121 research outputs found
The Law of Total Odds
The law of total probability may be deployed in binary classification
exercises to estimate the unconditional class probabilities if the class
proportions in the training set are not representative of the population class
proportions. We argue that this is not a conceptually sound approach and
suggest an alternative based on the new law of total odds. We quantify the bias
of the total probability estimator of the unconditional class probabilities and
show that the total odds estimator is unbiased. The sample version of the total
odds estimator is shown to coincide with a maximum-likelihood estimator known
from the literature. The law of total odds can also be used for transforming
the conditional class probabilities if independent estimates of the
unconditional class probabilities of the population are available.
Keywords: Total probability, likelihood ratio, Bayes' formula, binary
classification, relative odds, unbiased estimator, supervised learning, dataset
shift.Comment: 12 pages, 1 figure, new reference
Expected Shortfall and Beyond
Financial institutions have to allocate so-called "economic capital" in order
to guarantee solvency to their clients and counter parties. Mathematically
speaking, any methodology of allocating capital is a "risk measure", i.e. a
function mapping random variables to the real numbers. Nowadays
"value-at-risk", which is defined as a fixed level quantile of the random
variable under consideration, is the most popular risk measure. Unfortunately,
it fails to reward diversification, as it is not "subadditive". In the search
for a suitable alternative to value-at-risk, "Expected Shortfall" (or
"conditional value-at-risk" or "tail value-at-risk") has been characterized as
the smallest "coherent" and "law invariant" risk measure to dominate
value-at-risk. We discuss these and some other properties of Expected Shortfall
as well as its generalization to a class of coherent risk measures which can
incorporate higher moment effects. Moreover, we suggest a general method on how
to attribute Expected Shortfall "risk contributions" to portfolio components.
Key words: Expected Shortfall; Value-at-Risk; Spectral Risk Measure;
coherence; risk contribution.Comment: 18 pages, LaTeX with hyperref package, Remark 3.8 and references
update
Proving prediction prudence
We study how to perform tests on samples of pairs of observations and
predictions in order to assess whether or not the predictions are prudent.
Prudence requires that that the mean of the difference of the
observation-prediction pairs can be shown to be significantly negative. For
safe conclusions, we suggest testing both unweighted (or equally weighted) and
weighted means and explicitly taking into account the randomness of individual
pairs. The test methods presented are mainly specified as bootstrap and normal
approximation algorithms. The tests are general but can be applied in
particular in the area of credit risk, both for regulatory and accounting
purposes.Comment: 23 pages, some typos correcte
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