Fully Bayesian Unfolding (FBU)

• Fully Bayesian Unfolding (FBU) is a Bayesian approach to unfolding that estimates a complete posterior distribution over truth-level spectra rather than relying on point estimates.
• It models detector responses using Poisson likelihoods and incorporates explicit prior regularization, such as curvature penalties and hierarchical models, to enforce smoothness and positivity.
• FBU leverages advanced sampling techniques like MCMC and Hamiltonian Monte Carlo to explore high-dimensional, non-Gaussian posteriors, providing robust uncertainty estimates for applications from collider physics to neutron spectroscopy.

Searching arXiv for recent and foundational papers on Fully Bayesian Unfolding. Fully Bayesian Unfolding (FBU) is a Bayesian treatment of the unfolding, deconvolution, or detector-correction problem in which the unknown truth-level spectrum is represented explicitly as a vector of parameters and the result is the full posterior probability density over that spectrum rather than only a point estimate with an error matrix. In the formulation emphasized by Choudalakis, the posterior is defined in the multi-dimensional space of all possible spectra before smearing, and regularization is implemented by the prior rather than by an external stopping rule or an added penalty term (Choudalakis, 2012). The same designation has also been used in related contexts that propagate uncertainty in the response matrix, data, and latent spectrum by Monte Carlo integration, as in the improved iterative Bayesian treatment discussed by D’Agostini (D'Agostini, 2010). Across these variants, the central feature is the replacement of deterministic inversion by posterior inference.

1. Conceptual basis and problem setting

In unfolding, one seeks to infer a latent histogram or spectrum from data measured after detector smearing. The standard ingredients are a truth-level quantity, an observed detector-level histogram, and a response matrix encoding migration and inefficiency. In the Choudalakis formulation, the truth spectrum is denoted by $\mu\equiv(\mu_1,\dots,\mu_{N_t})$ with $\mu_t\ge 0$, the observed counts by $n\equiv(n_1,\dots,n_{N_r})$, and the expected reconstructed counts by

$R_r=\sum_{t=1}^{N_t}\mu_t\,P(r|t)+b_r,$

where $P(r|t)$ is the response matrix and $b_r\ge 0$ is a known background expectation (Choudalakis, 2012).

The same structure appears in later presentations with different notation. In collider applications, one writes the particle-level histogram as $\boldsymbol{t}=(t_1,\dots,t_{N_t})$, the measured histogram as $\boldsymbol{d}=(d_1,\dots,d_{N_m})$, and the response matrix as $\mathbf{R}\in\mathbb{R}^{N_m\times N_t}$ with entries $R_{ij}=\Pr(m_i\mid t_j)$ (Acosta et al., 8 Sep 2025). In neutron spectroscopy, the analogous objects are the unknown neutron spectrum $\mu_t\ge 0$0, detector response matrix $\mu_t\ge 0$1, and measured light-output counts $\mu_t\ge 0$2 (Zhu et al., 2019).

A defining point in the literature is that FBU should not be confused with D’Agostini’s iterative method. Despite a family resemblance in the use of Bayes’ theorem, FBU targets the full posterior density $\mu_t\ge 0$3 or $\mu_t\ge 0$4, whereas the iterative method updates a point estimate through repeated applications of Bayes factors and uses the iteration count as a regularizer (Choudalakis, 2012).

2. Likelihood models and posterior construction

The statistical core of FBU is the likelihood for observed detector-level counts induced by a truth-level spectrum through the response matrix. With Poisson fluctuations in each reconstructed bin and statistical independence across bins, the likelihood is

$\mu_t\ge 0$5

Together with a prior $\mu_t\ge 0$6, Bayes’ theorem gives

$\mu_t\ge 0$7

(Choudalakis, 2012).

Equivalent forms recur across implementations. In the modern collider notation,

$\mu_t\ge 0$8

and therefore

$\mu_t\ge 0$9

(Acosta et al., 8 Sep 2025).

