International Electronic Journal of Mathematics Education
Volume 4, Number 3, October 2009 www.iejme.com
Special issue on “Research and Developments in Probability Education” Manfred Borovcnik & Ramesh Kapadia (Eds)
GLOSSARY TO
PARALLEL DISCUSSION OF CLASSICAL AND BAYESIAN WAYS AS AN INTRODUCTION TO STATISTICAL INFERENCE
Ödön Vancsó
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FREQUENTLY USED TERMS IN THE ARTICLE For the basic concepts, we refer to Wikipedia, The Free Encyclopedia. The most relevant issues are assembled here for the ease of reading; links to Wikipedia are supplied to facilitate further study.
Wikipedia Glossary Bayesian statistics Bayesian statistics Posterior probability Credible intervals or Bayesian regions of highest density (RHD) Confidence intervals General issues Conceptual basis of confidence intervals Critique and comparison of Bayesian and classical methods Distinction between Bayesian frequentist intervals Alternatives and critiques to classical confidence intervals
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ISSN: 1306-3030 G-2 International Electronic Journal of Mathematics Education / Vol.4 No.3, October 2009
Confidence intervals
From Wikipedia, The Free Encyclopedia, http://en.wikipedia.org/wiki/Confidence_interval
“In statistics, a confidence interval (CI) is an interval estimate of a population parameter. Instead of estimating the parameter by a single value, an interval likely to include the parameter is given. Thus, confidence intervals are used to indicate the reliability of an estimate. How likely the interval is to contain the parameter is determined by the confidence level or confidence coefficient. Increasing the desired confidence level will widen the confidence interval.
A confidence interval is always qualified by a particular confidence level (say, γ), usually expressed as a percentage; thus one speaks of a "95% confidence interval". The end points of the confidence interval are referred to as confidence limits. For a given estimation procedure in a given situation, the higher the value of γ, the wider the confidence interval will be.
The calculation of a confidence interval generally requires assumptions about the nature of the estimation process – it is primarily a parametric method – for example, it may depend on an assumption that the distribution of errors of estimation is normal. As such, confidence intervals as discussed below are not robust statistics, though modifications can be made to add robustness – see robust confidence intervals.
Confidence intervals are used within Neyman-Pearson (frequentist) statistics; in Bayesian statistics a similar role is played by the credible interval, but the credible interval and confidence interval have different conceptual foundations and in general they take different values. As part of the general debate between frequentism and Bayesian statistics, there is disagreement about which of these statistics is more useful and appropriate, as discussed in alternatives and critiques.”
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Glossary to Vancsó G-3
Conceptual basis of confidence intervals
From Wikipedia, The Free Encyclopedia, http://en.wikipedia.org/wiki/Confidence_interval#Conceptual_basis
In this bar chart, the top ends of the bars indicate observation means and the red line segments represent the confidence intervals surrounding them.
“Interval estimates can be contrasted with point estimates. A point estimate is a single value given as the estimate of a population parameter that is of interest, for example the mean of some quantity. An interval estimate specifies instead a range within which the parameter is estimated to lie. Confidence intervals are commonly reported in tables or graphs along with point estimates of the same parameters, to show the reliability of the estimates.
For example, a confidence interval can be used to describe how reliable survey results are. In a poll of election voting-intentions, the result might be that 40% of respondents intend to vote for a certain party. A 90% confidence interval for the proportion in the whole population having the same intention on the survey date might be 38% to 42%. From the same data one may calculate a 95% confidence interval, which might in this case be 36% to 44%. At a given level of confidence, and all other things being equal, a result with a smaller CI is more reliable than a result with a larger CI. A major factor determining the width of a confidence interval is the size of the sample used in the estimation procedure, for example the number of people taking part in a survey. […]
In applied practice, confidence intervals are typically stated at the 95% confidence level.[3] However, when presented graphically, confidence intervals can be shown at several confidence levels, for example 50%, 95% and 99%.”
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Credible interval
From Wikipedia, The Free Encyclopedia, http://en.wikipedia.org/wiki/Credible_interval
“In Bayesian statistics, a credible interval is a posterior probability interval[1] which is used for interval estimation in contrast to point estimation. Credible intervals are used for purposes similar to those of confidence intervals in frequentist statistics and an alternative terminology is to use Bayesian confidence interval instead of "credible interval".[2] When dealing with more than one unknown quantity simultaneously, the term credible region is used.
