unbinned observables as compared to their binned $p_{T}^{Z}$ counterparts is
moderate, yet appreciable, for most operator pairs. The most stringent bounds
are those associated with unbinned observables, and the constraints provided
by the multivariate unbinned observable based on 7 kinematic features are
similar to those associated to the single-feature version trained on
$p_{T}^{Z}$. One observes strong correlations between all operators pairs
being considered, indicating quasi-flat directions: these will be broken
either by quadratic corrections or by adding other processes sensitive to the
same SMEFT operators, such as vector-boson scattering and diboson data [16].
The analysis of Fig. 5.6 indicates that for the $hZ\rightarrow
b\bar{b}\ell^{+}\ell^{-}$ process the chosen binning in $p_{T}^{Z}$ provides
the bulk of the information on the EFT operators, with minor additional
improvements provided by the unbinned multivariate analysis.

Therefore, for this specific process the formalism developed in this work
provides a diagnosis tool that determines when and under which circumstances a
binned analysis provides information on the EFT parameter space close to the
optimal one, represented here by the bounds associated to the unbinned
multivariate observable. This finding illustrates the two-fold applicability
of our formalism: whenever unbinned multivariate observables markedly improve
over their binned counterparts, they can be included in the global SMEFT fit,
otherwise one obtains quantitative confirmation that the information provided
by the specific settings of the binned observables is sufficiently close to
the optimal amount.

The results of the corresponding analyses of the impact of unbinned
observables in the $hZ\rightarrow b\bar{b}\ell^{+}\ell^{-}$ process in the
presence of quadratic EFT corrections are shown in Fig. 5.7 for the case of
two-parameter fits, and in Fig. 5.8 for the case where all relevant operators
are considered simultaneously, producing marginalized constraints on each
operator pair. As mentioned above, the latter is possible since quadratic
corrections break up the EFT parameter degeneracies that destabilize the
inference procedure in the linear case.

Considering first the results of Fig. 5.7, one finds that on the one hand the
improvement associated to the use of unbinned observables is more marked as
compared to the linear EFT case. On the other hand, the addition of other
kinematic features on top of $p_{T}^{Z}$ does not modify the discrimination
power of the unbinned observables, as also observed for the linear analysis.
For specific operator pairs, there is a clear reduction in the parameter
bounds based on the unbinned observables, for instance in the case of the
$\left(c_{\varphi W},c_{\varphi WB}\right)$ pair. We note that enhanced
discrimination in the presence of quadratic EFT corrections can be partially
attributed to energy-growing effects in the $p_{T}^{Z}$ distribution which are
more pronounced as compared to the linear level.

Another benefit associated to unbinned observables in this specific process is
that of breaking degeneracies leading to a double maximum structure in the
posterior distribution. Indeed, the unbinned observables removes the second
minimum present in the binned analysis arising from a bimodal distribution in
the ($c_{\varphi q}^{(3)}$, $c_{b\varphi}$)-plane and corresponding to the
scenario where the quadratic EFT contribution becomes of the same magnitude
and opposite sign as the linear term and hence canceling it. We also note
that, as opposed to the $t\bar{t}$ production case, for the $hZ$ process, even
with unbinned observables, one ends up with some operator pairs that exhibit
relatively large correlations which would thus only be broken by adding other
processes to the fit. The results of Fig. 5.7 highlight how in general the
relevance of considering unbinned observables depends not only on the
processes and on the order in the EFT expansion, but also on the specific
directions in the EFT parameter space in which one is interested.

The results of Fig. 5.7 display good agreement between unbinned observables
trained only on $p_{T}^{Z}$ and on the full set of $n_{k}=7$ kinematic
features, suggesting that a multivariate analysis does not appear to be
beneficial for this process. However, this interpretation turns out to be an
artifact of restricting the analysis to the case of two-parameter fits, and a
different picture emerges in the case of the marginalized analysis results
displayed in Fig. 5.8. The impact on the EFT parameter space of adopting
unbinned observables is more important now, leading in some cases to stringent
bounds improved by up to a factor 5 as compared to the binned results, for
instance for the $c_{\varphi W}$ operator. Furthermore, one can also
appreciate the improved constraints associated to the multivariate observable
based on the full set of kinematic features as compared to the single-feature
unbinned observable. We also note how the degenerate minimum associated to the
bottom Yukawa operator present in the binned analysis, now for all operator
pairs, goes away when unbinned observables are considered. The comparison of
Figs. 5.7 and 5.8 emphasizes that bounds obtained in two-parameter fits,
neglecting the effects of other operators, will in general overestimate the
EFT constraints associated to a given dataset.