Domain-specific variants may alter the forward model while keeping the Bayesian structure. In neutron unfolding, both a linear-Gaussian model,

$n\equiv(n_1,\dots,n_{N_r})$0

and a Poisson forward model,

$n\equiv(n_1,\dots,n_{N_r})$1

are presented, with the Poisson model preferred for low-count spectroscopy (Zhu et al., 2019). A separate regularized FBU formulation approximates the Poisson by a Gaussian for large counts and uses the practical likelihood

$n\equiv(n_1,\dots,n_{N_r})$2

with $n\equiv(n_1,\dots,n_{N_r})$3 (Baron, 2020).

This common structure suggests a broad definition of FBU: a posterior over binned truth-level quantities induced by a response model and a prior, with uncertainty represented by the full joint distribution rather than by an asymptotic covariance approximation.

3. Priors and regularization

In FBU, regularization is prior specification. Choudalakis writes the prior in the generic form

$n\equiv(n_1,\dots,n_{N_r})$4

inside a positivity box and zero outside, where $n\equiv(n_1,\dots,n_{N_r})$5 is a user-chosen regularization functional and $n\equiv(n_1,\dots,n_{N_r})$6 controls the strength (Choudalakis, 2012). For $n\equiv(n_1,\dots,n_{N_r})$7, the prior becomes flat in the allowed box. This choice is described as a common uninformative prior, often implemented as a constant density within a large hyper-rectangle $n\equiv(n_1,\dots,n_{N_r})$8.

Several explicit regularization functionals are given. The negative entropy functional is

$n\equiv(n_1,\dots,n_{N_r})$9

The curvature penalty is

$R_r=\sum_{t=1}^{N_t}\mu_t\,P(r|t)+b_r,$0

A relative first-derivative variation is also defined, with bin-width and bin-center corrections for non-uniform binning, and a Gaussian prior around Monte Carlo truth is given as

$R_r=\sum_{t=1}^{N_t}\mu_t\,P(r|t)+b_r,$1

(Choudalakis, 2012).

The regularized FBU extension adopts precisely a curvature penalty,

$R_r=\sum_{t=1}^{N_t}\mu_t\,P(r|t)+b_r,$2

with $R_r=\sum_{t=1}^{N_t}\mu_t\,P(r|t)+b_r,$3 (Baron, 2020). In that study, $R_r=\sum_{t=1}^{N_t}\mu_t\,P(r|t)+b_r,$4 is chosen by hand, and automatic $R_r=\sum_{t=1}^{N_t}\mu_t\,P(r|t)+b_r,$5-estimation via maximizing the Bayesian evidence or cross-validation is identified as future work.

In the neutron-spectrum formulation, regularization is hierarchical rather than fixed. The prior on $R_r=\sum_{t=1}^{N_t}\mu_t\,P(r|t)+b_r,$6 conditional on a hyperparameter $R_r=\sum_{t=1}^{N_t}\mu_t\,P(r|t)+b_r,$7 is

$R_r=\sum_{t=1}^{N_t}\mu_t\,P(r|t)+b_r,$8

where $R_r=\sum_{t=1}^{N_t}\mu_t\,P(r|t)+b_r,$9 is the discrete Laplacian (second-difference) matrix, and smaller $P(r|t)$0 forces more smoothness. The hyperprior is inverse-Gamma,

$P(r|t)$1

(Zhu et al., 2019).

A plausible implication is that the literature uses the term FBU across at least two regularization philosophies: explicit fixed priors or penalties on the unfolded spectrum, and hierarchical priors in which the smoothness scale is itself inferred from the data.

4. Posterior exploration, sampling, and summary statistics

Because the posterior is generally high-dimensional and non-Gaussian, FBU is implemented numerically. Choudalakis describes three posterior exploration strategies: grid sampling, uniform random sampling, and Markov Chain Monte Carlo (MCMC). Grid sampling is useful only for $P(r|t)$2, while uniform random sampling weights samples by $P(r|t)$3. The MCMC example uses Metropolis-Hastings with proposals

$P(r|t)$4

where $P(r|t)$5 is uniform in a small hyper-box, followed by acceptance probability

$P(r|t)$6

(Choudalakis, 2012).

