SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Zhongjie Yu
TU Darmstadt  
Darmstadt, Germany
Martin Trapp
Aalto University Espoo, Finland
Kristian Kersting
TU Darmstadt/Hessian.AI/DFKI
{yu,kersting}@cs.tu-darmstadt.de，martin.trapp@aalto.fi
In many real-world scenarios, it is crucial to be able to reliably and efficiently reason under uncertainty while capturing complex relationships in data. Probabilistic circuits (PCs), a prominent family of tractable probabilistic models, offer a remedy to this challenge by composing simple, tractable distributions into a high-dimensional probability distribution. However, learning PCs on heterogeneous data is challenging and densities of some parametric distributions are not available in closed form, limiting their potential use. We introduce characteristic circuits (CCs), a family of tractable probabilistic models providing a unified formalization of distributions over heterogeneous data in the spectral domain. The one-to-one relationship between characteristic functions and probability measures enables us to learn high-dimensional distributions on heterogeneous data domains and facilitates efficient probabilistic inference even when no closed-form density function is available. We show that the structure and parameters of CCs can be learned efficiently from the data and find that CCs outperform state-of-the-art density estimators for heterogeneous data domains on common benchmark data sets.
Probabilistic circuits (PCs [Choi et al., 2020]) have gained increasing attention in the machine learning community as a promising modelling family that renders many probabilistic inferences tractable with little compromise in their expressivity. Their beneficial properties have prompted many successful applications in density estimation [e.g., Peharz et al., 2020, Di Mauro et al., 2021, Dang et al., 2022, Correia et al., 2023] and in areas where probabilistic reasoning is key, for example, neuro-symbolic reasoning [Ahmed et al., 2022], certified fairness [Selvam et al., 2023], or causality [Zečević et al., 2021]. Moreover, recent works have explored ways of specifying PCs for more complex modelling scenarios, such as time series [Trapp et al., 2020, Yu et al., 2021b,a] or tractable representation of graphs [Wang et al., 2022].
However, while density estimation is at the very core of many machine learning techniques (e.g., approximate Bayesian inference [Murphy, 2012]) and a fun
damental tool in statistics to identify characteristics of the data such as  $k^{\mathrm{th}}$  order moments or multimodality [Silverman, 2018], even in the case of parametric families, densities are sometimes
Figure 1: Characteristic circuits provide a unified, tractable specification of joint continuous and discrete distributions in the spectral domain of probability measures.
not available in closed-form. For example, only special cases of  $\alpha$ -stable distributions provide closed-form densities [Nolan, 2013]. Fortunately, there exists a one-to-one correspondence between probability measures and characteristic functions [Sasvári, 2013], which can be understood as the Fourier-Stieltjes transform of the probability measures, enabling the characterisation of any probability measure through its characteristic function. Henceforth, the characteristic function of probability measures has found wide applicability in statistics, ranging from its use as a non-parametric estimator through the empirical characteristic function [Feuerverger and Mureika, 1977] to estimate heavy-tailed data, e.g., through the family of  $\alpha$ -stable distributions [Nolan, 2013]. However, even though the characteristic function has many beneficial properties, its application to encode high-dimensional data distributions and efficient computation of densities can be quite challenging [Nolan, 2013].
In this work, we bridge between the characteristic function of probability measures and PCs. We do so by examining PCs from a more general perspective, similar in spirit to their specifications as a summation over functions on a commutative semiring [Friesen and Domingos, 2016] or as a convex combination of product measures on product probability spaces [Trapp et al., 2020]. Instead of defining the circuit over density functions, we propose to form the circuit over the characteristic function of the respective probability measures, illustrated in Fig. 1. The resulting characteristic circuits (CCs) are related to recent works, which define a circuit over probability generating polynomials to represent discrete probability distributions [Zhang et al., 2021], in that both approaches can be understood as transformation methods. The benefits of using the spectral domain are manifold: (i) characteristic functions as the base enable a unified view for discrete and continuous random variables, (ii) directly representing the characteristic function allows learning distributions that do not have closed-form expressions for their density, and (iii) the moment can be obtained efficiently by differentiating the circuit. When modelling heterogeneous data, standard PCs do not naturally lend themselves to a unified view of mixed data but treat discrete and continuous random variables (RVs) conceptually differently. The difference arises as PCs model heterogeneous data domains after integration w.r.t. the base measure which, in the case of mixed domains, differs between discrete and continuous RVs. RVs distributed according to a singular distribution can typically not be represented in PCs at all. This dependence on the base measure is subtly embedded within PCs, resulting in challenges when it comes to learning these models in heterogeneous domains. In contrast, CCs provide a unified view compared to PCs by moving away from the dependence on the base measure, achieved by representing the distribution through its characteristic function, which is independent of the base measure.