Figure 5.8: Same as Fig. 5.7 now where the $95\%$ CL contours are obtained
from the marginalisation over the full posterior probability distribution,
rather than from two-parameter fits.

To summarize, as in the case of top quark pair production, for $hZ$ production
the use of unbinned multivariate observables is advantageous in constraining
the EFT parameter space. Our analysis indicates that the choice of $p_{T}^{Z}$
binning adopted by the STXS analysis is close to being optimal, especially in
the linear case, at least for the assumed integrated luminosity and for two-
parameter fits. When considering a full marginalized analysis in the presence
of quadratic EFT corrections, marked improvements associated first to the use
of unbinned observables and second to that of multiple kinematic features are
obtained. The use of unbinned observables also results in the removal of
degenerate minima corresponding to solutions where the quadratic EFT
correction approximately cancels out the linear one. The comparison between
the output of the two-parameter and the marginalized fits emphasizes the
crucial importance of carrying out global SMEFT interpretations, since setting
to zero a subset of operators that contribute to a given process is likely to
overestimate its impact in the EFT parameter space.

###  5.5 Methodological uncertainties

The results for the unbinned observables derived so far are obtained from EFT
parameter inference carried out with Eq. (3.19), the parametrization of the
likelihood ratio, as described in Sect. 5.1. As explained in Sect. 3.3, within
our approach, rather than a single best model, we produce a distribution of
models, denoted as replicas, each of them trained on a different set of Monte
Carlo events. Hence the end result of our procedure is Eq. (3.23), the
representation of the probability distribution of the likelihood ratio
$\\{\hat{r}^{(i)}_{\sigma}(\bm{x},\bm{c})\\}$ composed of $N_{\rm rep}$
equiprobable replicas. The spread of this distribution provides a measure of
methodological and procedural uncertainties associated e.g. to finite training
datasets and inefficiencies of the optimization and stopping algorithms.
Results presented in Sects. 5.2–5.4 are based on using the median of the
replica distribution $\\{\hat{r}^{(i)}_{\sigma}(\bm{x},\bm{c})\\}$ to
determine the bounds in the EFT parameter space, and here we assess the impact
of methodological uncertainties by displaying results for parameter inference
based on the full distribution of replicas of the likelihood ratio
parametrization.

Figs. 5.9 and 5.10 display the bounds in the parameter space obtained for
particle-level top-quark pair and $hZ$ associated production, respectively,
from the unbinned multivariate observables based on the complete set of
kinematic features and in the case of theory simulations that include
quadratic EFT corrections. We compare results for parameter inference based on
the median of the replica distribution
$\\{\hat{r}^{(i)}_{\sigma}(\bm{x},\bm{c})\\}$, which coincide with those of
Figs. 5.5 and 5.8, with results based on the full distribution of
replicas.444We note that for top quark pair production we display the 68% CL
contours in both cases. The reason is that for $N_{\rm rep}=25$ the evaluation
of 95% CL contours over the replica distribution becomes overly sensitive to
outliers. Namely, in the latter case one starts from the $N_{\rm rep}$
individual replicas of the profile likelihood ratio functions and performs
Nested Sampling inference on each of them, to subsequently combine the
resulting samples and estimate the posterior distribution by means of the KDE
method. In this manner, the differences between the contours shown in Figs.
5.9 and 5.10 provide an estimate of how methodological uncertainties
associated to the machine learning training procedure impact the derived
bounds in the EFT parameter space.

Figure 5.9: Same as Fig. 5.5 for the 68% CL contours in the EFT parameters
obtained with the unbinned multivariate observable for particle-level top-
quark pair production. We compare the bounds obtained from the median of the
replica distribution of the likelihood ratio parametrization, as done in Fig.
5.5, with the corresponding bounds obtained taking into account the full
replica distribution.  Figure 5.10:  Same as Fig. 5.9 for Higgs associated
production with a $Z$ boson, now for the 95% CL contours.

