwelettheclassofmodelsincludethesetofallcomputerprograms,a`model'identi¯ed withacomputerprogram(code)thatgeneratesthegivendata.However,ifweselectasmallerclass thenoncomputabilityproblemcanbeavoidedbutwehavetoovercomeanotherdi±culty:Howare wetode¯netheshortestcodelength?ItseemsthatinordernottofallbacktotheKolmogorov complexitywemustspelloutexactlyhowthedistributionsasmodelsaretobeusedtorestrictthe codingoperations.Inuniversalcodingwedidjustthatbydoingthecodinginapredictiveway,in Lempel-Zivcodebytheindexoftheleafinthetreeofthesegmentsdeterminedbythepastdata,and inContextCodingbyapplyinganarithmeticcodetoeach`next'symbol,conditionedonacontext de¯nedbythealgorithmasafunctionofthepastdata.Hereweadoptadi®erentstrategy:we de¯netheideaofshortestcodelengthinaprobabilisticsense,whichturnsouttosatisfypractical requirements.Todothatwemustbemoreformalaboutmodels.

statical modelling

b Aswediscussedabovetheproblemofmaininterestforusistoobtainameasureofboththe complexityandthe(useful)informationinadataset.Asinthealgorithmictheorythecomplexity istheprimarynotion,whichthenallowsustode¯nethemoreintricatenotionofinformation.Our planistode¯nethecomplexityintermsoftheshortestcodelengthwhenthedataisencodedwith aclassofmodelsascodes.Intheprevioussectionwesawthatthisleadsintothenoncomputability problemifwelettheclassofmodelsincludethesetofallcomputerprograms,a`model'identi¯ed withacomputerprogram(code)thatgeneratesthegivendata.However,ifweselectasmallerclass thenoncomputabilityproblemcanbeavoidedbutwehavetoovercomeanotherdi±culty:Howare wetode¯netheshortestcodelength?ItseemsthatinordernottofallbacktotheKolmogorov complexitywemustspelloutexactlyhowthedistributionsasmodelsaretobeusedtorestrictthe codingoperations.Inuniversalcodingwedidjustthatbydoingthecodinginapredictiveway,in Lempel-Zivcodebytheindexoftheleafinthetreeofthesegmentsdeterminedbythepastdata,and inContextCodingbyapplyinganarithmeticcodetoeach`next'symbol,conditionedonacontext de¯nedbythealgorithmasafunctionofthepastdata.Hereweadoptadi®erentstrategy:we de¯netheideaofshortestcodelengthinaprobabilisticsense,whichturnsouttosatisfypractical requirements.Todothatwemustbemoreformalaboutmodels.