In summary, our contributions are: (1) We propose characteristic circuits, a novel deep probabilistic model class representing the joint of discrete and continuous random variables through a unifying view in the spectral domain. (2) We show that characteristic circuits retain the tractability of PCs despite the change of domain and enable efficient computation of densities, marginals, and conditionals. (3) We derive parameter and structure learning for characteristic circuits and find that characteristic circuits outperform SOTA density estimators in the majority of tested benchmarks. $^{1}$
We proceed as follows. We start by discussing related work and review preliminaries on probabilistic circuits and characteristic functions. Consequently, we define our model characteristic circuits, discuss theoretical properties, and show how to learn the circuits' parameters and structure. We conclude by presenting an empirical evaluation and discussion of the new model class.
Characteristic functions (CFs) were originally proposed as a tool in the study of limit theorems and afterwards developed with independent mathematical interest [Lukacs, 1972]. The uniqueness between CFs and probability measures is recovered with Lévy's inversion theorem [Sasvári, 2013]. A popular application of the CF is in statistical tests [e.g., Eriksson and Koivunen, 2003, Su and White, 2007, Wang and Hong, 2018, Ke and Yin, 2019]. In practice, the CF of a distribution is in most cases not easy to estimate, and in turn, the empirical characteristic function (ECF) is employed as an approximation to the CF [Feuerverger and Mureika, 1977]. The ECF has been successfully applied in sequential data analysis tasks [Knight and Yu, 2002, Yu, 2004, Davis et al., 2021]. When handling high dimensional data, multivariate CFs, and ECFs were proposed for modelling e.g. multivariate time series [Lee et al., 2022] and images [Ansari et al., 2020]. Although a mass of work has been
Source code is available at https://github.com/ml-research/CharacteristicCircuits
developed for the applications of CF, less attention has been paid to estimating the model quality of CF itself. Therefore, modelling the multivariate CF remains to be challenging.
Probabilistic circuits (PCs) are a unifying framework for tractable probabilistic models [Choi et al., 2020] that recently show their power in e.g. probabilistic density estimation [Dang et al., 2020, Di Mauro et al., 2021, Zhang et al., 2021], flexible inference [Shao et al., 2022], variational inference [Shih and Ermon, 2020], and sample generating [Peharz et al., 2020]. When it comes to data containing both discrete and continuous values, a mixture of discrete and continuous random variables is employed in PCs. Molina et al. [2018] propose to model mixed data based on the Sum-Product Network (SPN) structure, casting the randomized dependency coefficient [Lopez-Paz et al., 2013] for independence test for hybrid domains and piece-wise polynomial as leaf distributions, resulting in Mixed Sum-Product Networks (MSPN). Furthermore, statistical data type and likelihood discovery have been made available with Automatic Bayesian Density Analysis (ABDA) [Vergari et al., 2019], which is a suitable tool for the analysis of mixed discrete and continuous tabular data. Moreover, Bayesian SPNs [Trapp et al., 2019] use a well-principled Bayesian framework for SPN structure learning, achieving competitive results in density estimation on heterogeneous data sets. The above-mentioned models try to handle the probability measure with either parametric density/mass functions or histograms, but yet could not offer a unified view of heterogeneous data. PCs will also fail to model leaves with distributions that do not have closed-form density expressions.
3 Preliminaries on Probabilistic Circuits and Characteristic Functions
Before introducing characteristic circuits, we recap probabilistic circuits and characteristic functions.
3.1 Probabilistic Circuits (PCs)
PCs are tractable probabilistic models, structured as rooted directed acyclic graphs, where each leaf node  $L$  represents a probability distribution over a univariate RV, each sum node  $S$  models a mixture of its children, and each product node  $P$  models a product distribution (assuming independence) of their children. A PC over a set of RVs  $X$  can be viewed as a computational graph  $\mathcal{G}$  representing a tractable probability distribution over  $X$ , and the value obtained at the root node is the probability computed by the circuit. We refer to Choi et al. [2020] for more details.
