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## Direct measurement in the absence of systematic errors

The first step consists in evaluating the uncertainty on a quantity measured directly. The most common likelihoods which describe the observed values are the Gaussian, the binomial and the Poisson distributions.
Gaussian:
This is the well-known case of `normally' distributed errors. For simplicity, we will only consider independent of (constant r.m.s. error within the range of measurability), but there is no difficulty of principle in treating the general case. The following cases will be analysed:
• inference on starting from a prior much more vague than the width of the likelihood (Section );
• prior width comparable with that of the likelihood (Section ): this case also describes the combination of independent measurements;
• observed values very close to, or beyond the edge of the physical region (Section );
• a method to give unbiased estimates will be discussed in Sections and , but at the cost of having to introduce fictitious quantities.
Binomial:
This distribution is important for efficiencies and, in the general case, for making inferences on unknown proportions. The cases considered include (see Section ):
• general case with flat prior leading to the recursive Laplace formula (the problem solved originally by Bayes);
• limit to normality;
• combinations of different datasets coming from the same proportion;
• upper and lower limits when the efficiency is 0 or 1;
• comparison with Poisson approximation.
Poisson:
The cases of counting experiments here considered2.15 are (see Section ):
• inference on starting from a flat distribution;
• upper limit in the case of null observation;
• counting measurements in the presence of a background, when its rate is well known (Sections and );
• more complicated case of background with an uncertain rate (Section );
• dependence of the conclusions on the choice of experience-motivated priors (Section );
• combination of upper limits, also considering experiments of different sensitivity (Section ).
• effect of possible systematic errors (Section );
• a special section will be dedicated to the lower bounds on the mass of a new hypothetical particle from counting experiments and from direct information (Section ).

Next: Indirect measurements Up: Evaluation of uncertainty: general Previous: Evaluation of uncertainty: general   Contents
Giulio D'Agostini 2003-05-15