1 Introduction
Probabilistic modeling is at the heart of modern computer science, with applications ranging from image recognition and image generation Pope and Lowe (2000); Radford et al. (2015) to weather forecasting Cano et al. (2004). Probabilistic models have a multitude of representations, such as probabilistic circuits (PCs) Choi et al. (2020), graphical models Koller and Friedman (2009), generative networks Goodfellow et al. (2014), and determinantal point processes Kulesza and Taskar (2012). Of particular interest to us are PCs, which are known to support guaranteed inference and thus have applications in safetycritical fields such as healthcare Aronsky and Haug (1998); Oniśko et al. (2000). In this work, we will focus on PCs that are fragments of the Negation Normal Form (), specifically s, s, s, and s Darwiche and Huang (2002). We refer to the survey by Choi et al. (2020) for more details regarding PCs.
Given two distributions and , a fundamental problem is to determine whether they are close. Closeness between distributions is frequently quantified using the total variation (TV) distance, where is the norm Lin et al. (2018); Canonne et al. (2020). Thus, stated formally, closeness testing is the problem of deciding whether or for . Determining the closeness of models has applications in AI planning Darwiche and Huang (2002), bioinformatics Rahmatallah et al. (2014); Städler and Mukherjee (2015); Yin et al. (2015) and probabilistic program verification Dutta et al. (2018); Murawski and Ouaknine (2005).
Equivalence testing is a special case of closeness testing, where one tests if . Darwiche and Huang (2002) initiated the study of equivalence testing of PCs by designing an equivalence test for s. An equivalence test is, however, of little use in contexts where the PCs under test encode nonidentical distributions that are nonetheless close enough for practical purposes. Such situations may arise due to the use of approximate PC compilation Chubarian and Turán (2020) and samplingbased learning of PCs Peharz et al. (2020a, b)
. As a concrete example, consider PCs that are learned via approximate methods such as stochastic gradient descent
Peharz et al. (2020b). In such a case, PCs are likely to converge to close but nonidentical distributions. Given two such PCs, we would like to know whether they have converged to distributions close to each other. Thus, we raise the question: Does there exist an efficient algorithm to test the closeness of two PC distributions?In this work, we design the first closeness test for PCs with respect to TV distance, called . Assuming the tested PCs allow polytime approximate weighted model counting and sampling, runs in polynomial time. Formally, given two PC distributions and , and three parameters (,,), for closeness(), farness(), and tolerance(), returns if and if with probability at least . makes atmost calls to the sampler and exactly 2 calls to the counter.
builds on a general distance estimation technique of
Canonne and Rubinfeld (2014) that estimates the distance between two distributions with a small number of samples. In the context of PCs, the algorithm requires access to an exact sampler and an exact counter. Since not all PCs support exact sampling and counting, we modify the technique presented in Canonne and Rubinfeld (2014) to allow for approximate samples and counts. Furthermore, we implement and test on a dataset of publicly available PCs arising from applications in circuit testing. Our results show that closeness testing can be accurate and scalable in practice.For some fragments, such as , no sampling algorithm is known, and for fragments such as , sampling is known to be NPhard Roth (1996). Since requires access to approximate weighted counters and samplers to achieve tractability, the question of determining the closeness of the PCs mentioned above remains unanswered. Thus, we investigate further and characterize the complexity of closeness testing for a broad range of PCs. Our characterization reveals that PCs from the fragments s and s can be tested for closeness in polytime via , owing to the algorithms of Darwiche (2001) and Arenas et al. (2021). We show that the approximate counting algorithm of Arenas et al. (2021) can be extended to loglinear s using chain formulas Chakraborty et al. (2015). Then, using previously known results, we also find that there are no polytime equivalence tests for PCs from and , conditional on widely believed complexitytheoretic conjectures. Our characterization also reveals some open questions regarding the complexity of closeness and equivalence testing of PCs.
The rest of the paper is organized in the following way. We define the notation and discuss related work in Section 2. We then present the main contribution of the paper, the closeness test , and the associated proof of correctness in Section 3. We present our experimental findings in Section 4, and then discuss the complexity landscape of closeness testing in Section 5. We conclude the paper and discuss some open problems in Section 6. Due to space constraints, we defer some proofs to the supplementary Section A.
2 Background
Let be a circuit over Boolean variables. An assignment to the variables of is a satisfying assignment if . The set of all satisfying assignments of is . If , then is said to be satisfiable and if , then is said to be valid. We use to denote the size of circuit , where the size is the total number of vertices and edges in the circuit DAG.
The polynomial hierarchy (PH) contains the classes (NP) and (coNP) along with generalizations of the form and where and Stockmeyer (1976). The classes and are said to be at level . If it is shown that two classes on the same or consecutive levels are equal, the hierarchy collapses to that level. Such a collapse is considered unlikely, and hence is used as the basic assumption for showing hardness results, including the ones we present in the paper.
2.1 Probability distributions