Each node in  $\mathcal{G}$  is associated with a subset of  $X$  called the scope of a node  $\mathsf{N}$  and is denoted as  $\psi(\mathsf{N})$ . The scope of an inner node is the union of the scope of its children. Sum nodes compute a weighted sum of their children  $S = \sum_{\mathsf{N} \in \mathrm{ch}(\mathsf{S})} w_{\mathsf{S},\mathsf{N}} \mathsf{N}$ , and product nodes compute the product of their children  $P = \prod_{\mathsf{N} \in \mathrm{ch}(\mathsf{P})} \mathsf{N}$ , where  $\mathrm{ch}(\cdot)$  denotes the children of a node. The weights  $w_{\mathsf{S},\mathsf{N}}$  are generally assumed to be non-negative and normalized (sum up to one) at each sum node. We also assume the PC to be smooth (complete) and decomposable [Darwiche, 2003], where smooth requires all children of a sum node having the same scope, and decomposable means all children of a product node having pairwise disjoint scopes.
3.2 Characteristic Functions (CFs)
Characteristic functions provide a unified view for discrete and continuous random variables through the Fourier-Stieltjes transform of their probability measures. Let  $\mathbf{X}$  be a  $d$ -dimensional random vector, the CF of  $\mathbf{X}$  for  $t \in \mathbb{R}^d$  is given as:
$$
\varphi_ {\boldsymbol {X}} (\boldsymbol {t}) = \mathbb {E} [ \exp (\mathrm {i} \boldsymbol {t} ^ {\top} \boldsymbol {X}) ] = \int_ {\boldsymbol {x} \in \mathbb {R} ^ {d}} \exp (\mathrm {i} \boldsymbol {t} ^ {\top} \boldsymbol {x}) \mu_ {\boldsymbol {X}} (\mathrm {d} \boldsymbol {x}), \tag {1}
$$
where  $\mu_{X}$  is the distribution/probability measure of  $X$ . CFs have certain useful properties. We will briefly review those that are relevant for the remaining discussion: (i)  $\varphi_{X}(0) = 1$  and  $|\varphi_{X}(t)| \leq 1$ ; (ii) for any two RVs  $X_{1}, X_{2}$ , both have the same distribution iff  $\varphi_{X_{1}} = \varphi_{X_{2}}$ ; (iii) if  $X$  has  $k$  moments, then  $\varphi_{X}$  is  $k$ -times differentiable; and (iv) two RVs  $X_{1}, X_{2}$  are independent iff  $\varphi_{X_{1},X_{2}}(s,t) = \varphi_{X_{1}}(s)\varphi_{X_{2}}(t)$ . We refer to Sasváří [2013] for a more detailed discussion of CFs and their properties.
Theorem 3.1 (Levy's inversion theorem [Sasvári, 2013]). Let  $X$  be a real-valued random variable,  $\mu_{X}$  its probability measure, and  $\varphi_{X} : \mathbb{R} \to \mathbb{C}$  its characteristic function. Then for any  $a, b \in \mathbb{R}$ ,
$a <   b$  ,wehave that
$$
\lim  _ {T \rightarrow \infty} \frac {1}{2 \pi} \int_ {- T} ^ {T} \frac {\exp (- \mathrm {i} t a) - \exp (- \mathrm {i} t b)}{\mathrm {i} t} \varphi_ {X} (t) \mathrm {d} t = \mu_ {X} [ (a, b) ] + \frac {1}{2} (\mu_ {X} (a) + \mu_ {X} (b)) \tag {2}
$$
and, hence,  $\varphi_X$  uniquely determines  $\mu_X$ .
Corollary. If  $\int_{\mathbb{R}}|\varphi_X(t)|\mathrm{d}t < \infty$ , then  $X$  has a continuous probability density function  $f_{x}$  given by
$$
f _ {X} (x) = \frac {1}{2 \pi} \int_ {\mathbb {R}} \exp (- \mathrm {i} t x) \varphi_ {X} (t) \mathrm {d} t. \tag {3}
$$
Note that not every probability measure admits an analytical solution to Eq. (3), e.g., only special cases of the family of  $\alpha$ -stable distributions have a closed-form density function [Nolan, 2013], and numerical integration might be needed.
