Textual Bayes: Quantifying Prompt Uncertainty in LLM-Based Systems
Authors: Brendan Leigh Ross
Noël Vouitsis
Atiyeh Ashari Ghomi
Rasa Hosseinzadeh
Ji Xin
Zhaoyan Liu
Yi Sui
Shiyi Hou
Kin Kwan Leung
Gabriel Loaiza-Ganem
Jesse C. Cresswell
Venue: ICLR 2026 Poster (2026)
Keywords: LLMs
large language models
uncertainty quantification
Bayes
Bayesian
Bayesian inference
Metropolis-Hastings
prompt engineering
autoprompting
TextGrad
Primary area: probabilistic methods (Bayesian methods, variational inference, sampling, UQ, etc.)
TL;DR: We quantify LLM prompt uncertainty by replacing prompt engineering with Bayesian inference using a novel MCMC method over text

Abstract: Although large language models (LLMs) are becoming increasingly capable of solving challenging real-world tasks, accurately quantifying their uncertainty remains a critical open problem—one that limits their applicability in high-stakes domains. This challenge is further compounded by the closed-source, black-box nature of many state-of-the-art LLMs. Moreover, LLM-based systems can be highly sensitive to the prompts that bind them together, which often require significant manual tuning (i.e., prompt engineering). In this work, we address these challenges by viewing LLM-based systems through a Bayesian lens. We interpret prompts as textual parameters in a statistical model, allowing us to use a small training dataset to perform Bayesian inference over these prompts. This novel perspective enables principled uncertainty quantification over both the model’s textual parameters and its downstream predictions, while also incorporating prior beliefs about these parameters expressed in free-form text. To perform Bayesian inference— a difficult problem even for well-studied data modalities—we introduce Metropolis-Hastings through LLM Proposals (MHLP), a novel Markov chain Monte Carlo (MCMC) algorithm that combines prompt optimization techniques with standard MCMC methods. MHLP is a turnkey modification to existing LLM pipelines, including those that rely exclusively on closed-source models. Empirically, we demonstrate that our method yields improvements in both predictive accuracy and uncertainty quantification (UQ) on a range of LLM benchmarks and UQ tasks. More broadly, our work demonstrates a viable path for incorporating methods from the rich Bayesian literature into the era of LLMs, paving the way for more reliable and calibrated LLM-based systems.

Published as a conference paper at ICLR 2026
TEXTUAL BAYES: QUANTIFYING PROMPT UNCERTAINTY IN LLM-BASED SYSTEMS
Brendan Leigh Ross'
NoËl Vouitsis Atiyeh Ashari Ghomi Yi Sui Shiyi Hou
Rasa Hosseinzadeh Ji Xin Zhaoyan Liu
Kin Kwan Leung Gabriel Loaiza-Ganem (brendan, noel, atiyeh, rasa, zhaoyan, amy, gloria, kk, gabriel, ji. ni..uuaatteer..o ca
Jesse C. Cresswell
Layer AI, Toronto, Canada
ABSTRACT
Although large language models (LLMs) becoming increasingly capable of solving challenging real-world tasks, accurately quantifying their uncertainty re- mains critical open problem-on that limits their applicability in high-stakes domains. This challenge is further compounded by the closed-source, black-box nature of many state-of-the-art LLMs. Moreover, -LM-based systems be highly sensitive to the prompts that bind them together, which often require sig- nificant manual tuning (i.e., prompt engineering). In this work, we address these challenges by viewing LLM-based systems through a Bayesian lens. We interpret prompts as textual parameters in a statistical model, allowing us use a small training dataset to perform Bayesian inference over these prompts. This novel per- spective enables principled uncertainty quantification over both the model's 's tex- tual parameters and its downstream predictions, while also incorporating prior be- liefs about these parameters expressed free-form text. To perform Bayesian in- ference-a difficult problem even for well-studied data modalities-we introduce Metropolis- Hastings through LLM Proposals (MHLP), a novel Markov chain Monte Carlo (MCMC) algorithm that combines prompt optimization techniques with standard MCMC methods. MHLP is a turnkey modification to existing LLM pipelines, including those that rely exclusively on closed-source models. Empir- ically. we demonstrate that our method yields improvements in both predictive accuracy and uncertainty quantification (UQ) on range of LLM benchmarks and UQ tasks. More broadly, our work demonstrates viable path for incorporating methods from the rich Bayesian literature into the era of LLMMs, paving the way for more reliable and calibrated LLM-based systems.
INTRODUCTION
Large language models (LLMs) have become increasingly embedded in our daily lives, with grow- ing adoption across domains such as customer support (Chaturvedi & Verma, 2023), code genera- tion (Wang et al. 2021 Chen et al. 2021), scientific research (Boiko et al. |0033 Schhmidal et al.) Rodriguzz & Williams) 2023). As their
2025 Yamada et al. 2025), and creative writing capabilities continue advance, there is also mounting interest in deploying them in agentic sys- tems, wherein they perform tasks autonomously on behalf of users (Wooldridge & Jennings [995] Xi et al. 2025). Despite their proliferation, however, trust in LLMSs remains limited, largely due to their propensity to generate hallucinated content (Maynez et al. 2020; et al. | (024) and their sus- ceptibility to adversarial attacks and jailbreaking (Wei et al. 2023 Zou et al.] 2023] Yan et al. 2024). These vulnerabilities in LLM--baed systems therefore must be addressed, especially to fully unlock high-stakes domains such as finance and medicine. A key step towards mitigating these risks is to reliably quantify the uncertainty of LLM-based systems. Accurate measures of uncertainty ensure that, when unable answer, LLM-based systems can abstain, defer to human experts, or augment their context with subroutines based on retrieval or reasoning (Lewis et al. 2020: Wei et al. 2022).
Eqqual contribution.

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Col
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In chain-of-thought (CoT) prompting (left), answers are generated by an LLM using single fixed prompt: this frequentist approach does not account for uncertainty about how the model should be prompted, causing potential issues such as overconfidence prompts from our Bayesian posterior and use each to generate answers from the LLM: this allows principled uncertainty quantification over both the prompts themselves and the resulting generated answers.
Figure I:
incorrect answers. In Textual Bayes (right), sample
Despite recent progress, uncertainty quantification (UQ) for L.Mss is far from solved and no consen- exists exactly what should be quantified (Kuhn et al. 2023 Wang & Holmes 2024 Yang iet al. 2024b). In this work, we propose to better quantify uncertainty in LLM-based systems by Viewing them through Bayesian lens. In light of a model, observational data, and one's prior be- liefs, Bayesian inference uses Bayes' rule to compute the distribution of possible model parameters. Commonly applied classical statisties and deep learning, Bayesian inference is a principled and mathematically grounded approach to UQ (Bernardo & Smith (009). Bayesian techniques have led to high-profile successes in methods like variational autoencoders (Kingma Welling/ 2014) and Bayesian neural networks (Blundell et al. 2015). As with their 20th century forebears (e.g.. (Duane et al., 1987 Saul et al.) 1996)), these methods estimate uncertainty over nigh-dimensional continuous variables. Here, we bring Bayesian methods into the age of LLMs. In LLM- MiMM-d--eedd-- sys- tems, the main variables of interest are prompts, since LLMs themselves are often black boxes that only be accessed via API. By treating prompts textual parameters in statistical model, as illustrated in Fig. I we can use Bayesian inference to estimate distributions over their values. These distributions rigorously quantify our uncertainty about the models themselves. Furthermore, they can be integrated into uncertainty estimates on the system's 's downstream outputs via easy-to- compute Monte Carlo estimates. To the best of our knowledge, we are the first perform Bayesian inference over the space of free-form prompts in LLM- systems.
Adapting Bayesian methods text has its challenges and advantages. On the one hand, textual vari- ables are discrete, making it difficult to apply traditional Bayesian deep learning techniques such as gradient- baase Markov chain Monte Carlo (MCMC) (Welling & Teh (01T) or variational inference (Saul et al. 1996). We address this obstacle with novel text-based MCMC method: Metropolis- Hastings through LLM Proposals (MHLP). On the other hand, textual variables are better suited conceptually to Bayesian modelling than nigh-dimensional continuous variables such as the weights of a deep neural network. Bayesian inference famously requires the specification of prior beliefs about variable; textual variables and as we show, prior beliefs can be readily incorporated into LLM-based systems as free-form text.
amenable to human priors than neural network weights,
To advance and justify proposed method, this work contains the following contributions:
1. We take a novel perspective on LLM-based systems in which prompts are viewed as Bayesian textual parameters 0 in model p(y a., 0). We show how this formulation leads to principled way to incorporate prior beliefs about 0 while quantifying our inherent uncertainty in the model.
2. To implement our Bayesian approach, we propose Metropolis-Hastee (MHLP), an MCMC algorithm to sample from intractable distributions over textual variables. MHLP has broad potential applications even beyond Bayesian inference.
through LLM Proposals
3. We propose novel metric of model calibration, semantic expected calibration error, for quan- tifying calibration, a form of UQ, on free-form textual outputs.
4. We systematically evaluate our method through standard LLM benchmarks and baselines, show- ing that it improves performance while providing state-of-the-ar UQ over model outputs.