A weight function assigns a positive rational weight to each assignment . We extend the definition of to also allow circuits as input: . For weight function and circuit , is the weighted model count (WMC) of w.r.t. .
In this paper, we focus on loglinear weight functions as they capture a wide class of distributions, including those arising from graphical models, conditional random fields, and skipgram models Murphy (2012). Loglinear models are represented as literalweighted functions, defined as:
Definition 1.
For a set of variables, a weight function is called literalweighted if there is a polytime computable map such that for any assignment
For all circuits , and loglinear weight functions , can be represented in size polynomial in the input.
Probabilistic circuits:
A probabilistic circuit is a satisfiable circuit along with a weight function . and together define a discrete probability distribution on the set that is supported over . We denote the p.m.f. of this distribution as:
In this paper, we study circuits that are fragments of the Negation Normal Form (). A circuit in is a rooted, directed acyclic graph (DAG), where each leaf node is labeled with true, false, or ; and each internal node is labeled with a or and can have arbitrarily many children. We focus on four fragments of , namely, Decomposable (), deterministic(), Structured (), and Prime Implicates(). For further information regarding circuits in , refer to the survey Darwiche and Marquis (2002) and the paper Pipatsrisawat and Darwiche (2008).
The TV distance of two probability distributions and over is defined as: .
and are said to be (1) equivalent if , (2) close if , and (3) far if .
Our closeness testing algorithm , assumes access to an approximate weighted counter , and an approximate weighted sampler . We define their behavior as follows:
Definition 2.
takes a circuit , a weight function , a tolerance parameter and a confidence parameter as input and returns the approximate weighted model count of w.r.t. such that
Tractable approximate counting algorithms for PCs are known as Fully Polynomial Randomised Approximation Schemes (FPRAS). The running time of an FPRAS is given by .
Definition 3.
takes a circuit , a weight function , a tolerance parameter and a confidence parameter as input and returns either (1) a satisfying assignment sampled approximately w.r.t. weight function with probability or (2) a symbol indicating failure with probability . In other words, whenever samples :
Tractable approximate sampling algorithms for PCs are known as Fully Polynomial Almost Uniform Samplers (FPAUS). The running time of an FPAUS for a single sample is given by .
In the rest of the paper denotes the set , represents the indicator variable for event , and
represents the expectation of random variable
.2.2 Related work
Closeness testing:
Viewing circuit equivalence testing through the lens of distribution testing, we see that the  equivalence test of Darwiche and Huang (2002)
can be interpreted as an equivalence test for uniform distribution on the satisfying assignments of
s. This relationship between circuit equivalence testing and closeness testing lets us rule out the existence of distributional equivalence tests for all those circuits for which circuit equivalence is already known to be hard under complexitytheoretic assumptions. We will explore this further in Section 5.2.Distribution testing:
Discrete probability distributions are typically defined over an exponentially large number of points; hence a lot of recent algorithms research has focused on devising tests that require access to only a sublinear or even constant number of points in the distribution Canonne (2020). In this work, we work with distributions over , and thus we aim to devise algorithms with running time at most polynomial in . Previous work in testing distributions over Boolean functions has focused on the setting where the distributions offer pairconditional sampling access Chakraborty and Meel (2019); Meel et al. (2020). Using pairconditional sampling access, Meel et al. (2020) were able to test distributions for closeness using queries, where is the ratio of the probabilities of the most and least probable element in the support.
3 : a tractable algorithm for closeness testing
In this section, we present the main contribution of the paper: a closeness test for PCs, . The pseudocode of is given in Algorithm 1.
Given satisfiable circuits and weight functions along with parameters , decides whether the TV distance between and is lesser than or greater than with confidence at least . assumes access to an approximate weighted counter , and an approximate weighted sampler . We define their behavior in the following two definitions.
The algorithm
starts by computing constants and . Then it queries the routine with circuit and weight function to obtain a approximation of with confidence at least . A similar query is made for and to obtain an approximate value for . These values are stored in and , respectively. maintains a sized array , to store the estimates for . now iterates times. In each iteration, it generates one sample through the call on line 7. There is a small probability of at most that this call fails and returns . only samples from one of the two PCs.
The algorithm then proceeds to compute the weight of assignment w.r.t. the weight functions and and stores it in and , respectively. Using the weights and approximate weighted counts stored in the algorithm computes the value on line 10, where is an approximation of the ratio of the probability of in the distribution to its probability in . Since was sampled from , its probability in cannot be 0, ensuring that there is no division by 0. If the ratio is less than 1, then is updated with the value otherwise the value of remains 0. After the iterations, sums up the values in the array . If the sum is found to be less than threshold , returns and otherwise returns .
The following theorem asserts the correctness of .
Theorem 1.
Given two satisfiable probabilistic circuits and weight functions , along with parameters and ,