Inspection of Figs. 5.9 and 5.10 and comparison with the corresponding bounds
displayed in Figs. 5.5 and 5.8 confirm that these procedural uncertainties, as
estimated with the replica method, do not modify the qualitative results
obtained from the median of the likelihood ratio. In particular, the
observation that bounds obtained using multivariate unbinned observables are
much stronger than those based on unbinned models based on one or two
kinematic features, which is common to both the $t\bar{t}$ and $hZ$ processes,
remains valid once replica uncertainties are accounted for. Furthermore, we
note that in principle it is possible to reduce the spread of the replica
distribution by training on higher-statistics samples and adopting more
stringent stopping criteria. In this respect, our approach provides a strategy
to quantify under which conditions the computational overhead required to
achieve more accurate machine learning trainings is justified from the point
of view of the impact on the EFT coefficients. This said, these results also
indicate that in realistic scenarios based on finite training samples
methodological uncertainties cannot be neglected, and should be accounted for
in studies of the impact of ML-based observables in EFT analyses.

##  6 Summary and outlook

The general problem of identifying novel types of measurements which, given a
theoretical framework, provide enhanced or even maximal sensitivity to the
parameters of interest is ubiquitous in modern particle physics. Constructing
such optimized observables has a two-fold motivation: on the one hand, to
achieve the most stringent constraints on the model parameters from a specific
process, and on the other hand, to provide bounds on the maximum amount of
information that can be extracted from the same process. Optimized observables
can hence be used to design traditional observables in a way that approaches
or saturates the limiting sensitivity by highlighting the best choices of
binning and kinematic features.

In this work, we have presented a new framework for the design of optimal
observables for EFT applications at the LHC, making use of machine learning
techniques to parametrize multivariate likelihoods for an arbitrary number of
higher-dimensional operators. To illustrate the reach of our method, we have
constructed multivariate unbinned observables for top-quark pair production
and Higgs production in association with a $Z$ boson. We have demonstrated how
these observables either lead to a significant improvement in the constraints
on the EFT parameter space, or indicate the conditions upon which a
traditional binned analysis already saturates the EFT sensitivity.

The ML4EFT framework presented in this work and the accompanying simulated
event samples are made available as an open source code which can be
interfaced with existing global EFT fitting tools such as SMEFiT, FitMaker
[17], HepFit [14], and Sfitter [79]. Its scaling behavior with the number of
parameters, together with its parallelization capabilities, make it suitable
for its integration within global EFT fits which involve several tens of
independent Wilson coefficients. While in its current implementation the
parametrized likelihood ratios are provided in terms of the output of the
trained network replicas, work in progress aims to tabulate this output in
terms of fast interpolation grids, as done customarily in the case of PDF
analyses [81, 82, 83, 84]. The resulting interpolated unbinned likelihood
ratios can then be combined with Gaussian or Poissonian binned measurements
within a global fit. While currently ML4EFT can only be used in combination
with simulated Monte Carlo pseudo-data, all of the ingredients required for
the analysis of actual LHC measurements are already in place.

The results presented in this work could be extended along several directions.
First, one could include experimental and theoretical correlated systematic
uncertainties, as required for the interpretation of measurements which are
not dominated by statistics. Second, other machine learning algorithms could
be adopted, which may offer performance advantages as compared to those used
here: one possibility could be graph neural networks [85, 86] which make
possible varying the kinematic features used for the training on an event-by-
event basis. Third, ML4EFT could be applied to more realistic final states
with higher order corrections, such as by means of NNLO+PS simulations of
$t\bar{t}$ and $hV$ [87, 88, 89], and accounting for detector effects to
bridge the gap between the theory predictions and the experimentally
accessible quantities. It would also be interesting to compare the performance
of different approaches to construct ML-assisted optimized observables for EFT
applications in specific benchmark scenarios.

Beyond hadron colliders, the framework developed here could also be relevant
to construct optimal observables for EFT analyses in the case of high-energy
lepton colliders, such as electron-positron collisions at CLIC [90, 91] or at
a multi-TeV muon collider [92, 93, 94]. As mentioned, statistically optimal
observables were actually first designed for electron-positron colliders,
where the simpler final state facilitates the calculation of the exact event
likelihood for parameter inference [21, 22, 23]. These methods may not be
suitable to the high multiplicity environment of multi-TeV lepton colliders,
in particular due to the complex pattern of electroweak radiation. Therefore,
the ML4EFT method could provide a suitable alternative to construct unbinned
multivariate observables achieving maximal EFT sensitivity at high-energy
lepton colliders.