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2 BACKGROUND AND TERMINOLOGY
2.1 LLM-BASED SYSTEMS
The central object in this work is the LLM-based system. The most common LLM-based system is one consisting of a single input I (e.g. question), a prompt 0 (e.g.. a system message defin- ing instructions for the model's behaviour, shared across all r), and output y (e.g., the model's predicted answer), which we denote
y LLM
We allow LLM 0) to any open- or closed-source model that we view as a random function of x and 6, whose randomness depends on the underlying LLM sampling strategy (e.g.. greedy, temperature, nucleus). In general throughout the work, capitalized, boldface function names will indicate random functions comprising
or more LLM calls.
LLM-based systems can be more complex than single- models. Many recent works have proposed to group LLM calls of arbitrary count and complexity into pipelines parameterized by the prompts used at each step (Khattab et al.) [024]] Zhuge et al.] [2024] Yuksekgonul et al.) [202]] Cheng et al. 2024 Hu et al., 2024). For example, Self-Refine (Madaan et al. 2023) iterates an initial I.L. output by alternating between an L.LM call providing feedback and one incorporating the feedback into refinement. Fully agentic systems integrate multiple LLM and tool calls to arrive at a final output. In full generality, we can describe a forward pass through an LLM-based system as
y LBS(2; (z; 0):
(2)
where LBS(;;0) 8) can be described as a directed acyclic graph with &o edges in which each edge ej corresponds to an LLM call LLM(-:;) parameterized by a prompt 0, and where we denote the combination of all prompts in the system 0 (0) (01.-.., 0x). Since each LLM call in the system is potentially random, y is a random function of x parameterized by 0 = (01...., 0k). The LLM- based system thus forms statistical model for y whose probability mass function we express as LBS(x;))
p(y| xz 0), where sampling
ply x, 0) is equivalent to computing
Unlike linear regressor neural network where 0 denotes continuous model parameters, for an LLM-based system 0 denotes textual parameters. From the statistical modelling perspec- tive, a natural next step is find the optimal value of 6. For example, given an i.idd dataset D = (1)1.,1)..... (In, }n)}. one might want to perform maximum-likclihone
g arg max ((D | () arg max II
p(y: I., 0).
(3)
The discrete nature of textual parameters prevents from applying gradient-basee based algorithms to maximize likelihood in L.L.-based systems. Prompt engineering can be understood as approximat- ing 0* by having human propose candidates 0 until adequate performance on small dataset is reached. However, this manual process lacks rigour, is lengthy and tedious, and does not scale well.
Past works have proposed heuristic approaches automatically optimize prompts 0 in LLM-basee systems (Zhou et al. 2022 Khattab et al. 2024 Zhuge et al.) 2024] Cheng et al. 2024 Here, we focus on iterative prompt optimization methods, which we can express mathematically as stochastic update function UPDATE applied iteratively to initial prompt (10);
g(t)
(PD)TE(((t-1)).
For simplicity, we assume that UPDATE is Markovian; i.e., not function of 0 values from earlier than -. 1. UPDATE, which consists of one or more LLM calls, is itself an LLM-bbsed system.
One particularly relevant prompt optimization method is TextGrad (Yuksekgonul et al. 2025). The TextGrad framework conceptualizes constructive feedback on prompts as textual gradients and pro- poses method for "backpropagating" feedback through an LLM-based system akin to backprop- agation in neural networks. Although this framework is highly analogous to backpropagation of gradients for continuous variables, it does not formally optimize model likelihood.
2.2 BAYESIAN INFERENCE
In this section, we briefly review Bayesian inference, For a more in-depth introduction, we refer the reader to MacKay 2003). Here, we allow ply x,, 0) to be any statistical model for some variable
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y given another variable From the Bayesian perspective, there is uncertainty about the true value of 0, and hence the point estimate 8 given by maximum-likclihodd may be an overly reductive way of summarizing dataset D. Bayesian statistics provides formal way of capturing this uncertainty.
First, we encode our prior uncertainty (beliefs) about the true value of as a prior distribution p(0). Then, having observed dataset D, update our beliefs about 0 using Bayes' rule
p(0 D) PD 0)
p(0) II. p(us Ee p(0') P[, 24, 21,0") 0*)
Iz. 0)
(5)
(D))
The posterior distribution p(0 DD) formally captures our uncertainty about 0 in light of (i) our prior beliefs and (ix) the observed data. Given p(0 'D) and a new unobserved datapoint Taews we can compute the posterior predictive distribution of ynew via
Pkynew
Fnew: D)= = Σ P(Unew Fneww p)p(@ ()) = Epe-p(8 (D) [p(Unew Tnew y 0)]
(6)
Eq. 6 formalizes predictive uncertainty in terms of uncertainty over A. Since it is expressed as an expectation, we can estimate via Monte Carlo sampling with draws from p(0 | D). The posterior predictive has immediate practical value: its value represents confidence in the prediction y/eww, and its variability (as measured by. e.g., variance or entropy) formally quantifies uncertainty.
The central challenge of Bayesian inference thus lies in sampling from the posterior p(0 D). As p(y | xz ()) or p(0) acquire even moderate complexity, sampling from p(0 D) quickly becomes intractable. In deep learning, Bayesian inference requires approximations such as gradient-based MCMC (Welling & Teh 2011), variational inference (Blundell et al., 2015). or Laplace approxima- tions (Ritter et al. 2018). We highlight that all of these approaches rely on the differentiability of p(O)p(D | 0) with respect to 0, so none can be readily applied to the context where 0 is a discrete prompt in an LLM-based system.
2.3 MARKOV CHAIN MONTE CARLO AND THE METROPOLIS HASTHINGS ALGORITHM
In Bayesian statistics, MCMC algorithms are common technique for tractably sampling from the posterior p(0 D) when only its numerator in Eq. 5 can be computed for any values and D. First, fix D and let g(t) p())p(D) 0) be the numerator of Eq. 5 Given an unnormalized density like g(0), an MCMC algorithm is a general-purposs technique that specifies a Markov chain Ee' अ(Ø p(0 D) as L co. In practice,
g(1), 002) ...., ((1),... .... whose distribution converges to
g(0)
by generating enough samples from the Markov chain, we can approximate sampling from p(0 D) without needing to evaluate it.
The Metrpolis-Heasting algorithm (MH) a generic and broadly applicable form of MCMC (for an introduction, see Robert (2013)). Starting with an initial sample g(D) MH iterates from sam- ple g(+ g(t) by generating new proposal g' from pre-defined proposal distribution q('' O)
Algorithm 1: Metropolis-Hastine
Require: ((0) q((' 0), 9(0);
for t < 1 to T do
Sample proposal: 6' ~ q(0' (((-1));
Compute acceptance probability:
and then either accepting it (i.e., setting p(1) := 0')
7= >=min (1, g(0)) g(01)((01-1) 10')
or rejecting it (i.e., setting 0i4)_ :- ((t-1) based on an acceptance probability 7 Alg.
g((t-)))) q(0' (((--1))
Sample random number: u ~ Uniform(0,
In MH, the main "tuneable hyperparameter". the choice of proposal distribution q(O' 8). is con- strained only by very mild regularity conditions. However, the choice of q has a pronounced ef- fect on the practicality of the algorithm, with poor choices (e.g., ones that perturb too mildly or too strongly at each step) taking an intractable amount of time to converge to the limiting distribution p(0 D). The importance of is such that some of the most popular MCMC algorithms (e.g., Langevin Monte Carlo and Hamiltonian Monte Carlo (Duane et al. 1987 Neal 1996)) are simply special cases of MH with highly specialized choices of 4. Our method, MHLPP will also fall into this category, being specialized for textual parameters 0.
if u < 7 then
Accept: ((0) 0':
else
Reject: ((1) + 0le-1);
return {0(t)) 1t=i
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The choice of should be driven by any information available about the desired limiting distribution p(0 D) [Rosenthal 2011). Indeed, the optimal q( Ø) would be equal to the desired limiting distribution itself; if this were possible, of course, there would be need to run MH in the first place. Nevertheless, we apply this intuition in Sec. 3 as we adapt MH to textual data.
3 TEXTUAL BAYES
In this section, we describe our method for Bayesian inference on LLMM-based systems. We begin with the setup described in Sec. 2.1 LLM-based system LBS( (2; 8) that gives rise to statistical model ply I, 0), where x is the input, y is the output, and 0 (0I,...,0y, Ox) represents all the textual parameters involved in the system. We assume that p(y xr, 0) can be evaluated for any LLM- based system we consider; to do this in full generality for closed source models, we require some see App. A.1 for more details. Our Bayesian inference algorithm will provide samples ((2), ((m)) p(0 DD,, which can in turn be used to quantify uncertainty over the system's 's outputs as per Eq. 6
approximations, including selective use of open-source likelihoods as surrogates
Textual priors To perform Bayesian inference, we must specify our prior belief's about 0 in the form of distribution p(0). Although 0 lies in an infinite and semantically complex space of discrete text, humans are well equipped to reason and express their beliefs about textual variables. For example, practitioner's 's prior about a prompt f might be that it should describe the purpose of the corresponding I.I. call, guidelines for how to solve the task at hand, and the expected structure of the output. To exploit this knowledge, we codify our beliefs about each parameter 03 as free- form numan-writter string of textual constraints 8j, and provide it to an L.M to model the resulting parameter
0j = -LLM(sj: "Generate an LLM prompt satisfying the given constraints.