If , then returns with probability at least .

If , then returns with probability at least .
The following theorem states the running time of the algorithm,
Theorem 2.
Let , then the time complexity of is in . If the underlying PCs support approximate counting and sampling in polynomial time, then the running time of is also polynomial in terms of and .
To improve readability, we use to refer to the distribution and to refer to .
3.1 Proving the correctness of
In this subsection, we present the theoretical analysis of , and the proof of Theorem 1(A). We will defer the proofs of Theorem 1(B) and Theorem 2 to the supplementary Section A.4.2 and Section A.4.3, respectively.
For the purpose of the proof, we will first define events and . Events are defined w.r.t. the function calls and , respectively (as on lines 4, 5 of Algorithm 1). and represent the events that the two calls correctly return approximations of the weighted model counts of and i.e. , and . From the definition of , we have .
Let denote the event that (Algorithm 1, line 7) returns the symbol in the th iteration of the loop. By the definition of we know that .
The analysis of requires that all calls and both calls return correctly. We denote this superevent as . Applying the union bound we see that the probability of all calls to and returning without error is at least :
(1) 
We will now state a lemma, which we will prove in the supplementary Section A.4.
Lemma 1.
We now prove the lemma critical for our proof of correctness of .
Lemma 2.
Assuming the event , let , then

If , then

If , then
Proof.
If , then . Using this fact we see that,
Thus we have that . We now divide the set of assignments into three disjoint partition and as following: ; ; . The definition implies that the indicator is for all assignments in the set , and is for all assignments in . Similarly takes value and for all elements in and , respectively.
Now we bound the magnitude of ,
For , we have that , and thus:
We can split the summation into three terms based on the sets in which the assignments lie. Some summands take the value in a particular set, so we don’t include them in the term.
Since we know that and , we can alter the second and third terms of the inequality in the following way:
Using our assumption of the event and Lemma 1, Since , we get . We can now deduce that if , then and if , then . ∎
Using to test PCs in general.
Exact weighted model counting(WMC) is a commonly supported query on PCs. In the language of PC queries, a WMC query is known as the marginal () query. Conditional inference () is another well studied PC query. Using and , one can sample from the distribution encoded by a given PC. It is known that if a PC has the structural properties of smoothness and decomposability, then the and queries can be computed tractably. For the definitions of the above terms and further details, please refer to the survey Choi et al. (2020).
4 Evaluation
To evaluate the performance of , we implemented a prototype in Python. The prototype uses ^{1}^{1}1https://github.com/meelgroup/WAPS Gupta et al. (2019) as a weighted sampler to sample over the input  circuits. The primary objective of our experimental evaluation was to seek an answer to the following question: Is able to determine the closeness of a pair of probabilistic circuits by returning if the circuits are close and if they are far? We test our tool in the following two settings:

The pair of PCs represent small randomly generated circuits and weight functions.

The pair of PCs are from the set of publicly available benchmarks arising from sampling and counting tasks.
Our experiments were conducted on a high performance compute cluster with Intel Xeon(R) E52690 v3@2.60GHz CPU cores. For each benchmark, we use a single core with a timeout of 7200 seconds.
4.1 Setting A  Synthetic benchmarks
Dataset
Our dataset for experiments conducted in setting A consisted of randomly generated 3s and with random literal weights. Our dataset consisted of 3s with variables. Since the circuits are small, we validate the results by computing the actual total variation distance using bruteforce.
Benchmark  Actual  Result  Expected Result  

15_3  0.75  0.94  0.804  R  A/R 
14_2  0.8  0.9  0.764  A  A 
17_4  0.75  0.9  0.941  R  R 
14_1  0.9  0.99  0.740  A  A 
18_2  0.75  0.9  0.918  R  R 
Results
Our tests terminated with the correct result in less than 10 seconds on all the randomly generated PCs we experimented with. We present the empirical results in Table 1. The first column indicates the benchmark’s name, the second and third indicate the parameters and on which we executed . The fourth column indicates the actual distance between the two benchmark PCs. The fifth column indicates the output of , and the sixth indicates the expected result. The full detailed results are presented in the appendix Section B.
4.2 Setting B  Realworld benchmarks
Dataset
We conducted experiments on a range of publicly available benchmarks arising from sampling and counting tasks^{2}^{2}2https://zenodo.org/record/3793090. Our dataset contained 100  circuits with weights. We have assigned random weights to literals wherever weights were not readily available. For the empirical evaluation of , we needed pairs of weighted s with known distance. To generate such a dataset, we first chose a circuit and a weight function, and then we synthesized new weight functions using the technique of one variable perturbation, described in the appendix Section B.1.
Benchmark  Result  (s)  Result  (s) 

