The construction of confidence belts s based on defining horizontal intervals according to a certain confidence level $C$. These can be constructed as central confidence intervals according to

$$ P(x<x_1|\mu)=P(x>x_2|\mu)=(1-C)/2 $$

or as upper confidence limit intervals

$$ P(x<x_1|\mu)=P(x>x_2|\mu)=(1-C)/2 $$

For a given measurement value of $x_0$ these then lead to an interval for $\mu$ with

$$ P(\mu\in[\mu_1,\mu_2])=C $$

This statement means that the unknown true value of $\mu,\mu_t$ lies within the interval $[\mu_1,\mu_2]$ in a fraction $C$ of the experiment conducted

Measurement of a constrained quantity

Bayesian construction with a normalisation that takes the physical limit of $\mu>0$ into account

$$ P(\mu|x)=\frac{e^{-(x-\mu)^2/2\sigma^2}}{\int_0^\infty e^{-(x-\mu')^2/2\sigma^2}d\mu'}(\mu>0) $$

This construction will then lead to one limit being zero, i.e. we set an upper limit

The issue of flip-flopping

❌ Describes choosing between central interval, upper limit etc depending on the measured value. Don’t do this!

The Feldman-Cousins method

it proceeds as follows ( example: Poisson with positive true value and background count $b$ )

• Consider the construction of a confidence interval of level $C$ for a given $\mu$
• For each value of $k$ there is a value of $\mu$ that maximises $P(k|\mu)$ and this value shall be called $\mu_\text{best}$
• In our case this is simply $\mu_\text{best}=\text{max}(0,k-b)$
• Define the ratio $R=P(k|\mu)/P(k|\mu_\text{best})$ given that $\mu_\text{best}$ maximises the probability distribution function this ratio is between 0 and 1
• Order all possible values of $k$ by their corresponding values of $R$ and starting from the largest $R$ add the values of $k$ to the confidence interval until the sum of their probabilities exceeds $C$

Application to Gaussian confidence belts

For a Gaussian (i.e. continuous) variable the procedure is very similar:

• For a Gaussian distributed variable $\mu$ with boundary condition $\mu\ge0$ find the value of $\mu$ for which $P(x|\mu)$ is maximal and call this $\mu_\text{best}$. This is given by $\mu_\text{best}=\text{max}(0,x)$

• Define the likelihood ratio

  $$ R(x)=\frac{P(x|\mu)}{P(x|\mu_{\rm best})} $$

• For a given $\mu$ find the interval $[x_1,x_2]$ such that $R(x_1)=R(x_2)$ and $\int^{x_2}_{x_1}P(x|\mu) \,\text dx=C$ with $C$ the desired confidence level. This integral gives the accepted range for each value of $\mu$

🗒️ Note: at $x=1.28$ the distribution transitions from a one-sided to a two sided interval