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Demographic Forecasting andthe Role of PriorsFederico GirosiThe RAND CorporationSanta Monica CA USA. ReferenceAll the material for this lecture can be foundat http gking harvard edu files s... Plan of the Lecture Demographic forecasting is a machine.learning problem Solving the problem in the Bayesian regularization framework A closer look at one dimensional priors A closer look at the smoothness parameter. Examples Demos Forecasting Mortality and Disease BurdenHas Important ApplicationsPension planningAllocation of public.health resourcesPlanning manpower needsGuidance for epidemiological Problem forecasting very short time series The forecasting problem is set as a.regression problem Mortality in country c age a and time t Regression coefficient Lagged covariates Typical Lagged Covariates. Lagged covariates Human capital Fat consumption Water quality Cigarette consumption. In most cases some pooling is necessaryRegressions cannotbe estimated separatelyacross age groups orcountries .17 separate regressions one for each age group Those who have knowledge do notpredict Those who predict do nothave knowledge.Lao Tzu 6th century BC The Standard Bayesian Approach 1 2 Likelihood P y exp 2 cat ca xcat . 2 cat Posterior P y P y P 1 2 P y exp 2 2 cat ca.cat H cat A Way Out We do need some sort of prior on the but we do not really have prior knowledge. BUT we do have knowledge on AND is related to X Strategy to build a prior Define a non parametric prior for as a functionof the cross sectional index age for example . Use the relationship between and X tochange variables and obtain a prior for What type of prior knowledge Mortality age profiles are smoothdeformations of well known shapes. Mortality varies smoothlyacross countries Mortality varies smoothly over A Good Prior on Discretizing age on a grid .H at a W aa a t a Only a Step Away from Prior on The matrix W is fully determinedby the order of the derivative n The template age profile can.be made disappear by subtractingif from the data Just need to substitute thespecification X And the Prior for is . n P exp Waa a Caa a aa C aa XaXa But What Does the Prior Really Mean . But What Does the Prior Really Mean Discretizing over age and fixing one year in time is simply a vector of random variables n P exp aWaa a . aa How do the samples from this prior look like Samples from prior with zero mean Samples from prior with non zero mean And what is the role of .Two important related identitiesE H E a tr W n The role of . determines the size of thesmoothness functional determines the average standarddeviation of the prior Samples from prior with non zero mean .varying the smoothness parameter Other Types of Priors a t Time H dtda n t . Time trends over age a t H dtda n m t a Dealing with Multiple Smoothness Parameters. Writing the priors is easy Estimating the 3 smoothing parameters is very Crossvalidation is hard to do with very shorttime series Some prior knowledge on the smoothing.parameters is needed Estimating the smoothness parameters Key observation the smoothness parameterscontrol ALL expected values of the prior Estimating the smoothness parameters. Sometimes we do have other forms of prior How much the dependent variables changesfrom one cross section or year to the nextF1 at a 1 t Estimating the smoothness parameters. Expected values of any function of can beestimated empirically by sampling the prior The following equations can be solved numerically Demo Deaths by Transportation Accidents in Transportation Accidents .no pooling Pooling Over Countries Transportation Accidents in ArgentinaNo Pooling Pooling Regularization theory is a powerful.framework that reaches beyond standardpattern recognition In some application it is important to payattention to the precise nature of the prior Prior knowledge applies to the.smoothness parameter too Mortality age profiles are well known andconsistent across countries and time Similar countries have similar mortalityFrance Greece.Italy IsraelChile Spain Before and After the CureRespiratory Infections in Belize.then the main points could come up one at a time in "build" fashion. Can we streamline the verbiage on this chart--maybe even make it into a kind of table with adult and child across the top and the important factors down the side? then the main points could come up one at a time in "build" fashion.