or70108UC10  A  23.2  R  22.82 
s641_15_7  A  33.66  R  33.51 
or5054  A  414.17  R  408.59 
ProjectService3  A  356.15  R  356.14 
s713_15_7  A  24.86  R  24.41 
or100102UC30  A  31.04  R  31.0 
s1423a_3_2  A  153.13  R  152.81 
s1423a_7_4  A  104.93  R  103.51 
or50510  A  283.05  R  282.97 
or60206UC20  A  363.32  R  362.8 
Results
We set the closeness parameter , farness parameter and confidence for to be and , respectively. The chosen parameters imply that if the input pair of probabilistic circuits are close in , then returns with probability atleast , otherwise if the circuits are far in , the algorithm returns with probability at least . The number of samples required for (indicated by the variable as on line 2 of Algorithm 1) depends only on and for the values we have chosen, we find that we require samples.
Our tests terminated with the correct result in less than 3600 seconds on all the PCs we experimented with. We present the empirical results in Table 2. The first column indicates the benchmark’s name, the second and third indicate the result and runtime of when presented with a pair of close PCs as input. Similarly, the fourth and fifth columns indicate the result and observed runtime of when the input PCs are far . The full set of results are presented in the supplementary Section B.
5 A characterization of the complexity of testing
In this section, we characterize PCs according to the complexity of closeness and equivalence testing. We present the characterization in Table 3. The results presented in the table can be separated into (1) hardness results, and (2) upper bounds. The hardness results, presented in Section 5.2, are largely derived from known complexitytheoretic results. The upper bounds, presented in Section 5.1, are derived from a combination of established results, our algorithm and the exact equivalence test of Darwiche and Huang (2002)(presented in supplementary Section A.1 for completeness).
5.1 Upper bounds
In Table 3 we label the pair of classes of PCs that admit a polytime closeness and equivalence test with green symbols and respectively. Darwiche and Huang (2002) provided an equivalence test for  s. From Theorem 1, we know that PCs that supports the and queries in polytime must also admit a polytime approximate equivalence test. A weighted model counting algorithms for s was first provided by Darwiche (2001), and a weighted sampler was provided by Gupta et al. (2019). Arenas et al. (2021) provided the first approximate counting and uniform sampling algorithm for s. Using the following lemma, we show that with the use of chain formulas, the uniform sampling and counting algorithms extend to loglinear distributions as well.
Lemma 3.
Given a formula (with a vtree ), and a weight function , requires polynomial time in the size of .
The proof is provided in the supplementary Section A.5.
5.2 Hardness
In Table 3, we claim that the pairs of classes of PCs labeled with symbols and , cannot be tested in polytime for closeness equivalence, respectively. Our claim assumes that the polynomial hierarchy (PH) does not collapse. To prove the hardness of testing the labeled pairs, we combine previously known facts about PCs and a few new arguments. Summarizing for brevity,
  

 
6 Conclusion and future work
In this paper, we studied the problem of closeness testing of PCs. Before our work, polytime algorithms were known only for the special case of equivalence testing of PCs; and, no polytime closeness test was known for any PC. We provided the first such test, called , that used ideas from the field of distribution testing to design a novel algorithm for testing the closeness of PCs. We then implemented a prototype for , and tested it on publicly available benchmarks to determine the runtime performance. Experimental results demonstrate the effectiveness of in practice.
We also characterized PCs with respect to the complexity of deciding equivalence and closeness. We combined known hardness results, reductions, and our proposed algorithm
to classify pairs of PCs according to closeness and equivalence testing complexity. Since the characterization is incomplete, as seen in Table
3, there are questions left open regarding the existence of tests for certain PCs, which we leave for future work.Broader Impact
Recent advances in probabilistic modeling techniques have led to increased adoption of the said techniques in safetycritical domains, thus creating a need for appropriate verification and testing methodologies. This paper seeks to take a step in this direction and focuses on testing properties of probabilistic models likely to find use in safetycritical domains. Since our guarantees are probabilistic, practical adoption of such techniques still requires careful design to handle failures. We are grateful to the anonymous reviewers of UAI 2021 and NeurIPS 2021 for their constructive feedback that greatly improved the paper. We would also like to thank Suwei Yang and Lawqueen Kanesh for their useful comments on the earlier drafts of the paper. This work was supported in part by National Research Foundation Singapore under its NRF Fellowship Programme[NRFNRFFAI120190004 ] and AI Singapore Programme [AISGRP2018005], and NUS ODPRT Grant [R25200068513]. The computational work for this article was performed on resources of the National Supercomputing Centre, Singapore (https://www.nscc.sg).
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Appendix A Proofs omitted from the paper
a.1 A test for equivalence
For the sake of completeness we recast the  circuit equivalence test of Darwiche and Huang [2002] into an equivalence test for loglinear probability distributions.
The algorithm:
The pseudocode for is shown in Algorithm 2. takes as input two satisfiable circuits defined over Boolean variables, a pair of weight functions and a tolerance parameter . Recall that a circuit and a weight function together define the probability distribution . returns with confidence 1 if the two probability distributions and are equivalent, i.e. . If , then it returns with confidence at least .
The algorithm starts by drawing a uniform random assignment from , where . Using the procedure given in Proposition 2 (in Section
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