Detector response and likelihood inference

Let $t_j$ be the expected produced signal count in true bin $j$, and let $R_{ij}$ be the probability that such an event passes selection and lands in reconstructed bin $i$. With that convention a column sum can be below one because missed events are omitted. If a different response matrix is normalized only among selected events, multiply a separate efficiency instead. Applying both conventions together would double-count inefficiency and bias the inferred rate.

Detector response depends on the process and its unresolved kinematics, not only on the plotted axis. Calibrations use control data and simulation to constrain energy scales, resolutions, and identification efficiencies. Forward folding predicts measured bins directly, while unfolding estimates a specified physical-level distribution through an inverse problem. Regularization can stabilize that inverse problem but introduces correlations and possible model dependence; unfolded points are not automatically independent raw measurements.

For disjoint event-count bins, a common likelihood is a product of Poisson probabilities with expectations $\nu_i$, multiplied by likelihood terms for auxiliary calibration and control measurements. Nuisance parameters encode quantities such as luminosity, background normalization, or energy response. A shared nuisance couples bins and channels; treating each appearance of a common calibration error as independent can artificially increase or decrease the claimed information. Write the generative assumptions before choosing an optimizer.

A likelihood is a function of parameters with the observed data fixed. It is not a normalized probability that a hypothesis is true. Profiling maximizes over nuisance parameters at each tested signal value, whereas marginalization integrates them with a stated prior in a Bayesian analysis. Both operations can be defensible, but they answer different questions. A plotted error bar should state its construction and coverage or posterior interpretation rather than inheriting meaning from the word "sigma" alone.

Large-sample likelihood approximations can simplify confidence intervals and significance calculations, but boundaries such as nonnegative signal strength and very sparse counts can invalidate a naive chi-squared rule. Pseudoexperiments or exact constructions can test coverage in the relevant regime. An auxiliary constraint also represents measured information, not a magical penalty that repairs a badly specified background shape. Validate predictions in control regions and examine goodness of fit alongside parameter estimates.

Searching many masses or selections changes the probability of finding an apparently unusual fluctuation somewhere. A local discrepancy and a global search significance answer different questions. Blinding and prespecifying procedures can reduce choices driven by a fluctuation, while independent data provide a stronger test. Report the fiducial definition, response assumptions, nuisance correlations, and treatment of repeated searches so another analyst can understand which physical claims the data actually support.