In this scenario, we expect μbg(θ) background events, distributed in energy as pbg(E∣θ). Similarly, we expect μsig(θ) signal events, distributed in energy as psig(E∣θ). θ denotes the parameters of this model, such as the signal strength, detector parameters, background rate, etc. Now, suppose we observe Nobs data points with energies X={Ei}.
Given such a model and data vector we would write Bayes' theorem as:
p(θ∣X)=p(X)p(X∣θ)p(θ).
Here, p(θ∣X) is the posterior distribution we want, p(X∣θ) is the likelihood function, and p(θ) is the prior. We can think of the denominator, p(X), as a normalisation constant; after all, probability distributions like our posterior have to integrate to 1, and our prior multiplied by our likelihood is not guaranteed to do so.
What is the explicit form of our likelihood? We can think about how we would write down p(X∣θ) explicitly. We would read this as the probability density of our data points, X, given a particular set of model parameters, θ. This tells us all we need to know!
Firstly, the total number of events should be Poisson distributed. We earlier stated that there are signal and background means μsig(θ) and μbg(θ); these summed together should be the Poisson rate parameter λ. In addition, we know the PDF as a function of energy for both the signal and the background components as psig(E∣θ) and pbg(E∣θ). (In a practical modelling problem, these will all be specified explicitly! See the colab notebook.)
We can thus write the likelihood as such:
p(X∣θ)=Poisson term for total number of eventsN!(μsig(θ)+μbg(θ))Ne−(μsig(θ)+μbg(θ))Likelihood for event energies ×μsig(θ)+μbg(θ)1i∏N(μsig(θ)psig(Ei∣θ)+μbg(θ)pbg(Ei∣θ)).
Cranmer, Kyle, Johann Brehmer, and Gilles Louppe. "The frontier of simulation-based inference." Proceedings of the National Academy of Sciences 117.48 (2020): 30055-30062.