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 $\sigma$ independent of $\mu$ (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 $\mu$ 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 considered^2.15 are (see Section ):

• inference on $\lambda$ 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 ).