15 Statistical Issues in Particle Physics
649
where p(A) is the probability or probability density of A, and p(A|B) is the
conditional probability for A, given that B has happened. This formula is acceptable
to frequentists, provided the probablities are frequentist probabilities. However
Bayesians use it with A = parameter (or hypothesis) and B = data. Then
p(parameter|data) ∝ p(data|parameter) × p(parameter),
(15.2)
where the three terms are respectively the Bayesian posterior, the likelihood function
and the Bayesian prior. Thus Bayes’ theorem enables us to use the data (as
encapsulated in the likelihood) to update our prior knowledge (p(parameter)); the
combined information is given by the posterior.
Frequentists object to the use of probability for physical parameters. Furthermore, even Bayesians agree that it is often hard to specify a sensible prior. For a
parameter which has been well determined in the past, a prior might be a gamma
function or log-normal or a (possibly truncated) Gaussian distribution of appropriate
central value and width, but for the case where no useful information is available the
choice is not so clear; it is easier to parametrise prior knowledge than to quantify
prior ignorance. The ‘obvious’ choice of a uniform distribution has the problem of
being not unique (Should our lack of knowledge concerning, for example, the mass
of a neutrino m ν be parametrised by a uniform prior for m ν or for m 2
ν or for log m ν ,
etc?). Also a uniform prior over an infinite parameter range cannot be normalised.
For situations involving several parameters, the choice of prior becomes even more
problematic.
It is important to check that conclusions about possible parameter ranges are
not dominated by the choice of prior. This can be achieved by changing to other
‘reasonable’ priors (sensitivity analysis); or by looking at the posterior when the
data has been removed.
15.1.3.3 Frequentist Approach: Neyman Construction
The frequentist way of constructing intervals completely eliminates the need for
a prior, and avoids considering probability distributions for parameters. Consider a
measurement x which provides information concerning a parameter μ. For example,
we could use a month’s data from a large solar neutrino detector (x) to estimate the
temperature at the centre of the sun (μ). It is assumed that enough is known about
solar physics, fusion reactions, neutrino properties, the behaviour of the detector,
etc. that, for any given value of μ, the probability density for every x is calculable.
Then for that μ, we can select a region in x which contains, say, 90% of this
probability. If we do this for every μ, we obtain a 90% confidence band; it shows
649
where p(A) is the probability or probability density of A, and p(A|B) is the
conditional probability for A, given that B has happened. This formula is acceptable
to frequentists, provided the probablities are frequentist probabilities. However
Bayesians use it with A = parameter (or hypothesis) and B = data. Then
p(parameter|data) ∝ p(data|parameter) × p(parameter),
(15.2)
where the three terms are respectively the Bayesian posterior, the likelihood function
and the Bayesian prior. Thus Bayes’ theorem enables us to use the data (as
encapsulated in the likelihood) to update our prior knowledge (p(parameter)); the
combined information is given by the posterior.
Frequentists object to the use of probability for physical parameters. Furthermore, even Bayesians agree that it is often hard to specify a sensible prior. For a
parameter which has been well determined in the past, a prior might be a gamma
function or log-normal or a (possibly truncated) Gaussian distribution of appropriate
central value and width, but for the case where no useful information is available the
choice is not so clear; it is easier to parametrise prior knowledge than to quantify
prior ignorance. The ‘obvious’ choice of a uniform distribution has the problem of
being not unique (Should our lack of knowledge concerning, for example, the mass
of a neutrino m ν be parametrised by a uniform prior for m ν or for m 2
ν or for log m ν ,
etc?). Also a uniform prior over an infinite parameter range cannot be normalised.
For situations involving several parameters, the choice of prior becomes even more
problematic.
It is important to check that conclusions about possible parameter ranges are
not dominated by the choice of prior. This can be achieved by changing to other
‘reasonable’ priors (sensitivity analysis); or by looking at the posterior when the
data has been removed.
15.1.3.3 Frequentist Approach: Neyman Construction
The frequentist way of constructing intervals completely eliminates the need for
a prior, and avoids considering probability distributions for parameters. Consider a
measurement x which provides information concerning a parameter μ. For example,
we could use a month’s data from a large solar neutrino detector (x) to estimate the
temperature at the centre of the sun (μ). It is assumed that enough is known about
solar physics, fusion reactions, neutrino properties, the behaviour of the detector,
etc. that, for any given value of μ, the probability density for every x is calculable.
Then for that μ, we can select a region in x which contains, say, 90% of this
probability. If we do this for every μ, we obtain a 90% confidence band; it shows