Later FBU implementations use Hamiltonian Monte Carlo. The regularized formulation employs Hamiltonian Monte Carlo with the No-U-Turn Sampler (NUTS), with potential energy $P(r|t)$7, kinetic energy $P(r|t)$8, a leapfrog integrator, and BuildTree control of path length. Typical tuning is stated as 1,000 warm-up steps, 2,000 sampling steps, with acceptance approximately $P(r|t)$9–$b_r\ge 0$0 (Baron, 2020). The modern Python implementation likewise highlights NUTS, building the probabilistic model in PyMC or a similar toolkit, defining each bin content as a random variable, and checking convergence diagnostics such as $b_r\ge 0$1-hat and effective sample size (Acosta et al., 8 Sep 2025).

The neutron-spectrum implementation alternates two conditional updates in an MCMC sampler. First, it samples $b_r\ge 0$2 by constrained Hamiltonian Monte Carlo, motivated by the facts that $b_r\ge 0$3 is high-dimensional and constrained to $b_r\ge 0$4. The conditional log-density is

$b_r\ge 0$5

Second, it samples the hyperparameter by conjugacy: $b_r\ge 0$6 (Zhu et al., 2019).

Posterior summaries are then extracted from the sample cloud. Choudalakis lists one-dimensional marginals, two-dimensional marginals, posterior mean, posterior median, posterior mode, and shortest 68% credible intervals (Choudalakis, 2012). In the neutron application, the posterior mean in bin $b_r\ge 0$7 is

$b_r\ge 0$8

and $b_r\ge 0$9 credible intervals are obtained from empirical quantiles of the sampled values in each energy bin (Zhu et al., 2019). The collider implementation similarly computes posterior means and covariances from MCMC samples and emphasizes that credible intervals and pairwise correlations are natural outputs (Acosta et al., 8 Sep 2025).

5. Relation to iterative Bayesian unfolding and iterative FBU variants

The principal methodological distinction in the literature is between FBU and D’Agostini’s iterative Bayesian unfolding. The iterative method starts from an initial guess and updates it according to

$\boldsymbol{t}=(t_1,\dots,t_{N_t})$0

followed by the classical update

$\boldsymbol{t}=(t_1,\dots,t_{N_t})$1

(D'Agostini, 2010). In this framework, a single pass leaves a memory of the prior, successive passes allow the prior to adapt to the data, and the stopping rule acts as regularization.

Choudalakis explicitly states that FBU should not be confused with D’Agostini’s iterative method. The differences singled out are that FBU yields the full posterior $\boldsymbol{t}=(t_1,\dots,t_{N_t})$2, the prior $\boldsymbol{t}=(t_1,\dots,t_{N_t})$3 plays the explicit role of regularization, there is no arbitrary iteration-stop criterion, multi-modal and highly non-Gaussian structure can be revealed, and positivity $\boldsymbol{t}=(t_1,\dots,t_{N_t})$4 is treated naturally (Choudalakis, 2012).

At the same time, some later work introduces iterative procedures within FBU itself, not as a substitute for posterior sampling but as a means of accelerating convergence or reducing wasted computation. The regularized FBU study proposes automatic two-stage phase-space estimation. Each bin initially receives bounds

$\boldsymbol{t}=(t_1,\dots,t_{N_t})$5

then, after sampling once, the marginal $\boldsymbol{t}=(t_1,\dots,t_{N_t})$6 is fit by a Gaussian $\boldsymbol{t}=(t_1,\dots,t_{N_t})$7 and the bounds are updated to

$\boldsymbol{t}=(t_1,\dots,t_{N_t})$8

Two iterations are reported to suffice in that study (Baron, 2020).

This suggests a useful distinction between iteration as an estimator-defining regularizer, which characterizes D’Agostini-style unfolding, and iteration as a sampling-domain refinement device inside an otherwise fully Bayesian posterior calculation.