In addition to applications to the SMEFT, our framework could also be relevant
to other types of theory interpretations of collider data such as global PDF
fits. In particular, PDFs at large values of Bjorken-$x$ are poorly
constrained [95] due to the limited amount of experimental data available.
This lack of knowledge degrades the reach of searches for both resonant and
non-resonant new physics in the high-energy tail of differential
distributions, as recently emphasized for the case of the forward-backward
asymmetry in neutral-current Drell-Yan production [96]. Given that
measurements of these high-energy tails are often dominated by statistical
uncertainties, the ML4EFT method could be used to construct unbinned
multivariate observables tailored to constrain large-$x$ PDFs at the LHC. Our
method could also be applied in the context of a joint extraction of PDFs and
EFT coefficients [97, 98, 99, 100, 101], required to disentangle QCD effects
from BSM ones in kinematic regions where they potentially overlap, such as
large Bjorken-$x$.

### Acknowledgments

We are grateful to Gaia Grosso, Wouter Verkerke, and Andrea Wulzer for
stimulating discussions. The work of J. t. H is supported by the Dutch Science
Council (NWO) via an ENW-KLEIN-2 project. The research of J. R. is partially
supported by NWO and by an ASDI grant of The Netherlands eScience Center. The
research of V. S. is supported by the Generalitat Valenciana PROMETEO/2021/083
and the Ministerio de Ciencia e Innovacion PID2020-113644GB-I00. The work of
M. M. is supported by the European Research Council under the European Union’s
Horizon 2020 research and innovation Programme (grant agreement n.950246) and
in part by STFC consolidated grant ST/T000694/1. The work of R. G. A. is
supported by the ERC Starting Grant REINVENT-714788.

##  Appendix A The ML4EFT framework

The ML4EFT framework presented in this work is an open source Python code
designed to facilitate the integration of unbinned multivariate observables
into fits of Wilson coefficients in the SMEFT. It is based on machine learning
regression and classification techniques to parameterize high-dimensional
likelihood ratios as required to carry out parameter inference in the context
of global SMEFT analyses. ML4EFT is made available via the Python Package
Index (pip) and can be installed directly by running

pip install ml4eft

or alternatively the code can be downloaded from its public GitHub repository,

https://github.com/LHCfitNikhef/ML4EFT.git

and then installed from source following the installation instructions. The
framework is documented in a dedicated website

https://lhcfitnikhef.github.io/ML4EFT,

where, in addition, one can find a self-standing tutorial (which can also be
run in Google Colab) where the user is guided step by step in how unbinned
multivariate observables can be constructed given a choice of EFT coefficients
and of final-state kinematic features.

In the same website we also provide links to the main results of this paper
presented in Sect. 5, including the likelihood ratio parametrizations that
have been obtained for the two processes ($t\bar{t}$ and $hZ$) considered
here. We also include there animations demonstrating the training of the
neural networks, such as e.g.

https://lhcfitnikhef.github.io/ML4EFT/sphinx/build/html/results/ttbar_analysis_parton2.html#overview

Additional unbinned multivariate observables to be constructed in the future
using our framework will be added to the same page. Furthermore, we also plan
to tabulate the neural network outputs using fast grid techniques so that
these observables can be stand-alone integrated in global fits without the
need to link the actual ML4EFT code.

##  Appendix B The unbinned Asimov data set

The Asimov data set was first introduced in [66] as an efficient method of
obtaining the distribution of the profile likelihood ratio $q_{\bm{c}}$, Eq.
(2.8), under the alternative hypothesis $\bm{c}^{\prime}\neq\bm{c}$ without
having to resort to computationally expensive MC simulations. Since then, it
has become part of the standard analysis toolkit at the LHC to determine e.g.
expected exclusion limits. The analysis of [66] considers the Asimov data set
applied to binned observables and here, by adopting a simple EFT-like toy
model, we present its generalisation to a continuous Asimov data set. This
makes possible determining expected exclusion limits in a sampling free way by
extending the results of [66] to unbinned observables. While we do not make
use of this unbinned Asimov data set in the results presented in this work,
this derivation can play an important role in the general discussion of
unbinned multivariate measurements for EFT studies.

