Role of information and its processing in statistical analysis

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Kim, Bryce
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preprint description
This paper discusses how real-life statistical analysis/inference deviates from ideal environments. More specifically, there often exist models that have equal statistical power as the actual data-generating model, given only limited information and information processing/computation capacity. This means that misspecification actually has two problems: first with misspecification around the model we wish to find, and that an actual data-generating model may never be discovered. Thus the role information - this includes data - plays on statistical inference needs to be considered more heavily than often done. A game defining pseudo-equivalent models is presented in this light. This limited information nature effectively casts a statistical analyst as a decider in decision theory facing an identical problem: trying best to form credence/belief of some events, even if it may end up not being close to objective probability. The sleeping beauty problem is used as a study case to highlight some properties of real-life statistical inference. Bayesian inference of prior updates can lead to wrong credence analysis when prior is assigned to variables/events that are not (statistical identification-wise) identifiable. A controversial idea that Bayesianism can go around identification problems in frequentist analysis is brought to more doubts. This necessitates re-defining how Kolmogorov probability theory is applied in real-life statistical inference, and what concepts need to be fundamental.