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In 1703, Jacob Bernoulli wrote to Gottfried Leibniz to discuss their shared interest in applying mathematics and probability to games of chance. Bernoulli speculated whether it would be possible to gather mortality data from gravestones and thereby calculate, by their existing practice, the probability of a man currently aged 20 years outliving a man aged 60 years.
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In real life applications, both kinds of uncertainties are present. Uncertainty quantification intends to explicitly express both types of uncertainty separately. The quantification for the aleatoric uncertainties can be relatively straightforward, where traditional (frequentist) probability is the most basic form.
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The form of Eq(12) is usually the goal of a sensitivity analysis, since it is general, i.e., not tied to a specific set of parameter values, as was the case for the direct-calculation method of Eq(3) or (4), and it is clear basically by inspection which parameters have the most effect should they have systematic errors. For example, if the ...
The statistical errors, on the other hand, are independent, and their sum within the random sample is almost surely not zero. One can standardize statistical errors (especially of a normal distribution ) in a z-score (or "standard score"), and standardize residuals in a t -statistic , or more generally studentized residuals .
It is used primarily as a visual aid for detecting bias or systematic heterogeneity. A symmetric inverted funnel shape arises from a ‘well-behaved’ data set, in which publication bias is unlikely. An asymmetric funnel indicates a relationship between treatment effect estimate and study precision.