Ülo Ennuste majandusartiklid

Statistikateadlase otsustusparadoks

Statistikateadlase otsustusteoreetiline paradoks

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Meie statistikateadlane olla väidetavalt väitnud:

matemaatilise statistika valdamine jätab tema hinnangul Eesti teadlaskonnas teinekord palju soovida.“*

Kuid kuna väitja kuulub ise kah Eesti teadlaskonda – järelikult – teinekord jätab tema enda matemaatilise statistika valdamine kah soovida.

Tõepoolest, seekord* on teadlane ära unustanud Bayes’i käsitluse (1763 vt Lisa) et referents-väärtuse kalibreerimisel on kah juba statistikat sisse arvestatud** ja unustanud ka ära et antud juhul üksiku mõõtmistulemuse täpsus on suure eelnevate vaatluste arvuga juba ette ära hinnatud ning mingi paar uut katset seda ei mõjuta***.

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*http://suusk.blogspot.com/ (muide – see blogi on teinekord väheusaldusväärne, üe)

**PM.ee 13.VII komm Veerpalu kaasust puudutavale artiklile:

Margus 13.07.2011 15:41

WADA referentstulemusse on statistika juba sisse arvestatud. Tolerants on 99,99% see tähendab võimalust ühte kahekümnetuhandikku, et positiivseks osutunud test on tegelikult negatiivne. Selles kontekstis pole Veerpalu testitulemuste kõikumine kuigi oluline. Mis loeb on see, et väärtus oli üle lubatud taseme. Kuna see tase on niivõrd konservatiivne, et soosib pigem süüdlasi kui süütuid, siis on selles valguses koolistatistikaga WADA vastu minek ette kaotatud lahing, kahjuks.

*** Otsustusteooria=ökonoomika+statistika

Tõepoolest, sest tuleb arvestada et mida otsustuse kvaliteedi tõstmine nt kas vaatluste arvu ja/või otsustusmudeli adekvaatsuse suurendamisega kulude-tulude osas kaasa toob.

Selge, et seega ei sobi sellesse teooriasse üldiselt Bernoulli’ (lihtsad puhtsageduslikud ning puhtalt lehelt) statistilised meetodid millede puhul nt üks uus katse/proov ei anna meetodile omaselt mingil juhul mingit informatsiooni juurde ja ka seniseks olemasolev informatsioon jääb arvestamata – alles suure arvu katsetega saadakse usaldusväärset teavet.

Otsustusteoreetiliselt sobivamad on Bayes’i mudelid (Lisa) mis võimaldavad vaatlust tehes arvestada juba olemasolevat aprioorset statistikat ning selle alusel ka iga üksik katse/proov omaette võib anda täiendavat aposterioorset teavet.

Arusaadavalt tuleb seejuures arvestada nii iga katse maksumust ning katse(te)le eelneva informatsiooni kogumise maksumust ning otsustuse kvaliteedi efekte.

Lisa (WikipediA)

Calculations with Bayesian probabilities

Main articles: Bayes’ theorem and Bayesian inference

Bayes’ theorem is one of the main tools for manipulating probabilities of any kind; that is, it is applicable no matter what interpretation is being placed on the probabilities being manipulated. Bayesian inference is a formal approach to making statistical inferences in cases where some of the probabilities are interpreted as representing beliefs, or knowledge, rather than having a frequency-based interpretation. While “Bayesian inference” makes uses of Bayes’ theorem, not all cases where Bayes’ theorem is applied should be labeled as “Bayesian statistics” or “Bayesian inference”.

The use of Bayes’ theorem in Bayesian inference may be described as follows. Let H denote a hypothesis; that a certain statement of supposed fact is true, or that a statistical parameter takes a certain value. Before observing data from a given experiment, one starts with some belief about whether the hypothesis H is true, expressed in the form of a probability, usually called the prior probability. Bayes’ theorem is used to determine what one’s probability for the hypothesis should be, once the outcome D from the experiment is known. The phrase “should be” is important here, as Bayes’ theorem is a condensation of the rules that anyone should apply to updating beliefs, provided that they are acting according to reasonable rules of requirements of rationality and consistency.[1][4] The probability of the hypothesis once the outcome from the experiment is known is called the posterior probability.

The posterior probability is proportional to the likelihood of the observed data, multiplied by the prior probability, and is given by Bayes’ theorem. Thus

 \operatorname{P}(H|D) = \frac{\operatorname{P}(D|H)\;\operatorname{P}(H)}{\operatorname{P}(D)},

where

The quantity \operatorname{P}(D) is the prior probability of witnessing the data D under all possible hypotheses, and it depends on the prior probabilities given to each of these other possible hypotheses. Given any exhaustive set of mutually exclusive hypotheses Hi,

\operatorname{P}(D) = \sum_i  \operatorname{P}(D, H_i) = \sum_i  \operatorname{P}(D|H_i)\operatorname{P}(H_i).\,

Here i can be considered to index alternative cases, of which exactly one is actually valid, and Hi is the hypothesis that case i is valid. Then \operatorname{P}(D, H_i) is the probability that both case i is valid and that the data from the experiment turn out to be what was observed. Since the set of alternative cases is assumed to be mutually exclusive and exhaustive, the above formula is a case of the law of total probability. In many cases, \operatorname{P}(D), which is a normalizing constant, need not be evaluated. As a result, Bayes’ formula is often simplified to:

\operatorname{P}(H|D) \propto \operatorname{P}(D|H)\;\operatorname{P}(H),

where \propto denotes proportionality.

juuli 14, 2011 - Posted by | Uncategorized

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