Empirical Characteristic Function (ECF). In many cases, a parametric form of the data distribution is not available and one needs to use a non-parametric estimator. The ECF [Feuerverger and Mureika, 1977, Cramér, 1999] is an unbiased and consistent non-parametric estimator of the population characteristic function. Given data  $\{\pmb{x}_j\}_{j=1}^n$  the ECF is given by
$$
\hat {\varphi} _ {\mathbb {P}} (\boldsymbol {t}) = \frac {1}{n} \sum_ {j = 1} ^ {n} \exp (\mathrm {i} \boldsymbol {t} ^ {\top} \boldsymbol {x} _ {j}). \tag {4}
$$
Evaluation Metric. To measure the distance between two distributions represented by their characteristic functions, the squared characteristic function distance (CFD) can be employed. The CFD between two distributions  $\mathbb{P}$  and  $\mathbb{Q}$  is defined as:
$$
\operatorname {C F D} _ {\omega} ^ {2} (\mathbb {P}, \mathbb {Q}) = \int_ {\mathbb {R} ^ {d}} | \varphi_ {\mathbb {P}} (\boldsymbol {t}) - \varphi_ {\mathbb {Q}} (\boldsymbol {t}) | ^ {2} \omega (\boldsymbol {t}; \eta) \mathrm {d} \boldsymbol {t}, \tag {5}
$$
where  $\omega(t; \eta)$  is a weighting function parameterized by  $\eta$  and guarantees the integral in Eq. (5) converge. When  $\omega(t; \eta)$  is a probability density function, Eq. (5) can be rewritten as:
$$
\operatorname {C F D} _ {\omega} ^ {2} (\mathbb {P}, \mathbb {Q}) = \mathbb {E} _ {\boldsymbol {t} \sim \omega (\boldsymbol {t}; \eta)} \left[ | \varphi_ {\mathbb {P}} (\boldsymbol {t}) - \varphi_ {\mathbb {Q}} (\boldsymbol {t}) | ^ {2} \right]. \tag {6}
$$
Actually, using the uniqueness theorem of CFs, we have  $\mathrm{CFD}_{\omega}(\mathbb{P},\mathbb{Q}) = 0$  iff  $\mathbb{P} = \mathbb{Q}$  [Sriperumbudur et al., 2010]. Computing Eq. (6) is generally intractable, therefore, we use Monte-Carlo integration to approximate the expectation, resulting in  $\mathrm{CFD}_{\omega}^{2}(\mathbb{P},\mathbb{Q})\approx \frac{1}{k}\sum_{j = 1}^{k}|\varphi_{\mathbb{P}}(t_j) - \varphi_{\mathbb{Q}}(t_j)|^2$ , where  $\{t_1,\dots ,t_k\} \stackrel{\mathrm{i.i.d.}}{\sim}\omega(t;\eta)$ . We refer to Ansari et al. [2020] for a detailed discussion.
4 Characteristic Circuits
Now we have everything at hand to introduce characteristic circuits. We first give a recursive definition of CC, followed by devising each type of node in a CC. We then show CCs feature efficient computation of densities, and in the end, introduce how to learn a CC from data.
Definition 4.1 (Characteristic Circuit). Let  $\mathbf{X} = \{X_{1},\ldots ,X_{d}\}$  be a set of random variables. A characteristic circuit denoted as  $\mathcal{C}$  is a tuple consisting of a rooted directed acyclic graph  $\mathcal{G}$ , a scope function  $\psi \colon \mathrm{V}(\mathcal{G})\to \mathcal{P}(\mathbf{X})$ , parameterized by a set of graph parameters  $\theta_{\mathcal{G}}$ . Nodes in  $\mathcal{G}$  are either sum (S), product (P), or leaf (L) nodes. With this, a characteristic circuit is defined recursively as follows:
1. a characteristic function for a scalar random variable is a characteristic circuit.
2. a product of characteristic circuits is a characteristic circuit.
3. a convex combination of characteristic circuits is a characteristic circuit.
Let us now provide some more details. To this end, we denote with  $\varphi_{\mathcal{C}}(t)$  the output of  $\mathcal{C}$  computed at the root of  $\mathcal{C}$ , which represents the estimation of characteristic function given argument of the characteristic function  $t \in \mathbb{R}^d$ . Further, we denote the number of RVs in the scope of  $\mathsf{N}$  as  $p_{\mathsf{N}} \coloneqq$
$|\psi(\mathsf{N})|$  and use  $\varphi_{\mathsf{N}}(t)$  for the characteristic function of a node. Throughout the paper, we assume the CC to be smooth and decomposable.