Let us consider a kinematic variable $x$ (e.g. the invariant mass) restricted
to $x\in X=[1,5]$ and distributed according to a probability distribution
$f_{\sigma}(x,c)$. The EFT toy model is chosen such that the effect of the EFT
on the kinematic variable is an enhancement in the tail of the distribution
$f_{\sigma}(x,c)$ with respect to the SM, as is common in many EFT scenarios.
The distribution $f_{\sigma}(x,c)$ is defined as

$f_{\sigma}(x,c)\equiv{{1}\over{\sigma(X,c)}}\left[f_{\sigma}^{(\mathrm{sm})}(x)+0.1c\cdot
f_{\sigma}^{(\mathrm{eft})}(x)\right],$ (B.1)

with

$f_{\sigma}^{(\mathrm{sm})}(x)={{1}\over{\sigma^{(\mathrm{sm})}}}{{(1+x^{2})\sqrt{x^{2}-1}}\over{x^{4}}}$
(B.2)

and

$f_{\sigma}^{(\mathrm{eft})}(x)={{1}\over{\sigma^{(\mathrm{eft})}}}{{(x^{2}-1)^{3/2}}\over{x}},$
(B.3)

where $f_{\sigma}^{(\mathrm{sm})}(x)$ and $f_{\sigma}^{(\mathrm{eft})}(x)$ are
both separately normalized by $\sigma^{(\mathrm{sm})}$ and
$\sigma^{(\mathrm{eft})}$ respectively such that the overall normalization
constant $\sigma(X,c)$ is given by

$\sigma(X,c)=1+0.1c.$ (B.4)

The number of expected events $\nu(c)$ is determined from $\sigma(X,c)$ and
the luminosity $L$, which we set to $L=1000$ unless otherwise specified. We
work with dimensionless parameters throughout this toy example.

In Fig. B.1(a) we show the distribution $f_{\sigma}(x,c)$ at $c=0$ and $c=1$,
corresponding to the SM and the EFT hypotheses respectively. One can see a
clear energy growing effect in the tail of the EFT distribution. The
histograms have been constructed using accept/reject sampling and contain 1M
samples from each hypothesis. These will serve as our parent data sets from
which we will take subsequent samples in the following.

(a)

(b)

Figure B.1: Left: the probability distribution $f_{\sigma}(x,c)$ from Eq.
(B.1) at $c=1$ and $c=0$, corresponding to the EFT and SM respectively. Right:
the MLE $\hat{c}$ under the SM follows a Gaussian distribution centered around
$c^{\prime}$ with standard deviation $\sigma$.

Now, taking the EFT at $c=1$ as our null hypothesis, the goal is to compute
the expected significance at which we can either accept or reject the null
assuming data generated under the SM (the alternative hypothesis). We adopt
the PLR from Eq. (2.8) as our test statistic,

$q_{c}=-2\left[\nu(\hat{c})-\nu(c)+\sum_{i=1}^{N_{\mathrm{ev}}}\log{{\nu(c)f_{\sigma}(x,c)}\over{\nu(\hat{c})f_{\sigma}(x,\hat{c})}}\right],$
(B.5)

whose distribution under the SM and the EFT we denote by
$\mathrm{pdf}(q_{c}|0)$ and $\mathrm{pdf}(q_{c}|c)$ respectively. One can
imagine repeatedly drawing toy datasets from the SM parent data set, computing
$q_{c}$, and evaluating its $p$-value under the null. The median p-value then
serves as a measure of expected sensitivity. Moreover, since the $p$-value is
monotonic with $q_{c}$, it is equivalent to report the $p$-value associated
with the median $q_{c}$ under the SM, denoted $\mathrm{Med}[q_{c}|0]$,

$p_{c,\mathrm{med}}=\int_{q_{c,\mathrm{med}}}^{\infty}\mathrm{pdf}(q_{c}|c)dq_{c}\qquad\text{with}\qquad
q_{c,\mathrm{med}}\equiv\mathrm{Med}[q_{c}|0].$ (B.6)

From the above discussion it becomes clear one not only needs
$\mathrm{pdf}(q_{c}|c)$, which follows a $\chi^{2}$-distribution as discussed
in Sec. 2.1, but also $\mathrm{pdf}(q_{c}|c^{\prime})$ with $c^{\prime}\neq
c$. At this point we employ Wald’s theorem [102], which states that for a data
set of size $N$ generated under $f_{\sigma}(x,c^{\prime})$, the PLR $q_{c}$
can be approximated by

$q_{c}={{(c-\hat{c})^{2}}\over{\sigma^{2}}}+\mathcal{O}\left(1/\sqrt{N}\right),$
(B.7)

where $\hat{c}$ follows a normal distribution with mean $c^{\prime}$ and a
variance $\sigma^{2}$, i.e. $\hat{c}\sim\mathcal{N}(c^{\prime},\sigma)$. Not
only does the approximation provided by Wald’s theorem depend on a large
sample size $N$ [102], but the approximation improves as
$c\rightarrow\hat{c}\simeq c^{\prime}$. This can be understood by Taylor
expanding Eq. (2.8) about $\hat{c}$. Defining $\Delta\equiv(c-\hat{c})$, we
find

$\displaystyle q_{c}=-2\log{{\mathcal{L}(c)}\over{\mathcal{L}(\hat{c})}}$
$\displaystyle=-2\log\left[1+{{1}\over{2\mathcal{L}(\hat{c})}}\left.{{\partial^{2}\mathcal{L}}\over{\partial
c^{2}}}\right|_{c=\hat{c}}\Delta^{2}+\mathcal{O}\left(\Delta^{3}\right)\right]$
$\displaystyle=-{{1}\over{\mathcal{L}(\hat{c})}}\left.{{\partial^{2}\mathcal{L}}\over{\partial
c^{2}}}\right|_{c=\hat{c}}\Delta^{2}+\mathcal{O}\left(\Delta^{3}\right)={{\Delta^{2}}\over{\sigma^{2}}}+\mathcal{O}\left(\Delta^{3}\right),$

where on the last line we have used that

${{1}\over{\sigma^{2}}}=-\left.{{\partial^{2}\log\mathcal{L}}\over{\partial
c^{2}}}\right|_{c=\hat{c}}.$ (B.8)

Let us define $q_{c}^{\mathrm{wald}}$ as the leading order approximation of
Eq. (B.7),

$q_{c}^{\mathrm{wald}}\equiv{{(c-\hat{c})^{2}}\over{\sigma^{2}}}.$ (B.9)

Recalling that $\hat{c}\sim\mathcal{N}(c^{\prime},\sigma)$, we note that
$q_{c}^{\mathrm{wald}}$ corresponds to a squared Gaussian random variable with
unit variance and non-zero mean. This is known to follow a non-central
$\chi^{2}$ distribution [66],

$\mathrm{pdf}(q_{c}^{\mathrm{wald}}|c^{\prime})={{1}\over{2\sqrt{q_{c}}}}{{1}\over{\sqrt{2\pi}}}\left[\exp\left(-{{1}\over{2}}\left(\sqrt{q_{c}}+\sqrt{\Lambda}\right)^{2}\right)+\exp\left(-{{1}\over{2}}\left(\sqrt{q_{c}}-\sqrt{\Lambda}\right)^{2}\right)\right]\;,$
(B.10)

with the non-centrality parameter $\Lambda$ given by

$\Lambda={{(c-c^{\prime})^{2}}\over{\sigma^{2}}}.$ (B.11)

Note how Eq. (B.10) reduces to a central $\chi^{2}$ distribution under the
null hypothesis, i.e. when $c^{\prime}=c$, as dictated by Wilk’s theorem.

The problem of obtaining $\mathrm{pdf}(q_{c}|c^{\prime})$ with $c^{\prime}\neq
c$ is now reduced to the question of determining the non-centrality parameter
$\Lambda$ to be used in Eq. (B.10). We can construct this using MC toys: for
example, in Fig. B.1(b) we show the distribution of $\hat{c}$ under the SM,
obtained using $10$K toy data sets containing $n_{\rm ev}$ samples each, where
$n_{\rm ev}$ is drawn from $\mathrm{Pois}(\nu(0))$ each time a new experiment
is run. The distribution in Fig. B.1(b) gives $\sigma=0.19$, resulting in
$\Lambda_{\mathrm{MC}}=27.12$. However, although constructing
$\mathrm{pdf}(\hat{c}|c^{\prime})$ using MC toys gives us the info we need to
extract $\Lambda$, it is computationally highly inefficient. This is
especially so in case one needs to perform scans along different $c^{\prime}$,
e.g. to find expected exclusion limits.

This is where the Asimov data set enters as an efficient tool to obtain
$\Lambda$ [66]. The Asimov data set is defined such that one recovers the true
value of $c$ during maximum likelihood estimation, i.e. $\hat{c}=c$. In the
case of a Poissonian likelihood $\mathcal{L}_{\mathrm{pois}}$ defined as in
Eq. (2.4), the MLE $\hat{c}$ satisfies

$\left.{{d\log\mathcal{L}_{\mathrm{pois}}}\over{dc}}\right|_{c=\hat{c}}=\left.{{d}\over{dc}}\left\\{-2\sum_{i=1}^{n_{\mathrm{bins}}}\left[n_{i}\log\nu_{i}(c)-\nu_{i}(c)\right]\right\\}\right|_{c=\hat{c}}=0\,.\\\
$ (B.12)

The Asimov data set is then constructed such that Eq. (B.12) implies
$\hat{c}=c$, which can only be true if $n_{i}=\nu_{i}(c)$. Therefore, we
define the Asimov data set under the hypothesis $c$ as

$\mathcal{D}_{\mathrm{Asimov}}\equiv\\{n_{i,A}\\}=\\{\nu_{i}(c)\\}.$ (B.13)

The non-centrality parameter $\Lambda$ can then be found by evaluating $q_{c}$
on the Asimov data set corresponding to the alternative hypothesis
$c^{\prime}$, i.e. $n_{i,A}=\nu_{i}^{\prime}$,

$\Lambda_{\mathrm{A,binned}}=q_{c}(\\{n_{i,A}\\})=-2\sum_{i=1}^{n_{\mathrm{bins}}}\left[\nu_{i}^{\prime}\log\left({{\nu_{i}}\over{\nu_{i}^{\prime}}}\right)-\nu_{i}+\nu_{i}^{\prime}\right],$
(B.14)

where we have used the fact that $\hat{c}=c^{\prime}$ and therefore
$\hat{\nu_{i}}=\nu_{i}^{\prime}$ by construction. The relationship between
$\Lambda$ and $q_{c}$ can be understood by comparing Eqs. (B.9) and (B.11) and
noting that they coincide when $\hat{c}=c^{\prime}$.

Now, one could wonder how Eq. (B.14) generalizes to the case of unbinned data.
An intuitive starting point is to study the behavior of
$\Lambda_{\mathrm{A,binned}}$ as one increases the number of bin insertions,
and by doing so we find that after $\sim 10$ bins,
$\Lambda_{\mathrm{A,binned}}$ converges to a stable value. Encouraged by this
convergence, we take the infinitely narrow bin limit of Eq. (B.14),

$\displaystyle\Lambda_{\mathrm{A,binned}}$
$\displaystyle=-2\sum_{i=1}^{n_{\mathrm{bins}}}\left[\nu_{i}^{\prime}\log\left({{\nu_{i}}\over{\nu_{i}^{\prime}}}\right)-\nu_{i}+\nu_{i}^{\prime}\right]$
$\displaystyle\rightarrow-2\int_{-\infty}^{\infty}\left[\nu(c^{\prime})f_{\sigma}(x,c^{\prime})\log\left({{\nu(c)f_{\sigma}(x,c)}\over{\nu(c^{\prime})f_{\sigma}(x,c^{\prime})}}\right)-\nu(c)f_{\sigma}(x,c)+\nu(c^{\prime})f_{\sigma}(x,c^{\prime})\right]dx\equiv\Lambda_{\mathrm{A,unbinned}},$
(B.15)

where we have sent $\nu_{i}(c)\rightarrow\nu(c)f_{\sigma}(x,c)dx$. Evaluating
the integral in Eq. (B.15) gives $\Lambda_{\mathrm{A,unbinned}}=22.11$.

In Fig. B.2 we select three benchmark points at which to visualize the
distribution $\mathrm{pdf}(q_{c}|c^{\prime})$: the SM ($c=0$) and the EFT at
$c=0.5$, $c=1$. Firstly, the green histogram shows the distribution of $q_{c}$
evaluated on data following the EFT hypothesis. The distribution is obtained
by repeatedly drawing samples of size $n_{ev}\sim\mathrm{Pois}(\nu(c))$ from
the parent EFT dataset and calculating $q_{c}$ as given by the PLR in Eq.
(2.8). The solid green curve indicates the distribution of a $\chi^{2}$ with 1
degree of freedom. We find excellent agreement between the green curve and
histogram, as expected from Wilk’s theorem.