For simplicity, we construct our prior p(0) = II p(0,) by assuming that all textual variables are independent, but this setup can be easily generalized by specifying joint constraints over multiple parameters 0 and modelling them in single LLM call.
Setropolis-Hasstinge through LLM Proposals Having constructed our prior p(0), we now need an algorithm to sample from p(O DD. A generally applicable MCMC method for text could have wide-ranging applications even beyond Bayesian inference. To this end, we propose Metropolis- Hastings through LLM Proposals (MHLP), text-specific variant of MH.
At the heart of MHIP is our proposal distribution. We could in theory achieve the correct limiting distribution through almost any arbitrary choice of q(' 0), like randomly replacing letters or words in 0. But it is easy to see that such a proposal would rarely change semantically and never converge in practice. Instead, to generate useful proposals, we turn to LLMs. Analogously to how Langevin Monte Carlo uses gradient computation to exploit differentiable structure on p(0 D), MHLP uses LLM calls to exploit linguistic structure on p(0 D). Ideally, q(6' () should be similar to p(0 (D) as possible. By this standard, as per the relationship p(0 | (D) αx p(D) | 0)p(O), samples 0' " q(0' 0) should roughly satisfy the following criteria: (i) 0' should satisfy all the constraints embodied by the prior p(0"), and (vi) 0' should provide strong downstream performance on 'D as measured by p(D 0').
We take inspiration from the prompt optimization methods discussed in Sec. 2. and use suggestions from LLMs to propose values of 0' that implement these guidelines. The observation underpinning MHLP is that iterative prompt optimization methods can be used to propose high-quality candidates 8'. Here we recall our formalization of prompt optimization as an iterated stochastic update function
UPDATE (Fq. 4). and sample from q(0' 0) by computing 0' = UPDATE(0) Note that since UPDATE itself an L.L.M system just like our model LBS, and so, like the model density p(y æ, 0). the value of q(0' 0) can be estimated by using an open-source model for the final LLM call of UPDATE and by using the approximations in App. A. Although MHLP is agnostic to the underlying prompt optimization method, we use TextGrad ( (Yuksekgonul et al. 2025 in our implementation. By analogy to numerical losses in standard gradient- based optimization, TextGrad
Some prompt optimization methods. such as the momentum variant of TextGrad YYuksekgonul et al. 2025). j. make updates based on a history of multiple past 0 values. MHI.P can take advantage of such methods by running multiple steps of the optimizer per accept/reject decision, akin to Hamiltonian Monte Carlo.
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optimizes objectives described in natural language. We can thus express criteria (1) and (ti) as ob- jectives in natural language and use TextGrad to propose improvements to based on these criteria. This choice of objectives is specific to Bayesian inference, but in MHLP they can be easily replaced or modified to suit any textual distribution, which broadens its potential impact. We demonstrate one such example in See. 4.2
Method summary We define MHLP as the variant of MH Agg. 1) acting on textual parame- ters 0 in which the proposal step ~ =q(0' (((-1)) is defined as a prompt optimization up- For our experiments, we implement UPDATE as a TextGrad step Alg. 2). Additionally, as is common practice in Bayesian deep learning (e.g.. Blundell et al. (2015); Daxberger et al. (202T)), we employ approximations for tractability, including a tempered posterior (Wenzel et al.] 2020) and stochastic minibatch estimates of certain quantities in Allg. 1 (rather than evaluating them exactly). Due to space constraints, approximation details are relegated to App.. A.
date 0' = UPDATE(((--))
Having a collection }m ~ p(O | D) of prompt samples, we can now put them to use at inference time to sample from the predictive posterior (Eq. 6). Given an input Inew, we generate
Jr=1
a set of samples (Neow 1m ~ p(yhew Winww) via
(r)
g(r) p(0) D),
p(ynew Tnew: ((r));
that is, by running Vnew = LBS( Nnew; ((r)) for each sampled prompt g(r). The variability of the resulting answer set [ynow 1m can be interpreted as the uncertainty of the LLM-based system.
4 EXPERIMENTS
In this section, we empirically evaluate our proposed Textual Bayes method. Specifically, we aim to answer the following question: how does Bayesian inference on the prompts of an L.-e-based system with MHLP algorithm translate into the system's downstream predictive performance and UQ abilities? In Sec. 4.1 we demonstrate that our method outperforms comparable basclines in accuracy, calibration, and abstention capabilities challenging LLM benchmarks. In SScc. 4.2 we adapt Textual Bayes to reducing hallucinations with conformal factuality (Mohri & Hashimoto 2024). distinct context from traditional Bayesian inference.
Implementation For each dataset, we use MHIP to generate samples ((1), (())....,(m)) ((m) p(0 (D) from a Markov chain of length T. To increase sample diversity we employ burn-in, in which a fixed number d of initial MCMC samples are discarded, and thinning. in which we take every h-th sample thereafter until m samples are obtained. Given a datapoint Tnew, we sample values of ynew using [Eq 6/and quantify uncertainty on the basis of these downstream outputs. Because our research focus is on compatibility with black-box LLMs, in this section we present experiments with GPT- 4o or, when possible, GPT-4o-mini, depending on the difficulty of the dataset. For details such as settings of d, h, m, and other dataset-specific hyperparameters, see App.
4.1
UNCERTAINTY QUANTIFICATION WITH TEXTUAL BAYES
We consider the canonical LLM-based system consisting of single LLM as defined by Eq. 1 Hallucinations in such systems occur when a model responds confidently with incorrect or
Setup
ungrounded information, an issue that can be combatted with calibration (Kadavath et al. [022] Wei et al. 2024). Calibration refers to the quality of model's confidence score, or the probability it assigns to the correctness of its provided answer; in other words, how well the model "knows what it knows". Here, we test calibration in downstream responses resulting from Bayesian inference over the LLM's prompt. We compute confidences by generating 10 responses from each system and measuring the frequency of each response. For MHLP, we initialize (()) to be a generic chain-of- thought (CoT) (Wei et al. 2022) prompt: "Answer the question. Think step-by-step."
Baselines We compare our method against four frequentist basclines. Paraphrasing and System- Message are two prompt perturbation methods proposed by Gao et al. (0224) These methods inject prompt stochasticity by rephrasing the question or system prompt in a question-answeringg con- text
To these add two additional baselines: (i) CoT refers sampling m predictions from
"Our implementation has minor differences from the cited paper. For further details see App. B.2.1

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then (BS(;;; ) and (it) TextGrad refers to first performing T steps of prompt optimization and
sampling predictions from LBS(2; ((T)) Both TextGrad and MHLP require one-time initial fixed cost incurred by prompt optimization and MCMC, respectively, and we use the same value of T for both. All methods use the same number of LBS calls during inference to ensure fair comparison from a computational perspective.
We reiterate Secc. in highlighting that we follow the common pipeline for quantifying uncertainty in two steps: (i) generate a diverse answer set y(!), ( and (it) summarize them into an uncer- tainty score. Because ours is a method for step (i), our baselines are methods designed specifically to do the same. This means
omit direct comparison to means of performing step (ii) such as semantic entropy (Kuhn et al. 2023) and other methods described in Innec.. 5) Although these are also UQ methods, they are orthogonal to our approach, and can be straightforwardly combined (for an example, see App. B.4). For direct comparison, all experiments in this section use confidence or semantic confidence (described below) as the means of summarizing the uncertainty in every set of answers.
Datasets We evaluate both predictive performance and model calibration on AIME 2024 (MA] 2024). SimpleQA (Wei et al. 2024), and QASPER (Dasigi et al.] (022), representing question- answering tasks that are closed-form, free-form, and free-form with context, respectively. We ran- domly select and fix 100 samples from each of SimpleQA and QASPER for all experiments and use all 30 available samples from AIME 2024. Notably, QASPER includes contextless questions, which are explicitly marked as unanswerable. We use these instances to assess our method's s ability to detect insufficient information and abstain from answering. See App. for further dataset details.
In Tab. I we report accuracy for all datasets us- ing exact-match on closed-form datasets and an LLM judge (Zheng et al. 2023) to assess seman- tie correctness on free-form datasets. In Tab. 2
Table 1: Accuracy (%) across datasets
Method
AIME
SimpleQA
QASPER
Paraphrasing
12.6 + 0.7
43.7 1 0.5
43.7 : 1.3
report the expected calibration error (ECE)
System Message
7.2
47.3 1 0.7
59.7 = 0.6
CoT
9.0
47.8
56.5
measure of model calibration (Nacini et al., 2015)
TextGrad
11.9
46.6 0.5
58.8 1.0
Guo et al. 2017). Additionally, for QASPER, we estimate a abstention ability on two types of unan- swerable questions: questions with no context, and those with a random context. We use the same confidence scores used to estimate calibration as an abstention metric and compute the ROC AUC of this score when used as a classifier of answer- ability. Results are shown in Tab. 3 All results are averaged over 10 independent runs with standard errors to account for stochasticity.
MHLP (Ours)
15.0
18.6#006
60.9
Table 2: ECE/ CCCE/SECE (%) across datasets
Method
AIME
SimpleQA
QASPER
Paraphrasing
21.1
0.7
11.61.11
Syste-MMesaage
1
0.1
23.9
CUT
31.5 4: 1.4
18.0000
26.2
0.67
TextGrad
27.4
17.7
110
21.6
MHLP (Ours)
22.0 +
1.0
15.4
0.6
17.7 1.1
Table Abstention ROC AUC (%)
QASPER
Methed
Semantic ECE Standard ECE cannot be applied to open-ended tasks since it requires confidence score, which is nontrivial to compute in general due to the variability of possible correct responses. To address this limitation, inspired by semantic en- tropy (Kuhn et al. 2023), we propose an extension of BCE based on semantic clustering. Our met- ric, semantic ECE (SECE), uses these clusters to estimate model confidence over free-form outputs.
No context
Random context
Paraphrasing
48.2 1.1
62.1
1.6
System-Measage
76.6 = 1.7
69.9 00.9+10 1.3
Coll
+
67.
TextGrad
66.6
1: 2.1
67.4 0.9
MHLP (Ours)
77.9
1.2
71.7
Specifically, for each input 2i, we sample m outputs: y, , ........ y/ (m). We then query LLM to group these outputs into semantic clusters. The empirical probability assigned by the model to each cluster is defined the proportion of the generated samples in that cluster. The maximum of these probabilities is then taken as the model's 's semantic confidence for input x;. Finally, we use this value as the confidence for standard ECE computation, enabling estimation of model calibration for free-form outputs.
Discussion Across tasks, MHLP the only method to consistently outperform the rest. It only trails in calibration (ECE) on AIME, but its accuracy exceeds the two best-calibrated methods by a substantial margin. We hypothesize this outperformance is due to the high gigh-posterior-vaat ples of 0 generated by MHLP; it effectively performs stochastic prompt optimization, incorporating quantitative performance into its accept/reject decisions. In contrast, TextGrad alone has no accep-
sam-
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Vreject scheme and thus "always accepts", leading to the inclusion of potentially less useful changes to the initial prompt. For qualitative examples and diagnostics of accept/reject decisions, see App.
4.2 CONFORMAL FACTUALITY WITH MHLP
Background Conformal factuality (Mohri & Hashimoto 2024) is method for providing statis- tical guarantees on the correctness of ILM-generated answers to open-ended questions based on conformal prediction (CP) (Vovk et al. 2005 [Shafer & Vovk 2008). Generally, CP techniques use a small set of ni labeled datapoints to calibrate a prediction threshold. In conformal factuality, given a question 2, an LLM generates an answer y which is broken into set of distinct claims {cI..... ct). Each claim is assigned a factuality score F(c; [/ceerrageee
generated by an LLM-based system- with larger values indicating increased confidence that e a factual claim. Then, after using CP to calibrate a threshold A, claims with F(c; 0) < X are filtered out, such that only high- soghcconfeeence claims are re- turned in the final answer y. CP guarantees that ğ contains only factual claims with high probability,
1
1 1 S P/c is factual V E ú) S 51-a-- n n+1
where the error rate is user-defined. The quality of final answers can be gauged through the fraction of claims which are retained, since longer answers with more claims are more useful Bet- ter calibrated expressions of confidence through F(c; 0) improve claim retention, and since MHLP enables better calibration, we can use it to design a better factuality score.
Baseline (GPT-4 frequency scoring) The best performing option for F(c; from Mohri & Hashimoto 2024) is frequency scoring. Five alternative answers y(P) generated for the same question x from GPT-4 (Achiam et al. 2023) using unit temperature and a manually crafted prompt 8. For each claim in the original answer y. the number of times it appears across the y i.e. its self-consistency (Wang al. 2023b Manakul al. 2023 is used as the score F(c;
Our method (MHLP frequency scoring) Like GPT-4 frequency scoring, our method estimates a claim's 's importance based on its frequency across alternative generations. However, instead of gener- ating with a single fixed prompt, we produce diverse alternatives by sampling different 0 via MHLP with zero temperature. Notably, in the factuality context, ground truth outputs {yr,.... yn are un- available, so the unnormalized posterior pD))p(D () is unavailable. To surmount this obstacle, we replace the unnormalized probability mass with a surrogate
g(0) Episs 2.0) [ Ecey F(c;
(10)
which estimate stochastically when running Alg. 1 One alternative answer y(l) is generated per sampled prompt. 'he ability to sample from this surrogate distribution underscores MHLP's versatility in situations beyond conventional Bayesian inference.
Dataset We use FactScore (Min et al. 2023), which is widely adopted for factuality tests of LLMs. Following Mohri & Hashimoto (2024). we focus on "person" entities from the biography genera- tion subset and extract subclaims from the generated biographies using the same extraction method across runs. We also follow Mohri & Hashimoto (2024) in using 50 samples for the calibration/test sets and performing 1000 random splits of calibration and test data for each value.
Implementation We initialize both scoring methods with the same prompt. For MHLP, we per- form sampling using a separate set of 100 samples from FactScore and obtain five prompt samples. Since there is no ground-truth answer in the open-ended QA setting, factuality is determined by decomposing answers into claims (as in Mohri & Hashimoto (2024) and annotating them using GPT web search tool. We use GPT-4 for answer generation, and m-o--mmm for claim generation, factuality annotation, frequency scoring, and MHLP proposals. See App. B for more details.
Results First, we verify that both scoring methods achieve the target coverage from Eq. 9 Fig. 2a shows that empirical factuality remains within the conformal bounds across all values of Fig. 2b compares the removal rate, with error bars showing the standard deviation of the average removal rate across the 1000 data splits. Our method consistently achieves lower removal, showing that MHLP scoring provides a better uncertainty estimation of the factuality of LLM outputs]
3 "iltering out all claims guarantces that ý does not contain false claims, but does not give a useful answer.
'The GPT-4 frequency scoring method shows slightly higher removal than reported by Mohri & Hashimoto
2024 likely due to our of stricter weh march-haseee factuality annotator.
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CT1 Eucarcy exiie
MBll (hepl soxiig Dattrnoleaadl
Average Porcent Removed
Emplrica Foctuaity
-4-
Essacnc KEII
0.5
94
0.5
6.7
08
0.5
0.9
Torget Faciuality
Empirical Factuality
(a) Empirical factuality vs. Target factuality 1 1-c a
(b) Average removal rate vs. Empirical factuality
Figure 2: Comparison of conformal factuality frequency scoring with fixed prompt (Mohri & Hashimoto
2024). and with prompts sampled through MHLP. (a) The empirical factuality achieved practice is consis-
tently within the bounds guaranteed by Eq. 2 (b) MHLP achieves the same level of empirical factuality as frequency scoring but removes fewer claims, indicating better calibrated confidence.
5 RELATED WORK
LLMs are applicable to a wide range of tasks and settings, which makes UQ inherently ambiguous there is no single, well-defined quantity that UQ aims to approximate. Although our main setting of interest is where we have access a pre-trained model and no fine-tuning is performed, we note that popular methods for UQ deep learning, such as ensembles (Lakshminarayanan et al., 2017) and Laplace approximations (Ritter et al. 2018 Kristiadi et al., Daxbrrerrrr et al. [2021), ), have been successfully ported over for UQ when fine-tuning LLMs (Wang et al.) [023a] Yang et al. 2024a).
Within our setting of interest, some approaches estimate uncertainty by analyzing the variability in
outputs generated by an LLM given the same input (Kuhn et al. 2023 Lin et al., 2024 Grewal
jet al., 2024 Wang & Holmes 2024 Qiu & Mikkullanenn 2024 Nikitin et al. 2024). others do
so by perturbing or modifying the input itself (e.g. by paraphrasing) (Hou et al.
2024 Gao et al.
2024 Abbasi Yadkori al. 2024 Zhang et al. 2024 Zhao al. Fen4]Feng et al. [2025a), and still
others rely on directly asking the model to express its own confidence (Kadavath et 2022 Yang al. 2024b). Unlike all these methods, we aim to quantify the uncertainty associated with LLM prompts. Other methods for UQ within in-context learning tasks with LLMs have also leveraged Bayesian ideas (Ling et al. 2024 Jesson et al. 2024. Tonolini et al., 2024 Feng et al.] 2025b), but we highlight that these works differ greatly from ours in that they do not directly perform Bayesian inference over free-form text and, once again, they do not quantify uncertainty over prompts.
Lastly, we mention another line of work performing Eatropolis-Hasttnng over text. Like ours, Faria et al. (2024) use LLMs to construct a proposal distribution within the MH algorithm. We nonetheless highlight many differences with this work: their method is applied machine translation and not to UQ they do not perform Bayesian inference, and their proposal is completely different and does not rely on prompt optimization methods. Also, concurrently to work, Faria & Smith (2025) build on top of Faria et al. (2024) by applying their proposal to a Bayesian formulation of the alignment problem wherein aligned model answers are sampled directly using MCMC. Investigating crossover applications of our and their proposals are potential directions for future work.
6 CONCLUSIONS, LIMITATIONS, AND FUTURE WORK
In this work, we propose Textual Bayes for quantifying uncertainty in LLM-based systems. Our work represents formalization of recent work conceptualizing LLM-based systems as models whose parameters are their prompts. Textual Bayes furthers this framework by performing Bayesian inference on these parameters, thus blending cutting-edge models with formal statistical frame- work for uncertainty quantification. To implement this framework, we propose through LLM Proposals (MHLP), novel MCMC algorithm for free-form text which finds applica- tions in Bayesian inference and beyond. We test these frameworks on several uncertainty quantifi- cation benchmarks and find that they consistently improve the frontier of accuracy and calibration. We also show that MHLP can be adapted to a factuality-basee objective, leading to more reliable factual claims quantified by the setting of conformal factuality.

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Although Textual Bayes and MHLP post strong performance against baselines, there remain avenues for improvement. First, MCMC is costly; despite equivalent inference cost to leading baselines, Tex- tual Bayes requires one-time expensive application of MHLP. This cost might be addressed, for example, by further engineering the underlying prompt optimization method or training small language model specifically for the task of generating proposals. Second, like many practical ap- plications of Bayesian inference, our method requires approximations, which will inevitably cause deviations from the true posterior. Third, our evaluations on free-form answering benchmarks re- quire LLM- based clustering. These techniques, though fairly applied across methods, are imperfect and stronger evaluation signal might be obtained with improved fine-tuning, prompt engineering, or human evaluation. Lastly, we expect future work to find broader applications for MHLP beyond Bayesian inference. For example, we could use MHLP to modulate the oulputs of LLM-basedd sys- tems in accordance with unnormalized functions quantifying objectives such as alignment or safety.
Code for this repo is available at https://gihuub.co//lyerrrr labs/textual-bayes Our method is described in full throughout [Sec. 3 and App. A.1 with exper- imental details in Sec. and App. B.2 The datasets we benchmark on are publicly available.
Reproducibility statement
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Yongchao Zhou, Andrei Ioan Muresanu, Ziwen Han, Keiran Paster, Silviu Pitis, Harris Chan, and Jimmy Ba. Large language models are human-level prompt engineers. In The Eleventh Interna- tional Conference on Learning Representations, 2022.
Mingehen Zhuge, Wenyi Wang, Louis Kirsch, Francesco Faccio, Dmitrii Khizbullin, and Jürgen Schmidhuber. GPTSwarm: Language Agents as Optimizable Graphs. In Forty-first International Conference on Machine Learning, 2024.
Andy Zou, Zifan Wang, Nicholas Carlini, Milad Nasr, Zico Kolter, and Matt Fredrikson. Universal and transferable adversarial attacks aligned language models, wrXiv:2307.150433 2023.
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A METHOD DETAILS
A.1 USEFUL APPROXIMATIONS
In general, MCMC can only be applied Bayesian inference when the g(0) calculable, where g(0) is defined by
n g = p(0)p(D - p(0) II p(v: / x,, 0).
(11)
In our context, certain terms in this equation are intractable and we instead estimate them stochasti- cally.
Mini-batch estimates of II P(u Tz. rz, 0) Computing the term IT P(y Ie. 0) requires eval- uating LBS(x;) for all i E (I....,7) n.. Training sets are often large enough make running full pass per MCMC step intractable. Instead, we use mini-batching. One way to mini-batch is to apply the MH adjustment of Seita al.] (2018). which involves a different accept/reject step; however, to avoid complexity size b, we make the estimate
use simple stochastic estimate of II:
Iz. 0) instead. Using batch p(y
b
I 24 0) =e II p(us 0),
(12)
==1
which is an unbiased estimate in log-space:
Σ log ply: | 12, 0) (IBBEe.))-UUnlomm)) (log(( Ix,
log log((1n; 1yiy;
(13)
b
where {x., y/, )i,).... (Zin, vi,)) is mini-batch of training datapoints. In experiments, we use batch size of
Tempered posterior A well-studies phenomenon in Bayesian deep learning is the cold posterior
effect wherein sampling from the "tempered" posterior (0 D) p(D 00)/Tp(() with 0 < T < often results better empirical performance than standard Bayesian posterior (i.e., = 1) (Wenzel et al. 2020, Aitchison 2021 Fortuin et al. 2022, Izmailov et al. 2021 Noci et al. 2021
Kapoor et al., | Nabarro et al. 2022). Following this practice, we apply a temperature т, making
the final estimate equal to
T1
II P(3 0)
II p(u,, Ity, 0)
(14)
For simplicity, we absorb the exponent into a single constant / = to and tune 3 for performance. As per the hyperparameter details below, often n/b the most effective, indicating hot posterior effect in our case.
Monte Carlo estimates of p(y: Fz. f) with surrogate models For a single LLM call y = LLM(x;; 9) that outputs answer directly, computing log p(y 2, is simple summing log- probabilities across every token in However, complex LLM-based systems that include interme- diate outputs and reasoning involve sources of stochasticity that not captured in log-probabilities associated with the tokens of
Let be a variable capturing all intermediate outputs in the computation of y LBS(2; 8). This includes internal LLM calls and reasoning text. We can express the generative process of sampling y as
p(z 0),
y p(y z, x, 0).
(15)
Then the probability p(y x. 6) required for MHLP would be computed as
ply x, 0) = Σ p(y z, x, 0)p(z x 0) = Es-p(sla,0) (p(y z, 0)]
(16)
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As suggested by the second equality, this intractable sum is again amenable to Monte Carlo esti- mates. We use single sample of to estimate the likelihood in MHLP. We note that one alternative would be to remove all stochasticity from 2 p(2 x, 0) by fixing seed setting the LLM tem- perature to 0 in the process of sampling 2; however, this would remove necessary stochasticity from the final model result and alter the underlying model.
When using closed-source model providers, bog-probabilities and thus the value of p(y 2, 0) itself is sometimes withheld. In this case, during MHLP we substitute the final LLM call in our LLM-based system with an open source model.
Lastly, we point out that all of the tricks above apply computing probability masses for any LLM- based system, including the proposal density q(O' () also required for MHLP.
A.2 UPDATES WITH TEXTGRAD
In our experiments, we implement UPDATE as a TextGrad step (Yuksckgonul et al. 2025). Given model output yppeed LBS(2; [z; 0), compute loss string & using a textual loss function LOSS
L = LOSS [z, Upred v)
LLM (2) Upred P:
(17)
Where is dataset-specific prompt describing how to evaluate a given model output, such as
Algorithm 2: TextGrad Update
Require: ((t-1);
will given question related scientiric research papers and an answer attempted by Tanguage model. Fvaluate the atteapted logtcal, very critical.
optimizer +
optimizet.zero-graana graad::
Be Do not
Upred LBS ((t-1));
salve the question. Just provide concise Feedback.
{ LOSS (r, y, Mpredd;
Question: { (ar)
(.BACWWARD
Actenpted answer: Upred )
0' 4- optimizer.STEP():
True answer: (M)
return
TextGrad implements autograd-style wrap-
per for all textual variables. This wrapper provides the method (.BACWWARDO. which "back- propagates" cariable-wise feedback from the evaluation L. The TextGrad optimizer.STEPP method
then incorporates this feedback to build a new parameter set 0'. Pseudocode is given in Alg. 2
To run Metropolis-Hastingg (Agg. 1,, we also need to compute the proposal density value g(',, I 0), where our case 0' UPDATE(0). Using the approximations described in App. A.1 we need only compute logits from the final call optinizer.STEP, and so always compute this step with an open-source LLM (Llama-3, --e.motroo------ -0B-Instrutt-HHH (Wang et al., 2025 Bercovich et al. 2025
B EXPERIMENT DETAILS
B.1 DATASET DETAILS
AIME MAA 2024 released under the MIT license, contains problems from the American Invi- tational Mathematics Examination (AIME) prestigious high school competition known for its challenging mathematical questions. Each answer is an integer. The exam consists of 29 to 30 ques- tions per year. For evaluation, we used the 2024 exam, which was not included in GPT's training data.
SimpleQA (Wei et al., 2024). released under the MIT license, is a benchmark that evaluates the ability of LLMs to answer short, fact-seeking questions. It covers wide range of topics, including science, history, geography, history, politics, etc. Both its questions and answers are short and direct. In our experiments, evaluated the models subset of 100 examples from the dataset.
QASPER (Dasigi et al. 2021), released under the CC-BY-4.0 license, a free-form question- answering dataset focused scientific research papers. It contains 5,049 questions across 1,585 papers in the field of Natural Language Processing. Each question is based on the content of spe- cific paper. In experiments, we provided the model with passage from the paper that contains
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the answer (i.e., the context), and then posed the question for it to answer using that context. We evaluated our model on 100 samples from this dataset under two different scenarios. In the first scenario, the context was entirely missing for 35 of the samples. In the second, 33 samples were provided with randomly selected context (Wen et al., (024) that did not contain the correct answer. In both cases, the model was expected abstain from answering.
B.2 HYPERPARAMETERS
In the following experiments. we use the OpenAl API for calls to GPT-4o-mini and GPT-4o. As our surrogate model for probability mass estimates (see App. A.I), we use Llama-3. 3.1 -Nemotron-70B-- Instruct-HF (Wang et al. 2025 Bercovich et al. 2025) through the Together AI API.
For LLM calls we use a temperature of We ensure m 10 final answers are sampled for each method. For the Chain-of-Thought baseline, we used the initial prompt and sampled 10 an- swers, then aggregated the resulting answers. For TexIGrad, we use the sample initial prompt but run TextGrad for given number of steps before sampling 10 answers from the final prompt. For MCMC, we sample 10 individual prompts from a single MCMC chain and sample answer from each. We tune the MHLP parameter 3 (see App. A) separately for each dataset. GPT-4o was ployed for clustering and LLM--based evaluation. Further hyperparameter details are shown below and in code (see especially the config files qasper yaml, aime. yaml, and simpleqa. yaml).
Table 4: Hyperparameters used for each dataset and method
Dataset
Method
Model
Steps (T)
Burn-in (d)
Thinning (h)
Chain-of-Thought
AIME
TextGrad
GPT-4o
60
MHLP
60
10
Chain-of-Thought
SimpleQA
TextGrad
GPT-4o
60
MHLP
60
100
Chain-of- Thouugh
QASPER
TextGrad
GPT-4o-mini
20
MHLP
20
100
2
2
For all methods, we fix string the end of the prompt describing standardized formatting instruc- tions for the model's 's final answer. We extract this answer and evaluate likelihoods p(y 2, 0) only on this value, relegating any reasoning beforehand to the variable (see App. A.1).
B.2.1 BASELINES
We adapted the perturber baselines from SPUQ (Gao al. 2024), specifically selecting the Para- phrasing and System Message perturbers for comparison. For all runs, we used GPT-4o-mini and GPT-4o for fair comparison. Our implementation differs from the original in several details:
Paraphrasing: Rather than using a single L.M call with JSON formatting to produce all para- phrases, we made separate LLM calls for each paraphrase (to avoid invalid JSON outputs from the LLM with the original prompt). We used the following prompt:
Suggest way to paraphrase the text triple quotes above. If the original text is a question, please take sure that your answer is also question. answer also has those options the order
If the original text has answer options, please nake
Answer should ONLY be the paraphrose and nothing else.
System Message: Instead of sampling with replacement from the available prompts, we expanded the set of system prompts and sampled without replacement. We appended these system prompts to the beginning the message chain to preserve any existing system prompts. This was crucial for maintaining the output format required by the evaluator (e.g., answers ending with Answer: <THE ANSWER>). The set of system prompts used was:
'you are helpful assistant*
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"you are question answering assistant'
"you are nice assistant"
"You are an Al support toel."
"You friendly helper.
to "You provide useful answers."
are
users."
"You are Kine AI agent.
YYou offer
information."
are snart assistant."
YYou help with nary tasks."
"You are reliable AI."
"You
are an
assistant."
"You try to useful."
"You are positive AI."
"You guide
are adept helper."
"You sinplify complex things."
"You are virtual guide."
"You
B.2.2 CONFORMAL FACTUALITY
Factuality annotation Since there is no ground-truth output for the biography generation task, assess the factuality of each generated answer by verifying its atomic sub-claims via web search. We use the GPT web search tool AP which allows the model to retrieve external evidence be- fore making a judgment. Each sub-claim is labeled factual (1) or not (0) based on the retrieved information. We call the API as follows:
response GPT_client. responses.. create podel="gpt-40--0 tools=[
"type": "web_ wee__search.previer" "saarch.conext.siie""" "low"
}
]. input=prompt,
response_content response output_text
Model and Prompt Setup We use GPT-4 for base biography generation and GPT-4o-mini for claim decomposition, factuality annotation, and frequency-based entailment scoring. Both the base- line frequency scoring and MHLP initialization use the same default system prompt: "You are helpful assistant. Write bio for people. For frequency scoring, generate five alternative answers using this prompt. All prompts are listed in T
Hyperparameter We run a single Metropolis- Hastings chain with T = 20 total steps, a burn-in of d = 4, and thinning interval of h 4, resulting in 4 sampled prompts. Together with the initial prompt, we obtain 5 prompts in total, which are used to compute frequency scores.
B.3 EXAMPLES
In this section, we explore some examples of how the algorithm runs.
First, for SimpleQA and QASPER, we exhibit several example questions and answers comparing results from Textual Bayes to TextGrad. We show how the 10 answers sampled by each method clustered, and the number of answers that fall into each cluster. Overall, we see that Textual Bayes's confidence levels are better calibrated to the model's correctness.
For AIME, we explore the algorithm's acceptance rate and individual accept/reject decisions over time.
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Subclaim Separator
Please breakdown the following return the output as a jsonl, where each line is subclaim: [O. AIMI, gpt-score: [coN]]. An The confidence score [OONF] should represent your confidence in the clain, where Is for clains that are very obscure or difficult for anyone to know, like the birthdays of non-notable people. If the input input is:
into
snall, independent clains (make not to and any infornation).
ebvious facts and results like 'The earth round' and '1+1=2'
shart,
only
Frequency scoring
You will get list clains piece text. For each clain, score whether text supports, contradicts, or unrelated to the claim. Directly return Jsonl, where each line is ["Id":[CLAIM,IO) "score": [SCORE]] "score":[SCORE] Directly return the ison1 with no explanation or other formatcing. For the [SCORE]. return for supports, for contradicts, and for unrelated. claims are:)n(clain..trinn]] (nynThe is:\yn(output)
Factuality Annotation
keys: ("subclain": "[CLAIM]", "factual"; (nReturn your enswer as a JSON array, where each "source": "source explanation" ) your response valid JSON array only, with additional text or fornatting. \nn Examples Din ("subclain": *claim 1", "factual": 1, "source"; "source"). \n ("subclaim": "clain 2", "factual": 8, "source": "source") An Jin
Please verify if each of these claims is factual. object
factuel.Anclaiin AnClains: (ntcleinsstext]
Table 5: Prompts for sub-claim separator, frequency scoring, and factuality annotation. Note both sub-claim separator and frequency scoring prompts are the same as used (Mohri & Hashimoto 2024
B.3.1 SIMPLEQA
The following examples selected from the SimpleQA dataset. The second example represents case where the LLM appears truly to not know the answer, our method quantifies uncertainty better by expressing much lower confidence (40%) than the TextGrad baseline.
Question:
According to Medland, Sarah E.; Loesch, Danuta Z.; Mizewski, Bogdan; Zhu, Gu; Montgomery, Grant W.; Martin, Nicholas G. (September 28, 2007), what chromosome location was identified as linked to the finger ridge counts of the ring, index, and middle fingers through multivariate linkage analysis?
Answer: 5q14. 1
Table 6: Counts per semantic cluster for TextGrad and our method
Semantic Cluster
TextGrad
Ours
5q14.1 5q14.3
I
1
15q14
O
21q22
1
O
3q26
O
5q13
0
1
5q35
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Question:
What the population of the town of Lesbury in Northumberland, England in the 2011 census?
Answer: 1007
Table 7: Counts per semantic cluster for TextGrad and our method
Semantic Cluster
TextGrad
Ours
1,154
1,118
1,057
1,205
O
1,264
1,386
0
1,122
984
1,187
1,112
B.3.2 QASPER
The following examples are selected from the QASPER dataset. Note that for the second example, the context given to the model is unrelated to the query, making the query unanswerable such that one would expect well-calibrated LLM to express a high degree of uncertainty.
Context: We begin with hate speech lexicon containing words and phrases identified by internet users as hate speech, compiled by Hatebase.org org. Using the Twitter API we searched for tweets containing terms from the lexicon, resulting in a sample of tweets from 33, 458 Twitter users. We extracted the time-line for each user, resulting in set of 85.4 million tweets. From this corpus then took random sample
tweets containing terms from lexicon by CrowdF Fower (CF) workers. Workers were asked to label each tweet as one of three categories: hate speech, offensive but not hate speech, or neither offensive nor hate speech. They were provided with our definition along with paragraph explaining it in further detail. Users appearing in a given tweet were instructed that the presence of a particular word, however offensive, did not necessarily indicate a tweet is hate speech. Each tweet was coded by three or more people. The intercoder-aggremeer decision for each tweet to assign label. Some tweets were not assigned labels as there was no majority class. This results in a sample of 24,802 802 labeled tweets.
them manually coded
asked to think not just about the words about the context in which they were used. They
provided by CF is 92%. We the majority
Question: How long is their dataset?
Answer: 85400000
Table Counts per semantic answer for TextGrad and our method
Semantic Answer
TextGrad
Ours
85.4 million tweets 24,802 tweets
9
4
Random Context: Figure FIGREF4 is the overview of the proposed method using character 3- gram embeddings (char3-MS-vec). illustrated this figure, our proposed method regards the sum of char3-MS-We and the standard word embedding as an input of an RNN. In other words, let INLIFFORMM be char INLINEFORMI -MS-vec and we replace Equation with the following: DISPLAYFORMO
Question:
they report results only on English data?
Answer: Unanswerable
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Table Counts per semantic answer for TextGrad and our method
Semantic Answer
TextGrad
Ours
Unclear not specified in context
0
Results are only on English data
0
1
Results are not only on English
9
3
Formatting error in answer
1
0
MHLP Acceptance Rate Over Time
Acceptance Rate
0.35
0.30
0.25
0.20
0.15
40
60
Steps
Figure 3: MHLP acceptance rate over time.
B.3.3 AIME
On the AIME dataset, we analyze the model's accept/reject decisions over time. In Fig. 3 we find that the model's acceptance over time closely matches the heuristic optimum of 0.234 prescribed by Gelman et al. (1977. Tab. I0 shows individual accept/reject decisions.
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Table 10: AIME MHLP Accept/Reject Decisions
Step
Current prompt
Proposed prompt
Accepted (Y/N)
Answer the math question. Think step-by-step.
Answer the math question by providing a step-by-step solution that explicitly considers all given conditions, such divisibility requirements terms of prime factor exponents. Employ appropriate combinatorial methods to account for problem constraints. Ensure accurate probability calculations based on correct combinatorial reasoning. Verify the simplification of your final answer, confirming the correct
Y
calculation of the greatest common divisor (GCD) if applicable. Present a clear, logical explanation for each step, and iteratively review and refine your solution to guarantee accuracy and clarity.
20
Provide a concise, step-by-step solution to the math question, ensuring clear understanding by first restating the problem in your words. Adopt systematic approach, setting up relevant equations or congruences that directly address the problem's 's conditions. Verify each step's s accuracy, correctly applying mathematical concepts such as modulo arithmetic, and maintain logical flow by simplifying expressions where necessary. Ensure comprehensiveness by considering all possible solutions, and guarantee accuracy by cross-ched coss-cheekiing key calculations, ultimately leading to a precise final answer that encompasses all valid solution values.
Provide a concise, step-by-step solution to the math question, ensuring clear understanding by first restating the problem in your words. Apply relevant geometric principles and formulas directly related to the problem, avoiding unnecessary assumptions and concepts. Structure your response in logical, connected manner, justifying each step. Verify calculations for accuracy, cross-checking key steps to guarantee precise final answer that encompasses all valid solution values, based solely on the provided information and pertinent mathematical concepts.
40
Provide a concise, sep-by-step solution to the math question, ensuring clear understanding by explicitly stating the problem's 's requirements and constraints. Justify your choice of coordinate systems and break down steps for deriving coordinates. Verify calculations, list and validate assumptions, and maintain coherent logical flow with explanations for each decision. For geometric problems, explicitly calculate and present coordinates of critical intersection points and provide detailed, logically sound methods for calculating areas. Finally, re-evaluate your approach, cross cross-reference with alternative methods if possible, and verify the correctness of your answer to ensure accuracy and relevance.
Provide a concise, step-by-step solution to the math question, ensuring clarity and accuracy. Analyze the problem's 's recursive definition (if applicahle), explicitly describing the transformation of key elements (e.g.. zeros, functions) at each step. Derive any formulas used for calculations from the given definitions. logically connecting each step. Include detailed breakdown of intermediate calculations and explanations for each decision. Where applicable, utilize visual aids or specific examples to illustrate complex transformations. Finally, gross-verify your approach by considering alternative methods or perspectives, and validate your final answer ensure accuracy and relevance.
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B.4 SEMANTIC ENTROPY
In the main text, we quantify uncertainty for our method and all baselines using confidence: the probability model assigns given answer (or estimates thereof). Confidence useful because it has a clear mathematical interpretation and can be used to assess calibration, but as outlined in Sec. there a numerous other ways to compute uncertainty scores from LLM-based systems.
A popular uncertainty score among these is semantic entropy (Kuhn et al., 2023). In Tab. 11 we check whether our performance is robust alternate ways of estimating model uncertainty by using semantic entropy an abstention score on the QASPER dataset, where unanswerable questions are those with the context removed. We find that the relative performances of methods in Sec. using confidence match those using semantic entropy.
Table 11: QASPER Abstention ROC AUC (%) with Semantic Entropy
Method
ROC
Parapbrasing+
50.0 1.4
System-Message+SE
68.1 1.7
CoT+SE
71.3 1.8
TextGrad+SE
70.2 1.1
MHLP+SE
78.2 1.1
24


Reviews:
[{"review_id": "kLXo6tIXCM", "reviewer": "Reviewer_wYh1", "summary": "This paper presents a Textual Bayes, a Bayesian framework for uncertainty quantification in LLM-based systems. The key idea of this work is how to treat prompts for LLMs as training textual parameters and perform Bayesian inference over them. For this, authors introduce a novel algorithm Metropolis–Hastings through LLM Proposals (MHLP), which leverages LLM-driven prompt optimization to generate proposal prompts and sample from a posterior distribution. It allows to estimate uncertainty over both prompts and outputs.\n\nExperimental results on QA and factuality benchmarks show that proposed framework, Textual Bayes, improves the following metrics: predictive accuracy, calibration (via lower Expected Calibration Error and a new semantic ECE metric). This method also improves factual reliability in conformal prediction settings by reducing the number of removed claims. Overall, the authors introduce novel integration of Bayesian reasoning with black-box LLM-based systems, enhancing uncertainty quantification in generative AI.", "questions": "1.\tThe paper mentions applying a “cold/hot posterior” using a likelihood temperature \\( \\tau \\) (or the equivalent scaling factor \\( \\beta \\)), but does not explain the choice or its impact. Could you clarify what specific temperature values were used, how they were selected or tuned, and whether model accuracy or calibration metrics (e.g., ECE) are sensitive to different temperature settings?\n2.\tCould you report standard diagnostics for the MHLP sampling process, such as acceptance rates, effective sample sizes, or any convergence checks (e.g., monitoring of likelihood or posterior stability)? Additionally, did you test whether using longer Markov chains (more MHLP steps) materially changes the results in terms of accuracy or calibration?\n3.\tHow do outcomes change with weaker/stronger or mis-specified textual priors \\\\(p(\\theta)\\\\)? Did you run any ablations that vary the prior constraints versus the data likelihood \\\\( p(D \\mid \\theta) \\\\) to assess sensitivity?\n4.\tCan MHLP handle multiple textual parameters \\\\( \\theta = (\\theta_1, \\ldots, \\theta_k) \\\\) within a single system pass, and have you tested this capability in agentic or multi-stage LLM pipelines?\n5.\tThe paper references the use of a “cold/hot posterior” and defines a tempered likelihood scaling factor \\\\( \\beta := \\frac{n}{\\tau b} \\\\), but the description appears inconsistent – Section 3 mentions a cold posterior, while Appendix A.1 describes an effective “hot” behavior when \\\\( \\beta < n \\\\).  Please clarify whether the posterior is cold, hot, or generally tempered; explicitly report the values of \\\\( \\tau \\\\) (or \\\\( \\beta \\\\)) used in experiments and how they were selected.  In addition, please interpret these quantities relative to the mini-batch size \\\\( b \\\\), not to the total dataset size \\\\( n \\\\), to make the scaling consistent and reproducible.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. Recasts prompts as Bayesian textual parameters in \\( p(y \\mid x, \\theta) \\) and performs inference over prompts \\(\\theta\\).\n2. Introduces MHLP, an MCMC scheme with LLM-driven proposals – both novel and well-motivated.\n3. Clear prior construction via free-form textual constraints.\n4. Proposal mechanism targets high-posterior prompts by respecting priors and improving data likelihood \\( p(D \\mid \\theta) \\).\n5. Demonstrates consistent gains in accuracy and calibration across QA benchmarks.", "weaknesses": "1. High computational cost due to MCMC with large LLMs. It is worth considering lighter models.\n2. The paper explicitly uses a cold/hot posterior and surrogate/open-source likelihoods for closed-source models, but doesn’t show sensitivity to these choices, which could bias the posterior and calibration.\n3. Evaluation limited to single-prompt systems; no empirical tests with multiple textual parameters \\((\\theta_1, \\ldots, \\theta_k)\\).\n4. Limited study of prior quality and initialization (limited analysis of sensitivity to priors and initialization)", "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": "JL8zlndsNY", "reviewer": "Reviewer_jbfF", "summary": "The authors propose Textual Bayes to quantify uncertainty in black-box LLM systems by treating prompts as Bayesian textual parameters. It introduces a new MCMC algorithm, MHLP, which uses the LLM to generate prompt proposals improving uncertainty quantification and predictive accuracy.", "questions": "- The argument for not computing an uncertainty score (such as SE), namely that the \"method is a means of generating a diverse answer set\" (line 314), is not clear. Why does MHLP lead to diverse answers? And why can't an uncertainty score be computed that is then compared to SE and other UQ methods?\n- Re. Evaluation point 2: What if CoT gets the same overall budget of MHLP? Would MHLP with matched compute still underperform this baseline?", "limitations": "", "rating": 2, "confidence": 3, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "- The central idea of treating the prompt ($\\theta$) as a textual parameter in a formal Bayesian model ($p(\\theta|\\mathcal{D})$) is interesting. It provides a principled, statistical approach to modeling the brittleness of LLM systems, namely their sensitivity to prompt engineering.\n- The proposed MHLP algorithm is theoretically well motivated. It repurposes prompt optimization techniques as MCMC proposal distributions for UQ even when using closed-source, black-box APIs.", "weaknesses": "**Motivation:** The claim \"we are the first both to quantify the uncertainty associated with prompts in LLM-based systems\" (line 79) is not correct! There exist multiple works that have already considered this, such as Abbasi-Yadkori et al. [1], Tonolini el al. [2], or Zhang et al. [3].\n\n**Method**: The authors introduce a sophisticated and complex theoretical methodology. However, the proposed practical implementation relies on crude approximations that compromise the Bayesian guarantees and that questions whether a much simpler, cheaper heuristic would perform just as well:\n1. Correction Factor: The required Hastings Correction is ignored (set to 1) because black-box LLMs provide no probabilities. This transforms the method from a theoretically sound sampler into an ad-hoc Metropolis algorithm.\n2. Acceptance Metric: The core quality measure (the likelihood term) is calculated by a different surrogate model. Moreover, to make the method computationally feasible, it is approximated over a singe sample (batch size set to 1 as stated in in line 283).\nThe complex MCMC process ultimately reduces to accepting a new prompt if a single sample yields a higher log-likelihood score under a surrogate model. \n\n**Evaluation**: \n1. The uncertainty being quantified is primarily over prompt choice rather than epistemic or aleatoric uncertainty in the traditional sense. This should be made more clear and also reflected in the title as it might be misleading.\n2. The authors claim that \"all methods use the same number $m$ of LBS calls during inference to ensure a fair comparison from a computational perspective\" (line 312). This is technically true but highly misleading: the methods use the same inference budget (10 calls per test instance) but they do *not* use the same total budget. MHLP gets 180-300 extra calls that CoT doesn't get.\n\n\n---\n[1] Y. Abbasi-Yadkori, Ilja Kuzborskij, A. György, and C. Szepesvari. To believe or not to believe your LLM: iterative prompting for estimating epistemic uncertainty. NeurIPS, 2024.\n\n[2] F. Tonolini, N. Aletras, J. Massiah, and G. Kazai. Bayesian Prompt Ensembles: Model Uncertainty Estimation for Black-Box Large Language Models. ACL, 2024.\n\n[3] Z. Zhang, A. Verma, F. Doshi-Velez, B. Low. Understanding the Relationship between Prompts and Response Uncertainty in Large Language Models. arXiv preprint arXiv:2407.14845, 2024.", "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": "UBp96GzB2o", "reviewer": "Reviewer_D3Rb", "summary": "This paper aims at uncertainty quantification in LLM-based systems. Specifically, The authors interpret prompts as Bayesian textual parameters in a statistical model and perform Bayesian inference over these prompts with introduction of  Metropolis-Hastings through LLM Proposals (MHLP). These prompts are then used to calculate uncertainty. They demonstrate improved predictive accuracy and uncertainty quantification (UQ) on question-answering tasks and also in a conformal prediction setting.", "questions": "1. How exactly do the authors use suggestions from LLMs to propose values of θ that implement the guidelines? (Line 254) Could the authors show the actual prompt?\n\n2. How are the textual constraints on prompts encoded in the prior, and how is it determined what constraints a valid prompt should satisfy? (Line 232)\n\n3. In line 817, what does “substitute the final LLM call in our LLM-based system with an open-source model” specifically mean? \n\n4. Missing citations: https://aclanthology.org/2024.naacl-long.390.pdf, https://arxiv.org/pdf/2412.09572", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "The paper proposes a novel approach to quantify the uncertainty associated with prompts through Bayesian inference.\n\nThe narratives are generally clear and easy to follow.", "weaknesses": "The empirical improvements in Tables 1–3 are relatively small compared to the substantial computational overhead reported in Appendix Table 4. Since the evaluation is also conducted on very small datasets, this further weakens the strength of the empirical evidence.\n\nI have concerns regarding the final sampled prompts. For example, I do not understand why the proposed prompt in step 20 is rejected. Another baseline that should be tested is just paraphrasing the prompt while keeping the question the same. The authors should also providing a full set of sampled prompts for at least one example which would help clarify this issue.\n\nThe justification regarding semantic entropy (lines 314–320) is not convincing. If multiple answers are semantically equivalent, they should be treated as a single answer. Similarly, in Tables 1 and 3 the evaluation should incorporate semantic clustering so that semantically equivalent answers are not counted as distinct ones.", "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": "0U8q5ur2cI", "reviewer": "Reviewer_dut8", "summary": "This paper proposes a method called Textual Bayes, which treats a set of prompts as textual parameters $\\theta$ in a Bayesian modeling $p(y|x, \\theta)$. Specifically, $\\theta$ contains a set of prompts, and the paper proposes an algorithm to optimize this set of prompts in order to maximize $p(D | \\theta)$, where D is the data, so that we view the optimized set of prompts $\\theta$ as faithful to the original prior (e.g., human-defined constraints), and best for model's performances on data $D$. The authors claim that using these prompts, we can better estimate the model's internal confidence by computing the expected probability over all prompts. \n\nThe core of this optimization algorithm is called Metropolis–Hastings through LLM Proposals (MHLP), which uses an LLM-powered optimization step (TextGrad) and the Metropolis–Hastings algorithm, which is a novelty of the paper. The paper also introduced semantic ECE, which can be used to calibrate free-form generations of LLMs. Experiments showed that the proposed Textual Bayes method led to better expected calibration error and better task performance across several datasets (AIME/SimpleQA/QASPER), and more reliable factual claims.", "questions": "See weaknessness.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "The overall setup offers a novel perspective on existing discussions in prompt selection, multi-step reasoning (e.g., self-reflection and TextGrad), and calibration, by treating textual prompts as Bayesian parameters. The authors provided a working recipe for this setup and demonstrated relatively promising results on calibration and factual correctness experiments. The paper is well organized, with a background section that helped me understand the overall setup in a short time. This paper may inspire future works to think more about abstracting LLM components into probabilistic frameworks.", "weaknesses": "1. The experiments are a little weak for the paper to be well-perceived as an uncertainty quantification work. The experiments are on three QA datasets with small sample counts. Some gains are modest and may not be statistically decisive (especially considering the variances). The baselines are a little weak, where the authors should consider prompt‑ensemble and decoding‑ensemble baselines (e.g., self‑consistency with diverse system messages/instruction styles, retrieval‑augmented ensembles, or perturbation and entropy-based methods). To showcase the effectiveness of the proposed prompt optimization, the authors should consider demonstrating examples and a human study, as well as comparing against simple baselines (e.g., having an LLM to \"decompose\" the original prompt into a diverse set of candidate system prompts that consider multiple angles).\n\n2. The authors should cite other papers that similarly abstract textual factors into Bayesian parameters in LLM settings for training, inference, and uncertainty quantification that share high-level motivations. Example: BIRD: A TRUSTWORTHY BAYESIAN INFERENCE\nFRAMEWORK FOR LARGE LANGUAGE MODELS", "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": "4BvOyiFRVK", "reviewer": "Reviewer_j29K", "summary": "The paper proposes to see the prompts as \"textual variables\" for LLMs.\\\nThey approximate the posterior distribution of the textual variable with MH technique.\\\nIn MH technique, prior is defined with human constraints, and proposal distribution is defined as Update, which is TextGrad in this paper.\\\nAfter this process, we can sample multiple textual variables (i.e., prompts) from the posterior distribution.\\\nLastly, by using these multiple prompts, we can approximate the posterior predictive distribution.", "questions": "1. Is there any warm-up steps for prompt sampling?\\\nIn MH, by TextGrad, the prompt is gradually adapted to the true distribution.\n\n2. $\\\\mathcal{D}$ is a single sample or an entire dataset?\\\nI think it should be an entire dataset.\\\nCan you elaborate more on how TextGrad updates a single prompt for the entire dataset?", "limitations": "", "rating": 4, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "1. The paper is well-motivated.\n- The large language models indeed show some vulnerability depending on the prompt, and prompt engineering is widely adopted.\n\n2. The proposed method to approximate the posterior distribution is sound and solid.\n- MH algorithm is a widely adopted technique to sample from an arbitrary distribution.\n- The textual prior defined in Eq.7 is understandable.\n- The proposal distribution defined with TextGrad is very interesting and makes sense for me.", "weaknesses": "1. Bayesian lens\n- The title of this paper is Title: \"TEXTUAL BAYES: QUANTIFYING UNCERTAINTY IN LLM-BASED SYSTEMS\".\\\nHowever, in my opinion, the paper only deals with \"prompt uncertainty\"\n- A true Bayesian lens would be the posterior over \"model parameter\", rather than over textual variables.\\\nFor example, if a user inputs the question with his own prompt, then this method cannot quantify the uncertainty.\\\nIn this sense, I think the prompt is more of an input rather than a parameter.\n\n\n2. \"To the best of our knowledge, we are the first both to quantify the uncertainty associated with prompts in LLM-based systems.\" could be an over-claiming.\n- [A] and [B] also quantify the uncertainty associated with prompts in LLM-based systems.\\\nIndeed, [B] defines \"prompt uncertainty\" in their paper, and tries to quantify it.\n- [A] Uncertainty Quantification with Pre-trained Language Models: A Large-Scale Empirical Analysis, EMNLP'22\n- [B] Uncertainty Quantification and Decomposition for LLM-based Recommendation, WWW'25\n\n\n3. Method section\n- It ends with samples ($\\theta$). What's next?\n- It would be better to link how those samples can be utilized for the posterior predictive distribution in Eq.6.\n\n\n4. Experiment\n-  A direct evaluation of \"uncertainty\" of the predictive distribution (Eq.6) would be needed.\\\nWe can compute the entropy of Eq.6 and adopt metrics like Kendall's tau to evaluate how well the uncertainty is related to the performance.\n- I think the calibration is more like a post-processing process.", "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 (Poster)