Product Nodes. A product node in a CC encodes the independence of its children. Let  $X$  and  $Y$  be two RVs. Following property (iv) of characteristic functions, the characteristic function of  $X, Y$  is given as  $\varphi_{X,Y}(t,s) = \varphi_X(t)\varphi_Y(s)$ , if and only if  $X$  and  $Y$  are independent. Therefore, by definition, the characteristic function of product nodes is given as:
$$
\varphi_ {\mathrm {P}} (\boldsymbol {t}) = \prod_ {\mathrm {N} \in \operatorname {c h} (\mathrm {P})} \varphi_ {\mathrm {N}} \left(\boldsymbol {t} _ {\psi (\mathrm {N})}\right), \tag {7}
$$
where  $\pmb {t} = \bigcup_{N\in \mathrm{ch}(P)}\pmb{t}_{\psi (N)}$
Sum Nodes. A sum node in a CC encodes the mixture of its children. Let the parameters of S be given as  $\sum_{\mathsf{N}\in \mathrm{ch}(\mathsf{S})}w_{\mathsf{S},\mathsf{N}} = 1$  and  $w_{\mathsf{S},\mathsf{N}}\geq 0,\forall \mathsf{S},\mathsf{N}$ . Then the sum node in a CC is given as:  $\varphi_{\mathsf{S}}(\pmb {t}) =$
$$
\int_ {\boldsymbol {x} \in \mathbb {R} ^ {d}} \exp (\mathrm {i} \boldsymbol {t} ^ {\top} \boldsymbol {x}) \left[ \sum_ {\mathrm {N} \in \operatorname {c h} (\mathbb {S})} w _ {\mathrm {S}, \mathrm {N}} \mu_ {\mathrm {N}} (\mathrm {d} \boldsymbol {x}) \right] = \sum_ {\mathrm {N} \in \operatorname {c h} (\mathbb {S})} w _ {\mathrm {S}, \mathrm {N}} \underbrace {\int_ {\boldsymbol {x} \in \mathbb {R} ^ {p s}} \exp (\mathrm {i} \boldsymbol {t} ^ {\top} \boldsymbol {x}) \mu_ {\mathrm {N}} (\mathrm {d} \boldsymbol {x})} _ {= \varphi_ {\mathrm {N}} (\boldsymbol {t})}. \tag {8}
$$
Leaf Nodes. A leaf node of a CC models the characteristic function of a univariate RV. To handle various data types, we propose the following variants of leaf nodes.
ECF leaf. The most straightforward way for modelling the leaf node is to directly employ the empirical characteristic function for the local data at each leaf, defined as  $\varphi_{\mathrm{LCF}}(t) = \frac{1}{n}\sum_{j=1}^{n}\exp(\mathrm{i}tx_j)$ , where  $n$  is the number of instances at the leaf L, and  $x_j$  is the  $j^{th}$  instance. The ECF leaf is non-parametric and is determined by the  $n$  instances  $x_j$  at the leaf.
Parametric leaf for continuous RVs. Motivated by existing SPN literature, we can assume that the RV at a leaf node follows a parametric continuous distribution e.g. normal distribution. With this, the leaf node is equipped with the CF of normal distribution  $\varphi_{\mathrm{L}_{\mathrm{Normal}}}(t) = \exp (\mathrm{i}t\mu -\frac{1}{2}\sigma^2 t^2)$ , where parameters  $\mu$  and  $\sigma^2$  are the mean and variance.
Parametric leaf for discrete RVs. For discrete RVs, if it is assumed to follow categorical distribution  $(P(X = j) = p_j)$ , then the CF at the leaf node can be defined as  $\varphi_{\mathrm{L}_{\mathrm{Categorical}}}(t) = \mathbb{E}[\exp (\mathrm{i}t x)] = \sum_{j = 1}^{k}p_{j}\exp (\mathrm{i}t j)$ . Other discrete distributions which are widely used in probabilistic circuits can also be employed as leaf nodes in CCs, e.g., Bernoulli, Poisson, and geometric distributions.