6. Applications, empirical behavior, and current extensions

FBU has been applied in both high-energy and nuclear contexts. In neutron spectroscopy with organic scintillators, a hierarchical Bayesian model is used to infer neutron fluence per energy bin from measured light-output spectra and detector response functions. The motivation given is that inferring neutron spectra is important for applications including nonproliferation and nuclear security, because differences in emitted spectra allow discrimination of fission sources in special nuclear material from other neutron sources (Zhu et al., 2019).

That neutron study benchmarks the hierarchical-Bayesian MCMC approach against GRAVEL, PIDAL, and Tik using simulated data for three sources: 2.5 MeV monoenergetic neutrons, and continuous spectra from $\boldsymbol{t}=(t_1,\dots,t_{N_t})$9Cf and $\boldsymbol{d}=(d_1,\dots,d_{N_m})$0AmBe. The accuracy metric is the Spectral Angle Mapper,

$\boldsymbol{d}=(d_1,\dots,d_{N_m})$1

where lower SAM indicates better shape recovery. The reported results are that, for all three sources, the hierarchical-Bayesian approach matches or betters Tik and GRAVEL in SAM; it requires no manual tuning of $\boldsymbol{d}=(d_1,\dots,d_{N_m})$2; under low-count regimes from $\boldsymbol{d}=(d_1,\dots,d_{N_m})$3 to $\boldsymbol{d}=(d_1,\dots,d_{N_m})$4 total neutrons it maintains robustness comparable to optimally tuned PIDAL; and only MCMC delivers point estimates plus full credible intervals without user-defined regularization (Zhu et al., 2019).

In collider unfolding, regularized FBU is illustrated on the top-quark pair pseudorapidity $\boldsymbol{d}=(d_1,\dots,d_{N_m})$5 spectrum, described as having a double-peak structure. The study reports that after one iteration, regularized unfolding with $\boldsymbol{d}=(d_1,\dots,d_{N_m})$6 agrees better with truth in low-statistics tails and reduces $\boldsymbol{d}=(d_1,\dots,d_{N_m})$7, while after two iterations differences between regularized and unregularized results all but vanish because the iterative bounds already concentrate on the correct region (Baron, 2020).

Recent work on Neural Posterior Unfolding positions FBU as a classical Bayesian baseline implemented in modern Python and contrasts it with amortized posterior estimation using normalizing flows. The comparison emphasizes that FBU samples a model anew for each dataset, provides an asymptotically exact posterior subject to sampler convergence, and uses an explicit prior, whereas Neural Posterior Unfolding learns an approximate posterior once and then reuses it for fast inference (Acosta et al., 8 Sep 2025). The same source reports a Gaussian-example timing comparison in which FBU took $\boldsymbol{d}=(d_1,\dots,d_{N_m})$8 per pseudo-experiment for 10,000 MCMC draws plus tuning, while the neural method required approximately $\boldsymbol{d}=(d_1,\dots,d_{N_m})$9 to train once and then $\mathbf{R}\in\mathbb{R}^{N_m\times N_t}$0–$\mathbf{R}\in\mathbb{R}^{N_m\times N_t}$1 to unfold 100 pseudo-experiments amortized (Acosta et al., 8 Sep 2025).

The consistent advantage attributed to FBU across these sources is direct access to the full posterior uncertainty of the unfolded spectrum, including correlations and non-Gaussian structure. The recurring limitations are computational cost in large dimensions, sensitivity to prior or regularization choice, and in some variants the use of Gaussian approximations that may break down at very low counts (Choudalakis, 2012, Baron, 2020). Potential extensions explicitly mentioned in the literature include automatic hyperparameter selection, alternative regularizers such as entropy or total variation, fully Poisson likelihoods without Gaussian approximation, variational inference or importance sampling for faster posterior approximation, and propagation of full uncertainty through downstream classification or material-identification algorithms (Baron, 2020, Zhu et al., 2019).