For example, a statement such as "following the experiment, a 90% credible interval for the parameter t is 35–45" means that the posterior probability that t lies in the interval from 35 to 45 is 0.9.
There are several ways of defining a credible interval from a given probability distribution for the parameter. Examples include:
• Choosing the narrowest interval, which for a unimodal distribution will involve choosing those values of highest probability density including the mode.
• Choosing the interval where the probability of being below the interval is as likely as being above it; the interval will include the median.
• Choosing the interval which has the mean as its central point, assuming the mean exists.
It is possible to frame the choice of a credible interval within decision theory and, in that context, an optimal interval will always be a highest probability density set.[3]”
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Distinction between a Bayesian credible interval and a frequentist confidence interval
From Wikipedia, The Free Encyclopedia, http://en.wikipedia.org/wiki/Credible_interval
“A frequentist 90% confidence interval of 35-45 means that with a large number of repeated samples, 90% of the calculated confidence intervals would include the true value of the parameter. The probability that the parameter is inside the given interval (say, 35-45) is either 0 or 1 (the non-random unknown parameter is either there or not). In frequentist terms, the parameter is fixed (cannot be considered to have a distribution of possible values) and the confidence interval is random (as it depends on the random sample). Antelman (1997, p. 375) summarizes a confidence interval as "... one interval generated by a procedure that will give correct intervals 95 % [resp. 90 %] of the time". [4]
In general, Bayesian credible intervals do not coincide with frequentist confidence intervals for two reasons:
• Credible intervals incorporate problem-specific contextual information from the prior distribution whereas confidence intervals are based only on the data.
• Credible intervals and confidence intervals treat nuisance parameters in radically different ways.
Many professional statisticians and decisions scientists as well as non-statisticians intuitively interpret confidence intervals in the Bayesian credible interval sense and hence "credible intervals" are sometimes called "confidence intervals". It is widely accepted, especially in the decision sciences, that "credible interval" is merely the subjective subset of "confidence intervals". In fact, much research in calibrated probability assessments never uses the term "credible interval" and it is common to simply use "confidence interval".
References
1. ^ Edwards, W., Lindman, H., Savage, L.J. (1963) Bayesian statistical inference in statistical research. Psychological Research, 70, 193–242. 2. ^ Lee, P. M. (1997) Bayesian Statistics: An Introduction, Arnold. ISBN 0-340-67785-6. 3. ^ O'Hagan, A. (1994) Kendall's Advance Theory of Statistics, Vol 2B, Bayesian Inference, Section 2.51. Arnold, ISBN 0-340-52922-9. 4. ^ Antelman, G. (1997) Elementary Bayesian Statistics (Madansky, A. & McCulloch, R. eds.). Edward Elgar, Cheltenham, UK. ISBN 978-1-85898-504-6. “
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Alternatives and critiques to classical confidence intervals
From Wikipedia, The Free Encyclopedia, http://en.wikipedia.org/wiki/Confidence_interval#Alternatives_and_critiques
“Main articles: Credible interval and Prediction interval
Confidence intervals are one method of interval estimation, and the most widely used in frequentist statistics. An analogous concept in Bayesian statistics is credible intervals, while an alternative frequentist method is prediction interval, which, rather than estimating parameters, estimates the outcome of future samples.
There is disagreement about which of these methods produces the most useful results: the mathematics of the computations are rarely in question – confidence intervals being based on sampling distributions, credible intervals being based on Bayes' theorem – but the application of these methods, the utility and interpretation of the produced statistics, is debated.
Users of Bayesian methods, if they produced an interval estimate, would in contrast to confidence intervals, want to say "My degree of belief that the parameter is in fact in this interval is 90%,"[8] while users of prediction intervals would instead say "I predict that the next sample will fall in this interval 90% of the time."
When confidence intervals are generated in order to perform statistical tests, there is an assumption inherent in the mathematics that only one test is being performed. When a study involves multiple statistical tests, this assumption is false: the probability of the confidence interval containing the parameter can be much less than the confidence level. For example, if a study involving 20 statistical tests produces one positive result, the confidence level of the interval may be 95% but the chance that that interval contains the parameter is actually only about 35%.”
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Author : Ödön Vancsó E-mail : [email protected]
Address : Institute for Mathematics, Eötvös Lóránd University, Budapest, Hungary