A similar procedure is used to obtain the blue histogram, this time
corresponding to the SM hypothesis. We repeatedly draw samples of size
$n_{ev}\sim\mathrm{Pois}(\nu(0))$ from the parent SM dataset and calculate
$q_{c}$ as given by the PLR in Eq. (2.8). The resulting distribution is
denoted by $\mathrm{pdf}(q_{c}^{\mathrm{exact}}|c)$. In contrast, the orange
histogram is produced using the approximation given by Wald’s theorem in Eq.
(B.9). For each sample drawn, we evaluate $q_{c}^{\mathrm{wald}}$ rather than
the full expression for $q_{c}$, resulting in the distribution
$\mathrm{pdf}(q_{c}^{\mathrm{wald}}|c)$. We notice a non-negligible difference
in the distributions of $q_{c}$ before and after employing Wald’s theorem. As
expected, this is due to the fact that the neglected $\mathcal{O}(\Delta^{3})$
term is in fact quite sizable, being on the order of a few percent. In Fig.
B.2(b) we show that the difference between
$\mathrm{pdf}(q_{c}^{\mathrm{wald}}|c)$ and
$\mathrm{pdf}(q_{c}^{\mathrm{exact}}|c)$ gets smaller when we move to a
smaller $\Delta$, corresponding to $c=0.5$.

Finally, we turn to the solid blue and orange curves. In orange, we plot the
distribution of the non-central $\chi^{2}$ as given by Eq. (B.10), setting the
non-centrality parameter $\Lambda$ to $\Lambda_{\mathrm{MC}}=27.12$. We find
good agreement between the orange curve and the orange histogram, as expected
from the fact that both employ Wald’s theorem. The blue curve, on the other
hand, indicates the non-central $\chi^{2}$ distribution evaluated at
$\Lambda_{\mathrm{A,unbinned}}=22.11$ as calculated using Eq. (B.15). In this
case the agreement between the blue curve and histogram is non-trivial: it
indicates that the non-centrality parameter evaluated on the unbinned Asimov
dataset provides an excellent description of the distribution of the PLR
$q_{c}$ evaluated on unbinned data. This is the case both at $c=1$ and
$c=0.5$, and provides a validation of the expression for
$\Lambda_{\mathrm{A,unbinned}}$ determined in Eq. (B.15). This agreement
suggests Eq. (B.15) includes some higher order contributions that are
otherwise omitted if one adopts instead Eq. (B.11). This was also conjectured
in [66].

(a)

(b)

Figure B.2: The distributions of $q_{c}$ under the EFT and the SM hypotheses
for $c=1.0$ (a) and $c=0.5$ (b). The blue curve agrees well with the
distribution $\mathrm{pdf}(q_{c}^{\mathrm{exact}}|c)$, indicating that the
distribution of $q_{c}$ is well described by the non-central $\chi^{2}$
evaluated with non-centrality parameter $\Lambda_{\mathrm{A,unbinned}}$. We
notice a disagreement between the distribution of $q_{c}^{\mathrm{exact}}$ and
$q_{c}^{\mathrm{wald}}$, meaning that neglecting the higher order corrections
in Wald’s theorem leads to sizable differences. On the right (b) it can be
seen that this difference gets smaller when $c\rightarrow c^{\prime}$, i.e.
$\Delta\rightarrow 0$.

Finally, Eq. (B.15) generalizes straightforwardly to the case of $n_{\rm p}$
model parameters $\bm{c}$, and to the case of a multi-dimensional vector
$\bm{x}$ of kinematic variables

$\Lambda_{\mathrm{A,unbinned}}=\int_{-\infty}^{\infty}\left[\nu(\bm{c}^{\prime})f_{\sigma}(\bm{x},\bm{c}^{\prime})\log\left({{\nu(\bm{c})f_{\sigma}(\bm{x},\bm{c})}\over{\nu(\bm{c}^{\prime})f_{\sigma}(\bm{x},\bm{c}^{\prime})}}\right)-\nu(\bm{c})f_{\sigma}(\bm{x},\bm{c})+\nu(\bm{c}^{\prime})f_{\sigma}(\bm{x},\bm{c}^{\prime})\right]d\bm{x}.$
(B.16)

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