$\alpha$ -stable leaf. In the case of financial data or data distributed with heavy tails, the  $\alpha$ -stable distribution is frequently employed.  $\alpha$ -stable distributions are more flexible in modelling e.g. data with skewed centered distributions. The characteristic function of an  $\alpha$ -stable distribution is  $\varphi_{\mathrm{L}_{\alpha\text{-stable}}}(t) = \exp(\mathrm{i}t\mu - |ct|^{\alpha}(1 - \mathrm{i}\beta \mathrm{sgn}(t)\Phi))$ , where  $\mathrm{sgn}(t)$  takes the sign of  $t$  and  $\Phi = \left\{ \begin{array}{ll} \tan(\pi \alpha / 2) & \alpha \neq 1 \\ -2 / \pi \log |t| & \alpha = 1 \end{array} \right.$ . The parameters in  $\alpha$ -stable distributions are the stability parameter  $\alpha$ , the skewness parameter  $\beta$ , the scale parameter  $c$ , and the location parameter  $\mu$ . Despite its modelling power,  $\alpha$ -stable distribution is never employed in PCs, as it is represented analytically by its CF and in most cases does not have a closed-form probability density function.
4.1 Theoretic Properties of Characteristic Circuits
With the CC defined above, we can now derive the densities, marginals, and moments from it.
4.1.1 Efficient computation of densities
Through their recursive nature, CCs enable efficient computation of densities in high-dimensional settings even if the density function is not available in closed form. For this, we present an extension of Theorem 3.1 for CCs, formulated using the notion of induced trees  $\mathcal{T}$  [Zhao et al., 2016]. A detailed definition of induced trees can be found in Appendix A.2.
Lemma 4.2 (Inversion). Let  $\mathcal{C} = \langle \mathcal{G},\psi ,\theta_{\mathcal{G}}\rangle$  be a characteristic circuit on  $RVsX = \{X_j\}_{j = 1}^d$  with univariate leave nodes. If  $\int_{\mathbb{R}}|\varphi_{\mathsf{L}}(t)|\mathrm{d}t < \infty$  for every  $\mathsf{L}\in V(\mathcal{G})$ , then  $\pmb{X}$  has a continuous
probability density function  $f_{\pmb{x}}$  given by  $f_{\pmb{X}}(\pmb{x}) =$
$$
\frac {1}{(2 \pi) ^ {d}} \sum_ {i = 1} ^ {\tau} \prod_ {(S, N) \in E (T _ {i})} w _ {S, N} \prod_ {L \in V (T _ {i})} \int_ {\mathbb {R}} \exp (- \mathrm {i} t x _ {\psi (L)}) \varphi_ {L} (t) \mathrm {d} t, \tag {9}
$$
and can be computed efficiently through analytic or numerical integration at the leaves.
Proof. Let  $\mathcal{C} = \langle \mathcal{G},\psi ,\theta_{\mathcal{G}}\rangle$  be a characteristic circuit on RVs  $\pmb {X} = \{X_j\}_{j = 1}^d$  with univariate leave nodes and  $p_{\mathbb{N}}$  the number of RVs in the scope of N. In order to calculate the density function of  $\mathcal{C}$ , we need to integrate over the  $d$ -dimensional real space  $\mathbb{R}^d$ , i.e.,
$$
f _ {\boldsymbol {X}} (\boldsymbol {x}) = \frac {1}{(2 \pi) ^ {d}} \underbrace {\int_ {\boldsymbol {t} \in \mathbb {R} ^ {d}} \exp (- \mathrm {i} \boldsymbol {t} ^ {\top} \boldsymbol {x}) \varphi_ {\mathcal {C}} (\boldsymbol {t}) \lambda_ {d} (\mathrm {d} \boldsymbol {t})} _ {= \hat {f} _ {\mathcal {C}} (\boldsymbol {x})}, \tag {10}
$$
where  $\varphi_{\mathcal{C}}(\pmb {t})$  denotes the CF defined by the root of the characteristic circuit and  $\lambda_{d}$  is the Lebesgue measure on  $(\mathbb{R}^d,\mathcal{B}(\mathbb{R}^d))$ . We can examine the computation of Eq. (10) recursively for every node.
Leaf Nodes. If  $\mathsf{N}$  is a leaf node  $\mathsf{L}$ , we obtain  $\hat{f}_{\mathsf{N}}(\cdot)$  by calculating:
$$
\hat {f} _ {\mathrm {L}} (x) = 2 \pi f _ {\mathrm {L}} (x) = \int_ {\mathbb {R}} \exp (- \mathrm {i} t x) \varphi_ {X} (t) \lambda (\mathrm {d} t), \tag {11}
$$
which follows from Theorem 3.1.
Sum Nodes. If  $N$  is a sum node  $S$ , then: