RefineStat: Efficient Exploration for Probabilistic Program Synthesis
Authors: Madhav Kanda
Shubham Ugare
Sasa Misailovic
Venue: ICLR 2026 Oral (2026)
Keywords: Probabilistic Programming
Constrained Generation
Primary area: probabilistic methods (Bayesian methods, variational inference, sampling, UQ, etc.)

Abstract: Probabilistic programming offers a powerful framework for modeling uncertainty, yet statistical model discovery in this domain entails navigating an immense search space under strict domain‐specific constraints. When small language models are tasked with generating probabilistic programs, they frequently produce outputs that suffer from both syntactic, and semantic errors, such as flawed inference constructs. Motivated by probabilistic programmers’ domain expertise and debugging strategies, we introduce RefineStat, a language model–driven framework that enforces semantic constraints ensuring synthesized programs contain valid distributions, well‐formed parameters, and then applies diagnostic‐aware refinement by resampling prior or likelihood components whenever reliability checks fail. We evaluate RefineStat on multiple probabilistic-programming code-generation tasks using smaller language models (SLMs) and find that it produces programs that are both syntactically sound and statistically reliable, often matching or surpassing those from closed-source large language models (e.g., OpenAI o3).

Published as a conference paper at ICLR 2026
REFINESTAT: EFFICIENT EXPLORATION FOR PROBA- BILISTIC PROGRAM S YNTHESIS
Madhav Kanda
Shubham Ugare
Sasa Misailovic
University of Illinois Urbana Chammaign (madhav3, sugare2, misailo] ) 1111111111 www.iiini........ .du
ABSTRACT
Probabilistic programming offers a powerful framework for modeling uncertainty, yet statistical model discovery in this domain entails navigating an immense search space under strict domain-specift constraints. When small language models are tasked with generating probabilistic programs, they frequently produce outputs that suffer from both syntactic, and semantic errors, such as flawed inference constructs. Motivated by probabilistic programmers' domain expertise and debugging strate- gies, introduce REFINESTAT, language model-driven framework that enforces semantic constraints ensuring synthesized programs contain valid distributions, well-formed parameters, and then applies diagnostic-aware refinement by resam- pling prior or likelihood components whenever reliability checks fail. We evaluate REFINESTAT multiple probabilistic- programming code-generation tasks using smaller language models (SLMs) and find that it produces programs that are both syntactically sound and statistically reliable, often matching or surpassing those from closed-source large language models (e.g.. OpenAl o3). Our code is available at uttttps://githhi http://teetuurree//eeeeeeeeeeeee
bobbabiistic-proraamming
INTRODUCTION
Scientific discovery often requires expressing complex systems as statistical models. Finding appro- priate models that are both interpretable and computationally efficient challenging. The vision of automating model discovery has a long-standing history. Past approaches have demonstrated success across various domains, such as identifying physical laws (Bongard & Lipson 2007: McKinney
et al. 2006 Linka et al. (202)), recovering the structure of nonlinear dynamical systems (Schmidt & Lipson 2009), performing structure-aware nonparametric regression (Duvenaud et al. 2013). and tackling unsupervised learning problems (Grosse, 2014). However, they typically relied on significant manual effort experts were required to define a domain-specific language (DSL) for representing models and engineer custom search algorithms for exploring compositions within that DSL.
Large Language Models (LLMs) have the potential to automate the model discovery by leveraging their extensive knowledge across various domains, enabling them propose modeling approaches that were traditionally developed by human experts. However, using LLMSs comes with significant challenges. Directly querying LLMs generate statistical models often produces semantically flawed and unreliable programs, particularly in probabilistic programming languages like PyMC and NumPyro that evolve rapidly. These bugs hinder the correct execution of programs and constrain the effective exploration of the search space of working solutions. Further, running LLMs is costly, as expressive models (e.g.. GPT-4) incur high API fees.
These limitations motivate Small Language Models (SLMs) as practical alternative. Despite recent gains on coding tasks, they still produce semantic bugs code that runs but violates the intended statistical meaning. For example, in PyMC (Salvatier et al. 2016), an SLM can generate code that places variance where a standard deviation is expected - pm. No Normal ( instead of pm. Normal ...., si sigma=s imma). This change encodes the wrong statistical model, often inflating uncertainty and even triggering errors such as S Sampl ingError. In addition, SLMs may produce other semantic mistakes, such as using invalid argument name sd in place of sigma, which raises a TypeError at model construction time. These cases illustrate the need for our constraints and Bayesian-workee workfloww checks (Gelman al. 2020 ensure correctness.
si sigma gmm * 2)

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Our Work: REFINESTAT We present REFINESTAT, novel probabilistic programming synthesis framework that efficiently guides a language model to generate probabilistic programs. REFINESTAT is the first to demonstrate that open-weight SLMs can synthesize reliable probabilistic programs in the Bayesian-workffon sense i.e., they satisfy standard checks, such as adequate effective sample size, low number of MCMC divergences and strong out-of-sample fit (Section presents the full list).
REFINESTAT produces reliable statistical programs through a two-phase approach: (1) semantically constrained generation and (2) diagnostic-aware refinement (Section 3). In this context, semantically constrained denotes adherence to programiingllnguage semantics (e.g.. distribution validity, pa- rameter consistency, proper data types), rather than the linguistic semantics of natural languages. Our semantic constraining ensures that synthesized probabilistic programs contain valid distributions with well-formed parameters, proper variable dependencies, and adherence to PyMC semantics. The diagnostic-awan refinement systematically resamples prior specifications or likelihood models when generated programs fail to meet established reliability criteria within the Bayesian workflow, thereby ensuring efficient search of probabilistic models using small language models.
We evaluate REFINESTAT on a suite of five representative probabilistic datasets, and five open- weight LLMs, with up to 8 billion weights. Our comparison shows that REFINESTAT significantly improves over directly querying LLMs in an unconstrained manner or only syntactic constraining with Syncode (Ugare al. 2024). We show that the programs generated by REFINESTAT often pass the diagnostic metrics that indicate high quality to represent and explain the data. We also show recent LLM-based generation algorithm BoxLM et 2024), which uses two GPT-4 LLM instances to iteratively propose likely program and refine it, respectively; yet REFINESTAT obtains those results with single small language model.
that the REFINESTAT'S performance is comparable
Contributions: The main contributions of this paper are:
Approach: We present REFINESTAT, a novel ML---based framework for synthesis of probabilistic programs that are semantically correct and have high predictive performance.
Constrained decoding: We propose using semantic constrained decoding to help generate syntac- tically and semantically valid probabilistic programs, at a small overall cost.
Iterative program search: We present an iterative refinement loop that leverages single, un- modified open-weights SLM to generate probabilistic programs with improved diagnostic metrics, refining statistical reliability by selectively resampling the likelihood and prior.
Evaluation: We demonstrate that REFINESTAT performs significantly better than baseline language models, in terms of different diagnostic metrics, and in some cases performs equally well as GPT-4 and hand written developer programs.
2 BACKGROUND
Language Models. Current autoregressive language models (LMs) operate on vocabulary V " of tokens. A tokenizer converts an input prompt On E E into a sequence of tokens & 12,2..... lk. The LM M : V' - IRI takes this sequence and outputs scores S over the vocabulary: S M(1, 12, 12,,..,. ...,. 4x). A softmax function transforms these scores into a probability distribution, from which f++1 is sampled. Appendix A. has more details on decoding and grammar-guides generation.
Bayesian Workflow. A robust Bayesian analysis follows an iterative workflow of model specification, posterior inference, diagnostic checking, and model comparison (Gelman et al. 2020). This process can be summarized as: (1) specify the model (likelihood and priors, in our case using an LLM), (2) perform posterior inference, (3) conduct posterior predictive checks and convergence diagnostics, (4) if diagnostics pass, estimate out-of-same of-sample fit (i.e.. how well the model would predict data not used in fitting), and (5) compare and rank models by their relative out-of-sample performance (with uncertainty). Further details about diagnostics and predictive evaluation in Appendix A.2
Probabilistic Programming. Statistical modeling aims to describe relationships between variables in data through joint probability distributions that capture both observed phenomena and underlying latent structure. In probabilistic modeling, we formalize this joint distribution p(z, =|n),, where x = Zi. represents n observed data points, 2 = 21:m denotes m latent variables, and corresponds to fixed model parameters. The inferential goal is to compute the posterior distribution p(2[z)) which quantifies uncertainty in the latent variables conditional on observed data. Probabilistic programming languages (PPL.) provide flexible computational substrate for specifying joint distributions p(a, z
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programs while leveraging automated inference methods (e.g., MCMC, variational inference) to compute the posterior p(= z) (van de Meent et al. 2021). Further details in Appendix A.3
Probabilistic Programming Diagnostics We briefly define the standard probabilistic programming diagnostics and metrics used our framework. Detailed formal definitions are in Appendix
1. P.g: Split-ft statistic for parameter d, measuring MCMC chain convergence.
2. F: The effective bulk sample size for the parameter o, estimating the sampling efficiency across the central mass of the posterior.
ES Tail effective sample size for parameter Φ. measuring sampling efficiency in tails.
4. Divergences( M): Count of divergent NUTS (Hoffman et al. 2014) transitions in model M.
(F) Bayesian Fraction of Missing Information for model M, assessing energy transition efficiency in Hamiltonian Monte Carlo (HMC) (Neal et al.) 2011 algorithm.
6. (M): Pareto shape parameter for observation i in PSIS-LOO (Pareto-smoothed importance sampling leave-one-out cross-validation), quantifying reliability of importance sampling estimates. PSIS-LOO approximates exact LOO predictive densities by smoothing raw importance weights with generalized Pareto fit to stabilize high-variance weights (Definition beloow.
7. elpd: Expected Log Pointwise Predictive Density under Leave-One-Out cross-validation, measur- ing model's s out-of-sample predictive accuracy.
To evaluate out- -fa-ppl predictive accuracy, we rely on the expected log pointwise predictive density under leave-one-out cross-validation (ELPD-LOO). direct computation of LOO requires refitting the model n times (once for each observation), which is often too costly in practice. To avoid this, we use Pareto-smoothed importance sampling leave-one-out volsdvaaiootion (PSIS- LOO) (Vehtari et al.) 2017), which provides a fast approximation to ELPD-LOO and also gives diagnostics influential observations via the Pareto i values:
Definition (PSIS-LOO (Vehtari et al. 2017)) Let D (u1)21 be the observed data and M model. Draw {0(5))5 from p(O D), compute raw importance weights w!!) 1/ply/ I o( and let w. denote the Pareto-smoothed weights, obtained by replacing the largest tail weights with generalized Pareto fit. Then the PSIS-LOO estimate is elpdpsisLLO
Ei_, E. (()) p(y: 1(())
Instead of refitting the model times, PSIS-LOO relies single full-data fit and stabilized importance weights. The generalized Pareto fit yields shape parameters k,, which assess the reliability of the approximation; following standard guidance, the estimate is considered unreliable if about 20% of the h; exceed 0.7.
REFINESTAT
Figure D presents REFINESTAT'S 's two main ideas. First, prune the search space of possible proba- bilistic programs by enforcing semantic validity during generation, mapping validation rules to nodes
Constrained Semantic
Bayevian Reliability Check
Gemeration
Pase-aliiee skilky
Tye ndiidty
Prior
Convergence
Diaribucce
valiity
Depestoccy
vallky
shverpence
LM
validay
Valdky
Semple
hralth
Predictive Welieity
(2)
Relax
(3)
Data
Prompt
Rellable
Statistical Medel
Figure I: REFINESTAT workflow: (1) A user provides data and prompt to the language model, which generates a probabilistic program. (2) Constrained semantic decoding enforces syntactic and semantic validity of the generated program. (3) A Bayesian reliability check diagnoses convergence, diver- gences, and predictive validity. failures are detected, the model is refined by backtracking and re- sampling priors likelihoods. (4) Upon passing checks, we get final reliable probabilistic program.
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1. (M) max (M) S an ) Published as a conference paper at ICLR 2026
in the partial parse tree and resampling problematic program fragments when constraints violated. Second, we implement diagnostic-aware refinement, systematically resampling components of sta- tistically unsound models to satisfy Bayesian Workflow guidelines. This integrated approach aims to improve semantic correctness, statistical reliability, and yield strong predictive performance. An illustrative example is provided in Appendix C.1
Problem Statement. Let 'D denote the dataset for given statistical modeling task. The objective of REFINESTAT is construct a statistical model M in probabilistic programming language that provides accurate predictive performance while quantifying uncertainty in fully Bayesian manner.
Our approach finding model M that explains the data will follow the standard Bayesian work- flow (Gelman et al. 2020) The key challenge to finding such a model M is to automatically compare various candidate models that an LLM produces. To automate this task, we will use a battery of diagnostics from statistical literature (Section)) 2 computed during the posterior inference in the standard Bayesian workflow (Gelman et al.) 2020).
Definition 2 (Bayesian Workflow Reliability Score) Let M be the set of candidate models, and fix thresholds oRR, Poulk, Stall, E. e. For each M E M, we define seven indicator functions 1, if eventt A holds, and set otherwise,
My) E (0, 1 by I[A| 00
2. (2(M) M[BMM((M) 7]:
3. (M = min ESS (M) 2 3balkk
4. (M = I[divergences(MM M) =0];
5. (M TI min F.S. a (M) 2 Beail
6. 86(M)
[Ra(M) S Tecd 2
7. (M = I (l) (M) is finite
Then the reliability score is B(M) = Σド= (M(
These diagmostics can be extracted directly from the MCMC engines (e.g. Stan (Carpenter et al. 2017a), PyMC (Salvatier et al. 2016)). Although elpd provides principled Bayesian measure of out-of-sample predictive accuracy, its Monte Carlo estimate can be unreliable if the sampler has not fully converged if the importance weights are unstable (Gelman et al., 1993). To mitigate
these risks, we consider elpd estimates for models that satisfy standard convergence thresholds, thus ensuring that predictive comparisons rest on reliable posterior samples.
We require each model to pass most of these seven checks: if any check fails, the corresponding (yM is zero, and the total score reflects how many diagnostics remain satisfactory. The final check concerns the availability of the elpd(M .M) estimate; if elpd(M) cannot be computed or is infinite, then SELPD (M) 0, and the model is treated as failing that diagnostic. A higher overall score indicates that more diagnostics have passed, so when elpd (M) ) is available, the resulting estimate can be trusted with greater confidence. We consider model reliable once its score exceeds a cutoff Ç. We use Ç = to allow marginal diagnostic failures while maintaining confidence in the reported elpd.
Definition 3 (Valid model space) For set of candidate models M, valid model space is:
Mvalid = {M M (BMM) B(M)
Definition 4 (REFINESTAT Objective) We finally define objective to identify the model that attains the highest ELPD-LOO estimate within the space of models Mvalid:
M arg max (I)),
3.1 SENANTICALIY
PROBABILISTIC PROGRAM GENERATION
We formalize the generation of semantically valid probabilistic programs via iterative constrained sampling. Let G = (N, T, P. P.))) be a context-free grammar with nonterminal symbols N,, terminal symbols T, production rules P, and start symbol So-
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For partial program with parse tree let F(x) denote the set of program fragments, where each fragment n E F()) is a rooted subtree of A corresponding to single syntactic statement. Validation functions operate fragment within program context и E II: db F(x) II {0.1). These
functions are conjunctions of individual correctness checks:
(n, π) = @ (n, T) A o (n, T) A... T (n,
Each fragment thus corresponds to single statement, possibly comprising multiple AST nodes.
Validity Predicates for Probabilistic Program Fragments. To ensure the correctness of synthesized probabilistic program fragments, we define three essential validation predicates. Let F(s) denote the set of all probability distribution functions invoked within fragment s. The validation predicates are:
1. Parse-ability: Φ (s. IT) = if the fragment conforms to the grammar G.
2. Distribution validity: (2 II) = II FFF)))) 1(f E EM} verifies that each probabilistic operation
exists in the available library .M of PPL.
3. Parameter validity: (s IT)
(EF(e) 1(P(f) S Paco(f)} confirms that operation parame- ters P(f) adhere the accepted specifications (f) essential for maintaining probabilistic semantics. i.e. we ensure that the provided parameters for any distribution are correct according the distribution's 's specification. For instance, Figure shoows parameter "sd" was invalid and resampled correctly as "sigma".
4. Dependency validity: (s IT) vanvss(s) 1 (all dependencies of defined before use)
ensures that random variables declared and initialized before they referenced.
Support validity: (s, II) [I {f) e Supp(f)} confirms that parameter values fall within the distribution's 's support (e.g., variance > 0, probabilities in [0, 1)-
6. Type validity: ( II) = II (EFG)) [(type(P(f) E T(f) checks that each parameter P(f)
has the expected type from the specification T(f), e.g.. ensuring numeric values for scale parame- ters, or integer values for counts.
The final predicate ((s, II) = Ai_, 0i(s, II) ensures that generated fragments satisfy all re- quirements of the probabilistic programming language (1,4,6) and the Bayesian model (2,3,5). Our generation algorithm leverages these prop- erties by maintaining global symbol table A - M mapping each alias a E AA its module or namespace E M.
Prompt
Provide the PyNC nodel deffinttion, for the following datasets
121, 125,
Ned schooli
pormol teeut,
Language Model
with po.Model O 2s R:
aigmami0t
Code Generation until now...
Generation proceeds via local rejection sam- pling on Sy: we repeatedly sample until finding s* where ((s", ((ss", II) = This iter- ative process continues until termination frag- ment is generated, ensuring every component the final probabilistic program satisfies all se- mantic constraints. Our local rejection sampling token-efficient and we backtrack and precisely resample the tokens that correspond the vio- lation of the constraints. The approach particularly effective for languages like PyMC, where maintaining consistent probabilistic variable scopes and dependencies is critical. Combining syntactic constraints with semantic validation enables efficient exploration of the program space while ensuring the probabilistic soundness of generated models.
SN
Step
da
pa. Normal ("ru', muec, HOR18:
+1
Noreol Caepty Mindy is aigeno-1
Figure 2: Constrained decoding in REFINESTAT fixing a TypeError from using sd instead of a sigma, as illustrated in Section
3.2 PROGRAM VALIDATION AND GUIDED RESAMPLING
Building on the validation predicate ④ we now formalize our constrained generation and refinement methodology. We introduce the beastrained-decodinn operator LCD : - B, which implements validation-guide guided sampling. Here, statement is individual syntactic unit, and code block B a (possibly statemmttt sequence of such statements. For any context C, this operator returns code block B satisfying (CC]IB) where denotes sequential concatenation of blocks;
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concretely, for two blocks A and B, A| is the program text formed by appending all statements of B immediately after those of A.
Algorithm 1 REFINESTAT Synthesis via DI|PIIC
D|[P|| C denotes the full probabillistic program with data D. prior P. and likelihood C. Dur- ing refinement we perform resampling via two steps: L < (Pil)) (likelihood resampling), P Loa(D) (D) (prior resampling), guaranteeing replaced blocks remain semantically valid.
Require: Rmax; a 3, (r)==11 K
1:
+ 0, l +0-0, V f 2, p - 1-,,45 2, +
2: while Rmax and |V] < do
3:
Prog + D
4:
~ (Prog) then
5:
r r -- continue
Before refinement begins, the data block D is taken directly from the user prompt, while the initial prior and likelihood blocks are generated
6: compute diagnostics di di...., dr. on Prog
7: Prass f- [ d3 2 23
8:
if P pass K then
via constrained decoding. i.e., P < (CD(D) and CCD(D I| P). Algorithm ! synthe-
9:
t Vu [Prog]: continue
10:
if l < a then
> likelihood resampling
sizes programs in two phases. First, it checks
11:
- Lot (D(PP)
semantic correctness via L ensuring parseabil-
12:
+ f+
ity, distribution validity, and parameter consis-
13:
else
b prior resampling
tency. Second, evaluates Bayesian diagnos- ties dj...., dp on the full program CI at least thresholds {Ty} must be met to accept a candidate. As shown in Figure [ if diagnostics fail, we perform one of two resampling steps to refine the program: (i) likelihood resampling replaces L under the data-prior context, addressing convergence or sampler-health issues; (ii) prior resampling replaces P under the data context, correcting prior-specification errors. We iterate until we collect B valid programs exhaust the budget Rmax. and return the program maximizing elpd.
14:
+ Led(D)
15:
rer+1
16:
return arg max (M) over all M
4 EXPERIMENTAL METHODOLOGY
We use PyMC, Python probabilistic programming library, to perform inference. We provide the same initial prompt while performing unconstrained generation and using REHNESTAAT. We prompt the model by providing it with the dataset, the necessary library and the text query. The exact format of the prompt is provided the Appendix [ As stated in Definition [ we assess model reliability using standard Bayesian diagnostics (Vehtari et al., 202 2017 Gelman et al. 1995). Further details on hyperparameters and experimental setup are provided Appendix
Datasets. We use five benchmark datasets from Stan PosteriorDB (Magnusson et al. 2024). mirroring the selection in prior research on automated statistical modeling (Li 20224)
Eight Schools (Rubin| 1981): This dataset originates from a study commissioned by the Educa- tional Testing Service, which examines the effects of coaching programs on test performance.
Dugongs (Unit) b): This dataset provides measurements on the ages and lengths of 27 dugongs.
Surgical (Unit a): This dataset comprises records on the number of cardiac surgeries performed on infants, along with the associated failure rates.
Peregrine (M Kery, 2011): This dataset tracks the breeding trajectory of the peregrine falcon population in the French Jura region from 1964 to 2003.
GP: This dataset contains simulated observations generated from a Poisson Gaussian Process.
Models. We experiment with a range of state-of-thee LLMs, spanning multiple parameter scales including Qwen2.5 (code-specific) (Hui et al. 2024). models from the Llama series, DeepScek. and Google's CodeGemma. We have used total of four models including, Llama3-8B (Grattation)
al. 2024), CodeGemma-77 7B (Team et al., 2024).
1ren2.5-Coder-75 (Hui et al.) 2024), and
D (hereafter we refer to it as "DQ-7B") (Guo et al. 2025
EXPERIMENTAL RESULTS
5.1
IMPROVED RUN RATE OVER UNCONSTRAINED AND SYNTAX-DRIVEN GENERATION
We conduct a comprehensive evaluation on diverse datasets, comparing REFINESTAT against both an unconstrained baseline and leading language models across suite of diagnostic and performance

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metrics. We used identical prompts across our framework, the Standard baseline (unconstrained), syntactically-constaaae constrained tool Syncode (Ugare et al., (024). and REFINESTAT.
Table I presents Run rates across different temperature settings, Run rate is the fraction of programs that successfully produce the samples from the posterior distribution. The problems that do not run successfully include those with runtime emros such as (1) numerical errors, e.g., inf/nan, (2) sampling issues due to unlikely prior parameterization, (3) other sampling warnings, e.g., failed to initialize chains, (4) static compilation issues.
REFINESTAT achieves success rates approxi- mately 40 percentage points higher than the Standard baseline and 30 points higher than Syn- code, demonstrating that our validation-guiide -ggided approach substantially enhances code genera- tion reliability by mitigating both syntactic and semantic error sources. These results show that REFINESTAT significantly enhances code generation reliability. The top root causes of failure are syntax errors, semantic errors, and sampler pathologies. We categorize these failures and provide detailed discussion in Appendix F.I
Table I: Run rates for Standard, SYNCODE, and REFINESTAT by temperature
Temp.
Standard
SYNCODE
REFINESTAT
0.2
0.10
0.21
0.45
0.3
0.11
0.21
0.50
0.4
0.11
0.21
0.50
5.2 COMPARISON GENERATED PROGRAM QUALITY TO UNCONSTRAINED BASELINE
To evaluate semantic correctness and diagnostic robustness, we compared programs generated by the base language model with those from REFINESTAT under identical prompts. Since observed that REFINESTAT consumes almost twice the number of tokens used by the Baseline (see Appendix G.I. baseline models five times with different seeds (2.5x tokens and compare the best program based on Bayesian Reliability Score and ELPD LOO to single REFINESTAT run. We repeat this process five times compute the mean and standard deviation for all metrics. In Table we report five representative metrics drawn from these diagnostics (for space reasons). Bold entries denote cases where REFINESTAT outperforms the corresponding baseline. Since the Reliability Score aggregates multiple diagnostic metrics, is highlighted most frequently;; when Reliability Scores are similar, we instead emphasize differences ELPD LOO. The symbol X indicates that no valid program was produced-i.e. the method failed to explore the search space sufficienlly to yield a correct result.
than REFINESTAT)
Except for instance, REFINESTAT matches exceeds the Standard Baseline in terms of the Bayesian Workflow Reliability score, and in some cases achieves up to twice the reliability, Moreover, REFINESTAT delivers substantially better performance
individual diagnostics, particularly diver- gences. Notably, DQ-7B, which failed on every dataset under the Standard Baseline, succeeded on all datasets when augmented with REFINESTAT. For example, on the Surgical dataset with Meta-Llama, the Standard Baseline produces over 1000 divergences, while REFINES TAT produces none.
We observed that in cases where the Standard baseline attains higher ELPD-LOO than REFINESTAT, closer inspection of the diagnostics exposes unreliable sampling. For instance, on the Meta-Llama
GP task the Standard model achieves superior ELPD score, but exhibits split- R 4.13 > 1 and a low reliability score (3), indicating severe convergence issues. In contrast, REFINESTAT may report a marginally lower ELPD yet maintains R R 1 and a higher reliability score, reflecting trustworthy posterior estimates. Furthermore, REFINESTAT delivers markedly lower variability in key diagnostics such R and the number of divergent transitions demonstrating its consistency and robustness. We further illustrate the types of structural changes introduced during refinement in Appendix C.2
5.3 COMPARISON OF GENERATED PROGRAM QUALITY TO BoxLM
Table 3: Comparison of ELPD LOO scores with BoxLM (Li
We compare REFINESTAT'S perfor- mance using the DQ-7B model (aver- aged over five runs) against three base- lines: the Expert stan programs from PosteriorDB (Magnusson et al. 2024 the BoxLM system introduced by Li et al. (2024), and programs generated
eet al. 2024), REFINESTAT using DQ-7B, and Expert values
Dataset
Expert
REFINESTAT w/
BoxLM GPT4
OpenA1-o3
DQ-7B((men 1: std)
(mean
std)
Light schools
30.70
-3.68 0.11
-30422
30.74 0.07
Dugongs
22.43
8.35 0.06
23.40
22.83
8.12
Peregrime
-112.60
-114.29
+: 2.76
-173.11
113..29
+ 10.33
Surgical
.3.73
46.51
GP
-26.33
-23.39 + 1.14
.34.5 + 13.28
by OpenAl o3. Since the code for BoxLM is not publicly available,
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Table 2: Comparison of Diagnostic Scores and ELPD-LOO for Standard vs. REFINESTAT
Dataset
Model
Variant
Reliab. Ser:
R_
ESS Bulk f
Diverg
Pareto k
ELPD
Mcan
Std
Mcan
Std
Mcan
Std
Mcan
Std
Mcan
Std
Mcan
Std
Standord
7.00
0.00
1.00
0.00
2261.00
0.00
0.00
0.00
0.00
0.00
-31.70
0.00
Metn-LLamo-3-88
RefineStat
7.00
0.00
1.00
0.00
2303.00
768.76
0.00
0.00
0.00
0.05
-31.77
0.61
CodeGemma-7B
Standard
x
x
K
X
x
X
X
X
X
X
X
RefineStat
7.00
0.49
1.00
0.00
2573.00
527.68
0.00
1.97
0.00
0.06
-31.46
1.84
8 Schools
Qwen Codeer
Standurd RefineStat
3.80
1.30
1.02
0.01
223.25
92.75
31.38
0.22
0.21
-31.31
0.68
5.00
0.45
1.01
0.01
256.50
883.83
34.50
86.89
0.00
0.05
-30.80
0.05
DQ-7B
Standard
x
x
X
x
x
X
X
x
x
X
RefineStal
5.00
0.98
1.02
0.02
219.00
2176.00
102.00
75.36
0.00
0.00
-30668
0.11
Meta- L-ama-3-88
Standard RelineStat
7.00
0.00
1.00
0.00
2167.67
884.79
0.00
0.00
0.01
0.02
-7.66
30.39
7.00
0.00
1.00
0.00
1696.00
284.33
0.00
0.00
0.02
24.51
CodeGemma-77 7B
Standard
5.70
2.30
1.04
0.06
1527.33
1325,54
285.67
494.79
0.00
0.00
3,90
7.71
RelineStat
7,00
0.00
1.00
0.00
1908.00
2066.23
0.00
0.00
0.04
0.02
8.07
15.42
Dugongs
Qwen-Coder-7B
Stundard
7.00
0.00
1.00
0.01
1788.67
123.43
0.00
0.00
0.04
0.00
8.15
0.29
StaineStal
7.00
0.00
1.00
0.00
1683.00
148.16
0.00
0.00
0.04
0.02
8.29
0.05
DQ-7B
Standard
X
x
X
X
X
X
X
x
x
x
X
RefineStat
7.00
0.00
1.00
0.00
2376.00
149.61
0.00
0.00
0.00
0.03
8.35
0.06
Mata-LLamaa--SB
Standard RefineStat
6.00
0.00
1.00
0.00
3774.00
0.00
7.00
0.00
0.00
0.00
114.966
0.00
7.00
0.00
1.00
0.00
3574.00
428.26
0.00
0.00
0.00
0.00
-173.00
4.91
CodeGemma-7B
Standard
7.00
0.00
1.00
0.00
4261.00
0.00
0.00
0.00
0.00
-172.91
0.00
RefineStat
6.50
0.53
1.00
0.00
2930.00
1343.79
0.50
0.53
0.00
0.00
-129.93
3.91
Peregrine Qwen-Coder-7B
Standard
7.00
0.00
1.00
0.00
4238.00
0.00
0.00
0.00
0.00
0.00
-173.11
0,00
RefineStat
7.00
0.00
1.00
0.00
4679.00
88.73
0.00
0.00
0.00
0.00
-172.98
0.12
Standard
x
x
x
x
X
x
x
x
x
x
x
X
DQ-7B
RelineStat
6.50
0.53
1.01
963.50
436.70
25.50
27.26
0.00
0.00
-114:29
2.76
Meta-LL LLama-3-88
Standard
5.50
1.30
1.01
0.01
1159.75
1202.11
1066.50
1267.61
0.65
0.44
-31.59
20.02
RelineStal
6.00
0.77
1.00
0.00
579.00
1272.12
0.00
5.75
0.00
0.36
.46.73
36.63
Standard
5.50
0.70
1.00
0.00
1230.00
503.46
6.00
5.66
0.46
0.65
-42.02
5.75
CodeGemma-78 7B
RelineStal
6.00
0.49
1.00
0.00
2026.00
450.46
0.00
0.49
0.00
0.12
-45.55
381.60
Surgical
Owen-Coder-7B
Standard
6.30
1.50
1.01
0.02
1332,75
784.13
37.50
75.00
0.23
0.46
-44.48
4.35
RelineStal
7.00
0.41
1.00
0.00
1642.00
1350.46
0.00
0.00
0.00
0.38
-46.55
2.79
DQ-7B
Standard
x
x
x
x
X
X
X
X
X
x
X
RefineStat
7.00
0.50
1.00
0.01
2800.00
1182.68
0.00
0.00
0.00
0.37
-46.51
3.49
Meta-III Lama--1-888
Standard RefineStat
3.00
0.00
4.13
0.00
4.00
0.00
1634.00
0.00
0.00
0.00
-21.61
0.00
6.00
0.49
1.00
0.00
1710.00
668.57
13:00
12.39
0.09
0.08
-152.30
139.07
CodeGemma-7R
Stundard
7.00
0.00
1.00
0.00
6034.00
0.00
0.00
0.00
0.18
0.00
-154.42
0.00
RefineStat
7.00
0.98
1.00
0.00
1752.00
1004.8 88
0.00
2.46
0.00
0.49
-22.76
126.08
GP
Qwen-Cader-7R
Standard RefineStat
x
x
x
X
X
x
X
X
X
X
X
X
x
x
x
x
x
X
X
X
X
X
Standard
x
x
x X
x
x
X
X
x
X
DQ-7B
RelineStat
6.50
0.53
1.01
0.00
816.50
89.27
30.50
32.61
0.00
0.00
-23.99
1.14
rely on the reported numbers from their paper for comparison. Because the dataset we use for the GP task not included in BoxLM's evaluation suite, we omit their result for that dataset.
Table 3 presents the ELPD LOO scores, showing that our framework consistently outperforms both BoxLM and OpenAI o3 on the PEREGRINE dataset, and surpasses OpenAI o3 on the GP task, while matching the performance of expert-written programs in most cases. Across the remaining datasets, our approach performs comparably to other baselines, with the exception of DUGONGS, where performance is slightly lower. These results demonstrate that REFinEStAT achieves performance comparable to, and in several cases better than, large language models like OpenAl o3 and multi-agent frameworks such as BoXLM.
5.4 ABLATION STUDY
Effectiveness of Semantic Validation Components. To evaluate the contribution of each validity predicate within REFINESTAT, we perform an ablation study by systematically disabling one com- ponent time and measuring the resulting compilation rate. This analysis is conducted across all models with ten different random seeds to ensure robustness. Notably, removing all validation predicate reduces the system to the behavior of SYNCODE.
8

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Published as a conference paper at ICLR 2026
Table 4 shows that parameter validity emerges as the most critical component, its removal results in substantial drop of 14.5 percentage points in compilation success. All checks contribute meaning-
Table 4: Ablation Study: Impact of Semantic Validation
Checks on Run rate
# Method
Run
Aun
REFINESTAT (all components)
50.0%
fully to overall performance: omitting distribution validity or parse-ability con- sistently reduce compilation rates (-9% and -5.5% respectively), underscoring the complementary role these validations play in ensuring soundness of generated programs.
2
w/o Parameter validity
35.5%
- -1.55%
3
w/o Distribution validity
26.5%
- -9%
4
w/o Parse-ability
21.0%
-5.5%
5
w/o grammar-guided generation
11.0%
-10.0 %
Memorization Effect. A number of studies (Dong et al. 2024a Golchin & Surdeanu 2025 2024
Li 2023) have highlighted concerns about the memorization effect in large language models, where models may reproduce previously seen content rather than demonstrating genuine synthesis. (Kong et al. 2025) addresses this issue by proposing code mutation to reveal potential memorization in program repair tasks, Inspired by this perspective, we introduce dataset and prompt modifications designed to preserve the semantics of the data while changing its presentation. We apply two systematic prompt modifications across all benchmarks when evaluating Meta-Llama 3-8B, with details provided in Appendix G.2 The modified versions match the original REFINESTAT on reliability, convergence, and predictive metrics. Additionally, PosteriorDB provides limited ground- truth PyMC programs: only the Eight Schools model is available, and it targets an outdated version of PyMC, while the remaining programs are provided in Stan. While these studies are limited in scope, and measuring the memorization effect is still an open problem, they give an indication that REFINESTAT'S effectiveness may not be the consequence of just memorization.
5.5 GENERALIZABILITY ACROSS PROBABILISTIC PROGRAMMING BACKENDS
Although our primary evaluation uses PyMC, the design of REFINESTAT is not specific to any single probabilistic programming library. To show generalizability of the framework, we apply REFINESTAT to NumPyro using the same prompting setup and the Qwen-2.5-Codee-7B model.
Across temperatures, REFINESTAT substantially improves the fraction of programs that success- fully compile and execute. Table 5 shows that the run rate more than doubles relative to Stan- dard unconstrained decoding, consistent with the improvements in PyMC3 experiment (Table [[. Table[]reporrts the metries for NumPyro generated pro- grams across all benchmarks, as those in Table 2 (for PyMC3). For every dataset, REFINESTAT attains equal or higher reliability than the Standard baseline. When both approaches achieve similar reliability, REFINES- TAT yields comparable or improved predictive perfor- mance. Notably, in the GP task, Standard decoding fails to produce any valid model, whereas REFINESTAT consistently generates executable programs with stable diagnostics. Overall, these results show that REFINESTAT'S semantic filtering and diagnostic- refinement generalize across PPL. backends, improving program validity and sampling quality beyond PyMC.
Table 5: Run-rate comparison when using NumPyro as the inference backend.
Temp
Standard
REFINESTAT
0.2
0.17
0.34
0.3
0.15
0.35
0.4
0.15
0.33
Table 6: Diagnostics and ELPD-LOO results for NumPyro using the generalizability of REFINESTAT.
demonstratin
Dataset
Variant
Reliab, Ser.
R L
ESS Bulk
Diverg.
J
Pareto
ELPD LOO
Mean
Sid
Mean
Sid
Mean
Sid
Mean
Sid
Meun
Std
Meun
Sid
8 Schools
Standard
7.00
0.00
1.00
0.00
1769.00
0.00
0.00
0.00
0.13
0.00
-31.83
0.00
REFinESTAT
7.00
0.00
1.00
0.00
1850.00
50.00
0.00
0.00
0.11
0.02
-3.100
0.05
Standard
7,00
0.00
1.00
0.00
1739.00
0.00
0.00
0.00
0.04
0.00
8.18
0.00
Dugongs
REFINESTAT
7.00
0.00
0.00
1747.00
11.55
0.00
0.00
0.04
0.00
8.19
Peregrine
Standard
6.00
0.00
1.00
0.00
3232.50
81.32
0.00
0.00
0.00
0.00
368. 21
0.04
REFINESTAT
6.67
0.47
1.00
0.00
1658.00
135.00
0.00
0.00
0.11
0.10
-76.25
122.44
Surgical
Standiad
6.00
0.00
1472.00
-245.14
REFINESTAT
7.00
0.00
1.00
0.00
1628.00
0.00
0.00
0.00
0.08
0.00
-50.37
0.00
GP
Standard
X
X
X
X
X
X
X
REFINESTAT
6.67
0.47
1.00
0.00
2344,33
115.46
0.00
0.00
0.09
0.13
-83.07
82.70

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Published as a conference paper at ICLR 2026
6 RELATED WORK
Researchers have presented probabilistic techniques for discovering model structures from observed data, including Bayesian networks (Mansinghka al.] 2006; Lowd & Domingos 2012). matrix- composition models (Grosse et al. 2012). Markov networks (Gogate et 2000 and deep probabilis- tic models (Gens & Domingos] 2013). Probabilistic programming languages (PPL.s) flexibly represent these models as programs but demand substantial domain and API expertise. Recent advances in LLM code generation make synthesizing probabilistic programs more accessible. REFINESTAT leverages this opportunity to advance the state of the art in LLM-based probabilistic program synthesis.
There have been many approaches for deterministic program synthesis, which is recently been dominated by LLM--baed approaches. Several works (Nori et al..
Probabilistic Program Synthesis:
2015 Saad al. 2019; Gerasimou al. 2015 Ceška al. 2019; Ellis al. 2015) synthesize proba- bilistic programs using classical machine learning or symbolic methods. Prior work has also proposed techniques for dehugging probabilistic programs (Dutta et al. 2019: 2022; Nussbaumer et al. 2026). which could provide further feedback for probabilistic program synthesis. (Gerasimou et al. 2015) uses genetic algorithm for probabilistic model generation. (Saad et al. 2019) presents techniques to automatically construct probabilistic programs using Bayesian inference over DSLs defined via probabilistic grammars, enabling qualitative structure discovery and quantitative prediction.
Most recently, (Li al. 2024) uses LLM for probabilistic program synthesis. The paper shows that with instances of GPT4 as the generator and critic (closed LLM) can find reasonable probabilistic programs from data. As our evaluation shows, REFINESTAT significantly improves the ability to find programs that fits the data (recall Table 3) and in contrast to (Li et al. 2024), runs only single small open LLM (< 8B weights), demonstrating the benefits of constrained decoding.
Program Synthesis with Constrained LLM Decoding: Recent advances in program synthesis have enabled constrained decoding approaches where LLMs generate code while adhering to formal language specifications. These constraints can be partially precomputed and enforced more efficiently
for regular (Deutsch et al. 2019 Willard & I Lour, 2023 Kuchnik et al.
2023) or context-free (Koo)
al. 2024; Ugare et al. 2024 Dong et al.J 2024b; [aanereee et al., 2025 Suresh et al. 2025
Firestone et al. 2025) languages, ensuring syntactic correctness. Recent grammar-constrained and
type-constrained approaches rely on a prefix property (Mündler et al.] 2025), where each partial program can be incrementally validated and completed into syntactic and/or well-typed program. In contrast, REFINESTAT targets properties that cannot be prefix-checker
since statistical reliability requires full posterior inference rather than purely static validation. For more dynamic program generation, Poesia et al. (2022) and Ugare et al. (2025) implement error-driven backtracking.
Recently, several works have explored probabilistic inference/programmiig in LM-constrained generation. Loula et al. (2025) guide generation with potential scores and grammar rules, constraining token emission but not model correctness. In contrast, REFINESTAT generates probabilistic programs, enforces PPL checks during decoding, and retains only those passing Bayesian diagnostics. In Grand ct al. (2025), a Planner writes an inference plan that LMs exccute to satisfy constraints. REFNNSTAT instead directly writes the probabilistic model and focuses the quality of the posterior. Ahmed eet al. (2025) adjusts next-token probabilities using a verifier so text matches high-level attributes. REFINESTAT aims for statistical validity, enforcing PPL semanties, and selecting the final program.
CONCLUSION AND LIMITATIONS
Conclusion. The main contribution of our work is to separate the task of generating probabilistic modeling through PPLs as fragments of priors and likelihood and to construct an LLM-based search procedure that automatically discovers the probabilistic program that satisfies the standard reliability metrics in the Bayesian workflow. We believe that our framework can be extended enforce arbitrary reliability criteria defined by domain experts for reliable generation in other domains that involve domain-specific languages and plan explore those in future work.
Limitations. While our framework incorporates key components of the Bayesian workflow (i.e., convergence diagnostics and predictive performance metrics), it does not include prior-vredictive or posterior- predictive checks, which often require manual inspection and domain-specin judg- posterior-prediicivve
ment (Gelman et al., (195). Thus, the reliability judgment is based on subset of available diagnostics, and the reported ELPD only partially reflect model adequacy in some cases. Further, our refinement strategy is effective in practice but does not guarantee convergence globally optimal program.
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Published as a conference paper at ICLR 2026
8 ACKNOWLEDGMENTS
This research was supported in part by NSF Grants No. CCF-1846354 and CCF-2313028.
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A APPENDIX
This appendix provides expanded technical details, evaluations, and examples supplementing the main paper. Below structured index:
A. Extended Background
1. Language Models
2. Bayesian Workflow
3. Probabilistic Programming Language
B. Probabilistic Program Metries
1. Basic Terminologies
2. Formal Definitions
C. Example
1. Illustrative Example
2. Examples of Structural Changes Introduced by REFINESTAT
D. Prompt Design
1. Prompt Template
2. Dataset-Speee Sataset-Speeific Prompt
E. Experimental Setup
Error Analysis
Search Space Limitations
2. Model Misfit and Sampling Failures
3. Call-Level Hallucinations
4. Termination Failures Due to Budget Constraints
G. Ablation Studies
1. Token Efficiency Analysis
2. Memorization Effect
A EXTENDED BACKGROUND
A.1 LANGUAGE MODELS
Decoding and Constraints. Various approaches for token selection include greedy decoding, sampling. and beam search, repeated until end-of-sequence (EOS) token or another stopping criterion is met. In constrained decoding, we may need to exclude specific tokens at certain positions. This is achieved using a mask m E {0, [VI, where indicates viable token and a discarded one. Decoding methods can then be applied
@ softmax(S).
Grammar-guided Generation Grammar-guided generation constrains model outputs to a formal grammar by using production rules of the form A - a, where A is nonterminal symbol and a is sequence of nonterminals and terminals (the actual tokens or characters that appear in the final output). Most programming languages can be described using context-free grammar, with rules that apply to nonterminal symbols independent of their context. Crammaa-guidde generation ensures that LM outputs follow the syntactic structure required for, e.g., code generation structured data formatting.
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A.2 BAYESIAN WORKFLOW
At its core, generative probabilistic program often follows DI structure: first fixing the observed data (D), then sampling latent variables under the prior (P p ((z)), and finally conditioning on the data via the likelihood (C p(r z)).
Modern probabilistic programming languages such as Stan (Carpenter et al. 2017b) and PyMC Sal- vatier et al. 2016) streamline this cycle by automating MCMC sampling and providing integrated diagnostics. These include prior predictive checks, convergence measures (e-g., split- R, effective sam- ple size, BFMI, divergent NUTS transitions) (Vehtari et al., 202 Hoffman et al. 2014 Betancourt
2017). and predictive accuracy metrics such as PSIS-LOO Vehtari et al. 2017).
These diagnostics guard against model mis-specification and poor sampling behavior, ensuring that only well-calibrated models are considered for inference or downstream decision decision-making. making.
A.3 PROBABILISTIC PROGRAMMING LANGUAGE
Probabilistic Programming Languages can be categorized internal Domain-Specific Languages (DSLs), which embed within a host language and reuse its syntax and tooling (e.g., PyMC (Salvatier al., 2016). NumPyro (Phan et al.) 2019), Pyro (Bingham al. (019]). or external DSLs that define their own syntax and compiler (c.g.. Stan (Carpenter et al. 2017b)). This representation enables automated model specification while leveraging existing inference algorithms (Gordon et 2014).
B PROBABILISTIC PROGRAM METRICS
B. BASIC TERMINOLOGIES
Before presenting examples for individual diagnostics, we briefly define few recurring terms that are used throughout:
Chain: An independent run of the sampler that generates a sequence of draws from the posterior distribution. Multiple chains are typically run to verify that results do depend on initialization.
Convergence: The state in which all chains are sampling from the same region of the posterior distribution. Lack of convergence suggests that the sampler has not fully explored the posterior.
Divergence: A warning issued by the Hamiltonian Monte Carlo (HMC) algorithm indicating that numerical integration failed to follow the posterior geometry accurately. Divergences often signal problematic parameterizations or highly curved posterior regions.
B.2 FORMAL DEFINITIONS
In our framework for valid statistical model synthesis, we employ several diagnostic metrics from probabilistic programming ensure model validity. Below, define each of these metrics formally.
Definition (Rt Statistic) The split- Rt statistic for parameter 6. denoted Ro. measures the con- vergence of Markov chains in MCMC sampling by comparing the between-chain variance the within-chain variance. Formally:
R
V
W
where V is the variance between chain means and W is the average variance within chains. Values close to 1.0 indicate convergence, while higher values suggest poor mixing of chains.
Definition (Effectivee Sample Size) The effective sample size (ESS) measures the equivalent num- ber of independent samples obtained from autocorrelated MCMC draws. For parameter dh
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define:
MN
bSSbu
(2)
T
MN
tSSta
(3)
Trail.d
where M the number of chains, N is the number of draws per chain, and t represents the autocorrelation time for bulk tail estimates respectively. Bulk ESS evaluates sampling efficiency across the central mass of the posterior, while tail ESS focuses on the distribution tails.
Definition 7 (Divergent Transitions) Divergent transitions, denoted divergences M) for model M, count the number of leapfrog steps in Hamiltonian Monte Carlo where the numerical approximation of Hamiltonian dynamics breaks down due to extremely high curvature in the posterior geometry. These indicate potential puthological geometries in the posterior distribution that may lead to biased inference.
Definition 8 (Bayesian Fraction of Missing Information) The Bayesian Fraction of Missing Infor- mation, BFMT(M) for model M, is defined as:
BFMI(M) = Var(AE)
Var(E)
where represents the energy (negative log probability density) and AE is the change in energy between consecutive HMC iterations. Low BFMI values indicate poor exploration of the target distribution.
Definition (Pareto Shape Parameter) The Pareto shape parameter R,(M) for observation in model M quantifies the reliability of importance sampling estimates used in PSIS-LOO cross- validation:
RM) shape parameter of Pareto distribution fitted to importance weights for observation i (5)
Values k. < 0.5 indicate reliable estimates, while R; > 0.7 suggest unstable estimates that may require more robust computational upprouches.
Definition 10 (Expected Log Pointwise Predictive Density) The Expected Log Pointwise Predic- tive Density under Leave- One-Out cross-validation, elpd(M) for model M, measures the model's ' out-of-sample predictive accuracy:
n
elpd(M) Σ log plys y-1)
(6)
where p(y y-1) is the predictive density for observation i after fitting the model to all other observations. Higher values indicate better predictive performance.
C EXAMPLES
C.I I.LUSTRATIVE EXAMPLE
We illustrate REFINESTAT on a standard Bayesian linear regression. Given a partial program P, we to find completion M that maximizes the Bayesian reliability score B(M). This example shows how each semantic and diagnostic check prunes or refines candidates.
Partial Program P
At timestep f, REFINESTAT generates the following partial program:
with Mo.Ml) as linear model:
alpha Normall" 0, 10)
beta aom.Normal(" 0, 10)
sigma pm..p..Hlaa... Haw.Haaaloooaasiiigiaa
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The LLM must complete the likelihood (Yobs).
LLM Proposals & Semantic Checks
Candidate likelihoods are checked against PPL. semantics shown in Table 7
Table 7: Semantic filtering of LLM-proposed likelihoods.
Proposal
Check
Outcome
Y_obs pm, Ext.orrmall ("y_obs",
mphalpha beta
sigmansigme, obsezvedry)
Distribution validity
Reject (halllucinate ExtNormal)
Y_obs pm. pm.Norrel ("y_obs", mumelphe beta adesigma, observeduy)
Parameter validity
Reject (deprecated sd vs. sigma)
y_o pm.Norma ("y_obs", mumelpha beta Xr sigma-sigma, obsecvedmy)
All checks
Accept
Diagnostic Checks & Guided Resampling We run NUTS on the accepted model and observe:
R = 1.2 (too high),
100 divergences.
These failures reduce the Bayesian reliability score B(M). REFINESTAT then resamples the likelihood or prior fragments (via the LLM) and retries inference until diagnostics (R, ESS, divergences, Pareto-k) fall within thresholds or the iteration limit is reached.
The final program M converges (R = 1), shows zero divergences, and yields reliable ELPD-LOO. By pruning invalid distributions early and resampling based on diagnostic triggers, REFNESTAA iteratively refines candidates into high-scoring M with robust statistical reliability.
C.2 EXAMPLES OF STRUCTURAL CHANGES INTRODUCED BY REFINESTAT
In addition to refining fixed model skeleton, REFINESTAT can introduce qualitatively new structural components during the resampling process. When the model families and diagnostic feedback indicates low reliability, REFINESTAT may alter the functional form of the model, add new parameters, change the dependency structure to explore alternatives with higher Bayesian reliability scores.
Below, we present three programs generated for the same dataset and initialization. These examples illustrate how REFINESTAT's refinement process modify terms in the likelihood, expand the parameter space, or even change which variable is treated as observed.
Program I: Standard linear regression
alpha "mmp"aaaa""alppha" mu=0, sigma=10) beta pm. "m.Nommal(""eta" mu=0, sigma=10) sigma = pm. Hal.HlaNNNoma""eee ("sigma", sigma=1)
mu alpha beta year likelihood = pormora ("ikelioooio mu=mu, sigma=sigma, observed=C)
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Program Nonlinear likelihood and an additional prior
alpha = an..oorma("alphha" mu=0, sigma=10) beta imn.Normal""btat" mu=0, sigma=10)
gamma pm. Won.Normal("aannnn mu=0, sigma=10)
sigma pm. Hin.Haallormaa("sigmaa sigma=10)
likelihood (alpha beta year + gamma year=*2) C+ sigma N
observed = pm..ormall "obseree""s""vvee""" mu=likelihood, sigma=sigma, observed=C)
Program 3: Changing the response variable and dependency structure
alpha mor.Noomaa""aphha: mu=0, sigma=10) beta pm. ww..Noomal"""tta", mu=0, sigma=10) gamma pm. NotnNoooo mu=0, sigma=10) sigma = pm. Haaaaaaallllllsiiinnnnni
sigma=10)
y_pred = alpha beta year + gamma *
y y_obs pm..Normal("y t_obs,,, mu=y_preee sigma=sigma, observed=N)
Summary of Structural Differences. Program uses a standard linear regression and models C as linear function of year. Program 2 expands the model by introducing new parameter (gammaa and applying nonlinear transformation involving year?, C, and N, leading to a different generative process. Program 3 changes the response variable entirely, modeling N instead of C, and modifies the dependency graph by incorporating C as a covariate in the linear predictor.
These examples show that REFINESTAT capable of exploring alternative model families and introducing new structure, such as additional priors, nonlinear link functions, or altered observational targets whenever such modifications remain semantically valid.
D PROMPT DESIGN
We use the same prompt across both the baseline, and REFINESTAT for experimentation pur- pose. To standardize the prompt across different datasets, template in which the fields (descript, intripttionn iion) and ( Semplate_coom } are replaced with the dataset-specific description and code snippet, respectively.
Prompt Template
Template prompt:
Complete the PyMC model definition within the "with po.Model() as m:' block below.
Your output must define a complete Bayesian model with appropriate priors, likelihood, and then sample the posterior using, 'pm.sample(1000, tune 1000, chains 4, return_ inttm_i_ffreneeaaaa True, ddata_kwarge [["log _og__lliklhhoo.. True) Do not include any extra commentary or text outside the code. Follow best practices for expert-level Bayesian modeling.
# Description: (Description)
(Template_Codee Code]
The placeholders (description) and (template_coml code) are automatically substituted for each dataset. Below are the (description) and (template_code) respectively for each dataset:
Note:
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D.1
EIGHT SCHOOLS
Description: A hierarchica cal model for the 8-schools s data.
Template Code
import pyme as pm import numpy as np
import arviz az import haapppoolib..yploo. as plt
# Data
y mp.array( (128, 8, -3, 7, 18, 12))
sigma np.array(115, 10, 16, 11, 9, 11, 10, 18))
with pm.Model0 m:
D.2 DUGONGS
Description: growth model for dugengs with missing data.
Template Code
import pyme as pm import numpy np import arviz as az import matplotlib.pyplot as plt
# Data
X = mp.array(11, 1,5, 1.5, 1.5, 2, 8,,,,,,,,,,,,, 9.5, 9.5, 10, 12, 12, 13, 13, 14.5, 15.5, 15.5, 16.5, 17, 22.5, 31.5)) y= np.array([118, 1.85, 1.87, 2.0, 2.15, 2.26, 2.47, 2.19, 2.26, 2.4, 2.39, 2.41, 2.5, 2.32, 2.32, 2.43, 2.47, 2.56, 2.65, 2.47, 2.64, 2.56, 2.7, 2.72, 2.57))
with podell as m:
D.3 SURGICAL
Description:
The mortality rates in 12 hospitals performing cardiac surgery on babies.
Template Code
import pymc as pm import numpy as np import arviz as az import natplotlib.ppylot as plt
# Given Data
N 12 # Number of observations n mp.7rray([47, 148, 119, 810, 211, 196, 148, 215, 207, 97, 256, 360])
np. 18, 8, 46, 8, 13, 31, 14, 8, 29, 24))
with Model
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D.4 GP
Description: Simulates data from Poisson GP model.
Template Code
import pyme as pm import numpy np
import arviz as az import matplotlib.pyploo as plt
# Given Data
N = 11 # Number of observations
X np.array([-10, -8, -6, -4. 0, 2, 6, 10])
y np.array([4.7590) 75506, 1.59423, 2.99548, 5.27501, 1.66472, 2.24347, 2.8914, 4.08681, 1.60588, 0.802364, 3,92136))
k np.array([40, 37, 29, 12, 4,3.9, 19, 77, 82, 33))
with pm.ModelO m:
D.5 PEREGRINE
Description: Simulated population counts of peregrines in the French Jura over years
Template Code
import pyme as pm import numpy as import arviz as az import matplotlib.pyplor as plt
# Data
nyears 40 # Number of years year np.array([-0.55, -0.9, -.88 -0.7 -07, 00000-000000---------00000------
-0.55, -0.5,, -0.5, -0.4,
-0.35, -0.3, -0.25, -0.2. -0.15, 0.1. 0, 0.05, 0.1, 0.15, 0.2, 0.25, 0.3, 0.35, 0.4, 0.45, 0.5, 0.55, 0.6, 0.65, 0.7, 0.75, 0.8, 0.85, 0.9, 0.95, 1))
C= mp.array 42, 35, 55, 61, 19, 49, 51, 48, 73, 49, 54, 43, 59, 30, 24, 62. 51, 47, 14, 27, 45, 20, 26, 19])
42, 73, 48, 49, 19, 72, 30, 18, 31, 71, 63,
N ([,3r 83, 53, 95, 24, 62, 91, 64, 57, 97, 56, 74, 66, 28, 92, 40, 46, 96, 91, 75, 71, 100, 72, 77, 64, 68, 43, 32, 97, 92, 75, 84, 22, 58, 81, 37, 45, 39])
with pm.Model() as m:
E MERTI
We the convergence threshold to ag 1.05 for split-R, allow ESSbulk Boulk 400, and adopt a relaxed cutoff Stall = 100 for ESStail to accommodate lower sampling efficiency in the tails. For leave-one-our validation, (M) must be finite, with at least Pareto shape values below L.a = 0.7. These thresholds are used consistently when computing the reliability score across all models. We STANDARD unconstrained generation baseline.
0.8 of data points having
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Further, based on preliminary experiments we have chosen / to be 4, to be 2, and Rmax as 100 for all experimental purposes.
We experiments a 48-core Intel Xeon Silver 4214R CPU with 2 NVidia RTX A5000 GPUs. REFINESTAT is implemented using Py Torch (Paszke et al. 2019). and Itergen library (Ugare et al. 2023) for refining the parser-guided LLM generation infrastructure. We run all experiments for 10 seeds to reduce result randomness, and use a temperature range of 0.2 to 0.4.
F
ERROR ANALYSIS
F.1
RUN RATE FAILURES
We categorized the failures by their causes in different methods found during the run rate experiment:
The Standard baseline exhibited frequent syntax enrors (e.g., unmatched delimiters, missing imports) and invalid API calls.
Syncode eliminated many basic syntactic mistakes but still suffered semantic errors, such as incorrect distribution parameter names, type mismatches, referring to deprecated API functions (e.g., calling pm. samp lempleerprorr from earlier PyMC release), and inventing non-existent methods like pm. randon_coefs
Stat in contrast, often produced models whose samplers failed to explore the correct posterior modes, leading to chains stuck in low-density regions or divergent transitions, failures stemming from the model definitions rather than our decoding framework; REFINESTAT reduced both syntactic and semantic errors and avoided sampler pathologies by enforcing grammar and parameter validity during decoding.
F.2 OTHER FAILURE MODES
While REFINESTAT significantly improves the syntactic correctness and statistical validity of gener- ated probabilistic programs, we analyze the remaining failure cases to better understand the limitations of the framework and identify directions for future improvement.
Search Space Limitations. Despite the use of resampling based mechanism for semantic validity, the language model occasionally reintroduces previously rejected code fragments. For instance, may repeatedly generate outdated invalid syntax such the use of sd instead of s sigma in PyMC model definitions:
mu
pm. Normal ("mu", mu=0, sd=10)
This behavior likely stems from the model's prior exposure deprecated APIs in its pretraining corpus and reflects the difficulty of escaping local attractors in the search space.
Model Misfit and Sampling Failures. Despite generating semantically correct code, some models fail during posterior inference due to numerical instabilities inherent in the model specification. A common manifestation of this issue is the PyMC error:
SamplingErrox Initial evaluation of model at starting point
failed!
This eтror often arises when certain mathematical operations within the model, such exponentiation or logarithms, result in undefined or non-finite values. For instance, exponentiating large numbers can lead to overflow, while taking the logarithm of zero or negative numbers is undefined. These numerical issues can cause the log-prop probbabiiyl evaluations to return NaN or inf, thereby preventing the sampler from initializing properly.
The model occasionally hallucinates invalid function names or API calls not present in the target probabilistic programming language. For example:
Call-Level Hallucinations.
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pm. ExtNormal ('ex', mu=0)
Such hallucinations highlight mismatch between the syntactic plausibility and the executable validity of generated code, reinforcing the need for grounded semantic constraints during decoding.
Termination Failures due to Budget Constraints. In practice, we impose limits on the maximum number of iterations or generated tokens to maintain tractability. some instances, these constraints are reached before valid program is synthesized, resulting in truncated or incomplete code outputs.
SamplingError Initial evaluation of the model at the starting point failed due to numerical instabil- ities (overflow/NaNs).
Indentation and commenting failures: always close string comments.
Wrong indentation of python like function codes; inability
G ABLATION STUDY
G.1 TOKEN EFFICIENCY ANALYSIS
To evaluate the computational cost associated with framework, measure the number of tokens consumed in generating a program under Itergen, Baseline (Unconstrained generation), REFINESTAT without Refinement Loop (ReFInEStAT w/o RL), and REFINESTAT using Meta-L.Lama-3-8B The token usage across five runs is recorded for each of these methods, from which we report the mean and standard deviation.
The Table 8 presents the results across models and datasets, with the final column ("Token Ratio") reporting the ratio of token usage by REFINESTAT relative the baseline. On average, REFINESTAt consumes twice the number of tokens consumed by Baseline. While this reflects the added cost of our refinement mechanism, the overhead varies across settings. For instance, some cases such as Dugongs, REFINESTAT uses fewer tokens than the baseline due to early convergence. Conversely, high multipliers (e-g.. Eight Schools, GP) reflect continued refinement due to unmet stopping conditions, even if the generated program is already of high quality.
Table 8: Comparison of Itergen, Baseline, and REFINESTAT Variants with Multipliers.
REFINESTAT w/o RL
Dataset
Itergen
Baseline
REFinEStAt
Token
Mean
Std
Mean
Std
Mcan
Std
Mcan
Std
Ratio
Eight Schools
98.8
5.3
606.8
386.3
147.6
36.7
1503.0
409.5
2.5x
Dugongs
209.3
225.9
747.4
495.0
294.4
86.3
419.0
87.1
0.6x
GP
131.3
8.9
663.6
410.9
155.0
21.2
1660.0
253.5
2.5x
Peregrine
132.5
21.4
884.0
538.9
180.8
103.6
1740.0
418.0
2.0x
Surgical
97.3
4.3
624.4
260.6
113.2
9.2
1275.0
1348.1
2.0x
Average Token Ratio
1.9x
G.2 MEMORIZATION EFFECT
To further stress-test our approach against memorization, we performed two controlled prompt modifications across all datasets using Meta-Ilama 3-8B.
Anonymized Prompt (REFINESTAT-AP): All metadata and dataset names were removed, leaving only the raw dataset.
Syntactic Obfuscation (REFINESTAT-SO): All numerical values were transformed into exponential notation (e.g., 3.28e2 instead of 328) prevent exact string matches with any potential training data.
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Anonymized Prompt and Syntactic Obfuscation (REFINESTAT-AP-SO) Combined variant using both Anonymized Prompt, and Syntactic Obfuscation.
Table 9: Comparison of Diagnostic Scores and ELPD-LOO for REFINESTAT variants.
Datasel
Variant
Reliab, Score
ft +
ESS Bulk
Divergences.
Pareto & 4
ELPD LOO f
Mcan
Std
Mean
Std
Mean
Std
Mean
Std
Mean
Std
Mean
Std
REFINESTAT
7,00
0.00
1,00
0.00
1696.00
284.33
0,00
0.00
0.00
0.02
8.42
24.51
REFINESTAT-AP
1073.00
0.00
4.24
Dugongs
REFINESTAT
7.00
0.00
1,00
0.00
1753.50
68.95
0.00
0.00
0.04
0.00
8.31
0.09
RERNESTAT-AP-SO
7.00
0.00
1,00
0,00
2257.00
731.23
0.00
0.00
0.00
0.00
1.79
7.09
REFINESTAT
7.00
0.00
1,00
0,00
2303.00
768.76
0.00
0.00
0.00
0.05
-31.77
0.61
REFINESTAT-AP
-3.62
Eight Schools
REFINESTAT-S
7.00
0.00
1,00
0.00
2470.50
32.61
0.00
0.00
0.06
0.06
-31661
0.01
RERNESTXI-AP-SO
7.00
0.00
1,00
0,00
2187. 50
252.83
0.00
0.00
0.00
0.00
-31.60
0.04
REANNESTAT
7.00
0.00
1,00
0,00
3574,00
428. 26
0.00
0.00
0.00
0.00
-173.00
4,91
REFINESTAT-AP
-173.14
Peregrine
REFINESTAT
7.00
0.00
1.00
0.00
1812.50
24.05
0.00
0.00
0.00
0.00
-132.88
8:20
RURNESTAI-AP-SO
6.00
0.00
1,00
0.00
1812.50
24.05
0.00
0.00
0.00
0.00
-140.88
8.20
REANESTAN
6.00
0,49
1,00
0.00
1710.00
668.57
13,00
12.39
0.09
0.08
-152.30
139.07
REFINESTAT-AP
-21.24
GP
REFINESTAT-S 7.00
0.00
1.00
0.00
1135.00
0.00
0.00
0.00
0.00
0.00
24.99
0.00
RURNESTAI-AP-SO 6.50
0.53
1,00
0,00
1140.00
2,
69.00
73.76
0.00
0.00
-23.01
0.47
REANNESTAT
6.00
0.77
1,00
0,00
579.00
1272.12
0.00
5.75
0.00
0.36
-46.73
36.63
REFINESTAT-AL
44..1
Surgical
REFINESTAT
7.00
0.00
1.01
0.01
1311.00
742.99
0.00
0.00
0.04
0.04
66.22
19.85
RURNESTXI-AP-SO
7.00
0.00
1,01
0.01
1139.00
638.22
0.00
0.00
0.04
0.04
-69.27
24.20
As shown in Table 9 both variants match the original REFINESTAT on reliability, convergence, and predictive metries. The table also reports combined variant using both modifications, which performs comparably. This consistency suggests that REFINESTAT'S effectiveness stems from learning from provided data rather than memorization.
24


Reviews:
[{"review_id": "azlbsNGSh7", "reviewer": "Reviewer_PPbt", "summary": "The submission proposes a framework for constrained generation of probabilistic programs using a small language model and is based on both static and inference-based statistical checks to resample and improve the program. The framework is evaluated on several language model using a set of public datasets from statistical literature and shows solid performance.\n\nThe claimed contributions are \n\n* a method for SLM based generation of probabilistic programs ensuring synatcic and semantic correctness and predictive performance\n\n* the use of constrained decoding for generation of probabilistic programs\n\n* iterative resampling of probabilistic programs using an unmodified small language model\n\n* an evaluation baseline of probabilistic program generation based on a number of criteria outlining syntactic, semantic, and statistical correctness.", "questions": "How easy/challenging is it to retarget RefineState to Stan?", "limitations": "", "rating": 8, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The submission presents a non-trivial view on incorproration of Bayesian workflow into Large/Small language model pipeline. It is good to see Bayesian workflow to be formalized and automated for data-driven probabilistic program search. The method is described in detail, supported by clean source code to reproduce experiments, and thoroughly evaluated.", "weaknesses": "Major:\n\n1. I am concerned about evaluating any language model based method on datasets broadly represented in the statistical literature. While the authors argue that through ablation studies they demonstrates that the strengths of RefineStat come from learning rather from memoization, independent empirical evidence of this does not seem to be very convincing. \n\n2. I am wondering whether PyMC is special in any way to specifically support the approach, how easy to apply the approach to other inference frameworks (Stan, Pyro/NumPyro, DynamicHML.jl, Turing.jl etc). I do not think PyMC is  sufficiently justified in the paper as the 'natural' choice.\n\nMinor: while the authors make a very respectable effort to define everything, the clarity of definitions and notation throughout the paper can be improved. For example:\n\nLines 130-132 define PSIS-LOO in terms of Pareto smoothed ways. Lines 134-136 that follow mention the fitted Pareto shape parameters k_i, not defined previously. Either more details should be provided about Pareto smoothing in the definion, or fewer details in what follows. As it is now, the reader stumbles on this and unless happens to be familiar with pareto-smoothed cross-validation, has to read the cited paper to understand what is going on.\n\nLines 173-180 define the score as the sum of 7 'binary indicators'. The 'indicators' are defined as booleans rather than indicators, and booleans cannot be summed up in mathematics (they can in Python). Some sound notation should be used so that s_j is 1 if the condition holds and zero otherwise. Then, in lines 187-192 are quite confusing, at least on the first reading, giving an example when elpd cannot be computed and then immediately saying that the higher the score is, the more the value of elpd can be trusted.\n\nLines 220-223 talk about how std= is invalid and RefineStat correctly replace std with sigma, but in the accompanying Figure 2 there is sd= rather than std=.\n\nPedro Domingos appears as Domingos in line 440 and as Pedro in lin 442 in citation anchors.\n\nThese are just a few examples, there are quite some more such glitches, and this impairs the reader's ability to understand the paper. The paper should be proofread and fixed.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "gqvK9p6cVP", "reviewer": "Reviewer_oWB5", "summary": "The paper presents a method for generating a probabilistic program using a \"small\" LLM, where during generation process certain semantic checks are enforced. This method can potentially improve the reliability of the final output. A variety of techniques are currently being proposed to check correctness at decoding time,  this paper differs due to its focus on probabilistic models and their properties. The method is evaluated on a  set of benchmarks against baselines including unconstrained generation by powerful LLMs as well as prior syntax-constrained generation methods.", "questions": "1. Can you explain the key differences between this and constrained decoding (e.g. \"Type-Constrained Code Generation with Language Models\" in PLDI 2025). Are there new challenges or just adaptation to probabilistic settings.\n2. Can you explain how synthesis algorithms works using an illustrative example?\n3. Based on the experimental results, the method doesn't seem \"ready\" for real-world applications. That's ok, but can you articulate what advances will be needed for this?", "limitations": "", "rating": 4, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The problem is well motivated and the basic approach is an appealing way to address the challenges.", "weaknesses": "The authors cite a number of papers proposing constrained decoding for classical programs, but a meaningful comparison is missing. The technique, as a result, seems incremental. The experimental results are limited and not particularly impressive.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "ZgcTav7Me3", "reviewer": "Reviewer_uUBR", "summary": "The paper proposes REFINESTAT, a probabilistic program synthesis framework that allows small open-weight language models to automatically generate Bayesian models that are both executable and statistically trustworthy.\n\nREFINESTAT has two core components:\n\n- Semantically constrained generation.\nInstead of allowing an LLM to free-generate PyMC code, REFINESTAT enforces a set of domain-specific validity predicates at generation time. These constraints check: parseability, valid distribution names, correct parameter names/signatures, shape/type compatibility, support (e.g. positive-only scales), and dependency ordering between random variables. If a generated fragment violates one of these constraints, that fragment alone is locally rejected and resampled, rather than discarding the entire program. This prunes away statistically nonsensical or sampler-hostile code before it ever reaches inference.\n\n- Diagnostic-aware refinement.\nFor each candidate model, REFINESTAT runs inference in PyMC (using NUTS/HMC) and computes standard Bayesian workflow diagnostics. These are aggregated into a Bayesian Workflow Reliability Score, which counts the number of diagnostics that pass. If the score is below a threshold, REFINESTAT selectively resamples just the likelihood block or just the prior block, keeping the rest of the model fixed, and tries again. A model is accepted only if it reaches sufficient reliability and good predictive performance.\n\n\nThe framework is tested on five canonical PosteriorDB problems (Eight Schools, Dugongs, Surgical, Peregrine, Poisson GP) using several open models and compared to:\n\n- naive prompting (unconstrained generation),\n- syncode (syntax-only constrained decoding),\n- multi-agent GPT-4 systems such as BoxLM,\n- expert-written Stan models,\n- and proprietary baselines like OpenAI o3.", "questions": "During refinement, can REFINESTAT introduce qualitatively new structure (e.g., add a GP term, change link functions, etc.), or does it mostly resample priors/likelihoods within the same initial skeleton? Empirically, how often does refinement change the model family, versus just adjusting hyperparameters? If it does not change structure mid-loop, do you consider that a limitation or is it a deliberate design choice?\n\nWhen diagnostics fail, what feedback does the LLM get?\n\nAre the thresholds fixed globally across all datasets/LLMs, or tuned per task?\n\nCan you quantify average resource requirements? How many complete HMC/NUTS inference runs does REFINESTAT perform per dataset, on average, before it finds an acceptable model? What’s the typical wall-clock time order of magnitude per dataset on your hardware (minutes, hours)?\n\nFor a moderately larger dataset, is the loop still tractable, or does it become prohibitively slow?\n\nIt would be helpful to understand how naive prompting + the same diagnostic-aware refinement loop, but without semantic constraints, does. This ablation would isolate the marginal benefit of semantic-constrained decoding and strengthen the argument that both components are essential.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "Prior constrained decoding work in LLM code generation typically enforces syntactic validity or DSL grammar. REFINESTAT goes further and embeds statistical semantics: valid distribution calls, correct parameter names and shapes, valid supports (e.g., positive scale parameters), and dependency ordering between latent variables and likelihood terms. \n\nInstead of treating generation as one-shot code synthesis, REFINESTAT adopts the classical Bayesian workflow loop: propose a model $\\rightarrow$ run inference $\\rightarrow$ diagnose $\\rightarrow$ revise. It encodes this loop in a way that is easily enforceable by turning convergence/efficiency diagnostics.\n\nCompeting systems like BoxLM rely on multi-agent GPT-4 setups (generator + critic). REFINESTAT shows that structured guidance, combined with principled diagnostics, can enable a single, small open model to match or beat these heavier pipelines on some benchmarks.\n\nREFINESTAT increases run rate from $\\approx$10–11% (unconstrained)/ $\\approx$21% (syntax-only) to $\\approx$45–50%. Ablation studies show that removing specific semantic checks (especially parameter validity) sharply degrades success, establishing that each constraint is important.\n\nA weaker open model (DeepSeek-R1-Distil-Qwen-7B), which fails with naive prompting, becomes capable of producing usable models on all datasets when run through REFINESTAT.", "weaknesses": "All experiments are on five classic PosteriorDB tasks (Eight Schools, Dugongs, Surgical, Peregrine, Poisson GP). These are relatively low-dimensional, reasonably clean, and heavily studied.\nThe author doesn’t use any messy, modern, high-dimensional problems:\n- hierarchical GLMs with a large number of predictors,\n- data with missing values or covariate drift,\n- settings where NUTS is brittle.\n\nThe paper claims progress toward automated statistical model discovery, but current evidence is confined to highly curated, small-scale Bayesian exemplars.\n\nAll experiments are in PyMC. The authors do not demonstrate portability. The semantic predicates (distribution names, argument schemas, support typing, etc.) are currently PyMC-specific.\n\nThe refinement loop fixes models by resampling either the likelihood or the prior block when diagnostics fail, and continues iterating until it passes a threshold. This is a clever local repair heuristic, but it also has a limitation: it assumes the proper global structure (hierarchy, link function, GP terms, etc.) was already present in the initial model draft.\n\nThe paper has to quantify how often the system gets stuck in such local minima (tinkering with scale priors on a fundamentally wrong structure) or how frequently it must abandon a structure entirely.\n\nThe thresholds are hand-picked. The paper does not provide a sensitivity analysis to show robustness.\n\nThe method repeatedly runs full NUTS/HMC inside the loop to evaluate diagnostics and to drive refinement. That’s the expensive part.\nThe paper does discuss token usage (and claims REFINESTAT avoids GPT-4-scale inference costs by using only small open models). Still, it does not report: (i) the number of complete inference runs per dataset per accepted model, and (ii) wall-clock time.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "Ql1t7zYc4S", "reviewer": "Reviewer_J73j", "summary": "The paper introduces RefineStat a new framework for automatically synthesizing probabilistic programs using \"smaller language models\" (around 8B parameters). RefineStat introduces two checks for generating better probabilistic programs: 1. semantic and syntax checks to ensure the language models generates valid PyMC code (this is done using constrained decoding). 2. statistical diagnostic checks that ensure the results from posterior inference are reliable.\n\nEmpirically the paper presents results on synthesizing programs for some PosteriorDB datasets. RefineStat is evaluated using multiple open source language models as different base models.\n\nOverall, I am in favor of accepting the paper because I think it provides a useful building block for setting up an automated workflow for synthesizing probabilistic programs. I do have some reservations about the empirical validation though which i would like the authors to address in the rebuttal.", "questions": "L130: If this definition is taken from Vehtari et al, 2017 then the reference should be part of the definition.\n\nAlgorithm 1 misses any description for how $\\mathcal{D}$, $\\mathcal{P}$, and $\\mathcal{L}$ are initially sampled.\n\nTable 1: What does it mean for a value to be bold in the table?\n\nTable 3: The table caption should state which metric is shown, I assume it is ELPD LOO? Additionally, could you show the standard deviation of the metrics (if I understand correctly these values are averages over 5 runs?)", "limitations": "", "rating": 8, "confidence": 3, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "The paper is very well written. The main arguments are easy to follow and the methodology is explained clearly. Both checks (semantic/syntax checks and statistical diagnostics) are sensible for any automated workflow in building Bayesian probabilistic models. I think the paper can serve as a useful stepping stone for building automated model building workflows.", "weaknesses": "There's a couple of weaknesses which I would like the authors to respond to in the rebuttal period. I don't think any of them should disqualify the paper from being accepted but I would appreciate some further discussion on the points listed below.\n\n1. The paper motivates the use of small language models by the potential large API cost of using bigger language models. However, there is not a rigorous cost comparison of the RefineStat approach vs just calling the OpenAI-o3 model.  At least for the OpenAI model it would be good to provide the cost of the API calls.\n\n2. Similarly, why is there no comparison with OpenAI-o3 in Table 1?\n\n3. I appreciate the fact that the paper explicitly accounts for the fact that the empirical validation might be impacted by the fact that the PosteriorDB models could be contained in the training set of the language models. However, I find the empirical checks to be a bit unconvincing (although I am not an expert in the dataset memorization literature). Looking at the Table in Appendix G there is sometimes quite a lot of variation in the achieved scores (e.g. for Dugongs, GP and Surgical) and additionally, this check was only done on the LLama model whereas the evaluation in Table 3 uses the DQ-7B model as a base model. So the whole experimental setup is a bit inconsistent.\n\nAt the same time, dataset leakage is an unsolved problem as far as I am aware. So while this evaluation might be flawed, I think it shouldn't disqualify the paper from getting accepted.\n\n4. The OpenAI-o3 model seems to be competitive with the RefineStat approach (on 8 schools, dugongs and surgical, if I am reading Table 3 correctly). So the big question is whether we really need RefineStat or whether training a bigger model will also improve probabilistic program synthesis performance. I don't see a reason though why RefineStat couldn't be used with a big model (other than cost) so it would have been nice to see some results for using OpenAI-o3 as a base model in RefineStat (not sure whether the constrained decoding is possible with a closed source model and/or it might be too expensive).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Oral)