Hallucination Detection on a Budget: Efficient Bayesian Estimation of Semantic Entropy
Authors: Kamil Ciosek
Nicolò Felicioni
Sina Ghiassian
Venue: Accepted by TMLR ()

Abstract: Detecting whether an LLM hallucinates is an important research challenge. One promising way of doing so is to estimate the semantic entropy (Farquhar et al., 2024) of the  distribution of generated sequences. We propose a new algorithm for doing that, with two main advantages. First, due to us taking the Bayesian approach, we achieve a much better quality of semantic entropy estimates for a given budget of samples from the LLM. Second, we are able to tune the number of samples adaptively so that `harder' contexts receive more samples. We demonstrate empirically that our approach systematically beats the baselines, requiring only 53% of samples used by Farquhar et al. (2024) to achieve the same quality of hallucination detection as measured by AUROC. Moreover, quite counterintuitively, our estimator is useful even with just one sample from the LLM.

Published in Transactions on Machine Learning Research (09/2025)
Hallucination Detection on a Budget: Efficient Bayesian Estimation of Semantic Entropy
Kamil Ciosek Spotify
kamilo@spotifu. com com
Nicolò Felicioni Spotify
nicolof com
Sina Ghiassian Spotify
sinag @spootfyy.oomm com
Reviewed on OpenReview:
https: // openreview. net/ forum? id= J2N2RuNdbC
Abstract
Detecting whether an LLM hallucinates an important research challenge. One promising way of doing so to estimate the semantic entropy (Farquhar al., 2024) of the distribu- tion of generated sequences. We propose new algorithm for doing that, with two main advantages. First, due to taking the Bayesian approach, achieve much better qual- ity of semantic entropy estimates for a given budget of samples from the L.M. Second, we are able to tune the number of samples adaptively so that 'harder' contexts receive more samples. We demonstrate empirically that our approach systematically beats the basclincs, requiring only 53% of samples used by Farquhar et al. (2024) to achieve the same quality of hallucination detection as measured by AUROC. Moreover, quite counterintuitively, our estimator is useful even with just one sample from the LLM.
1 Introduction
Detecting hallucinations in LLMs is task of huge practical significance (Ji et al. 2023). An important subset of hallucinations, called 'confabulatory', amounts to the model making up confabulations or statements with made-up meanings (Filippova 2020 Maynez et al. 2020). Recently, semantic entropy (Farquhar et al., 2024) model exhibits this lype hallucination
has been introdueed as an imporlant indicator for detecting Semantic Entropy is based on two principles. The first one to measure a type of Shannon entropy of the soquences generated by model, reflecting the idea that large entropy indicates confounding knowledge. The second principle is to do the measurement in the space of meanings rather than directly operating on raw token sequences. By doing so, one can leverage the insight that in many cases, distinct token sequences can have the same meaning. It turns out that combining these insights to estimate semantic entropy and then thresholding on value amounts to a highly competitive hallucination detection method Farquhar et al. 2024).
lack of
While semantic entropy is a state-of-the art method of hallucination detection, computing has high cost. It first requires the generation of several independent answers to the same question and then quadratic number of calls to the function determining if two meanings the same. In fact, the work of Farquhar ct al. (2024) used ten generations per prompt, which is prohibitively expensive in many practical cases. Wo address This bottleneck by making semantic entropy estimation much cheaper. We achieve this by leveraging
insights from Bayesian literature about entropy estimation (Wolpert Wolf [994 Hausser & Strimmer 2009 Archer et al. 20144), building up a probabilistic belief about the underlying distribution over meanings and reasoning about how the belief in the space of meaning distributions affects the belief in the space of possible values of the entropy. We further study a novel, adaptive, setting, where 'harder' prompts are

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afforded larger budget of samples. In this setting, the efficiency of our estimator be increased even further.
Contributions We develop
system for measuring semantic entropy, based Bayesian principles. Compared to the work of Farquhar al. (2024), outperform all other ways of measuring semantic entropy, reducing the sample complexity measured as the number of LLM generations needed to achieve the same performance by 47% on average across datasets. We also release several datasets for semantic entropy estimation, enabling researchers without access to GPU resources to work on even better estimators.
2 Preliminaries
Language Generation Denote with X the of prompt the LLM can be asked respond One instance of prompt X would be
x Where is the Biffel Tower?'
Denote with Sx the set of all reply sequences given prompt Denote the probability of the LLM generating sequence E Sx in response to the prompt E X with p(s)x). One possible continuation is
s = 'I's Paris.'
We model both and random variables, denoting them in bold.
Meanings Semantic entropy is always conditioned on a prompt. We are going to consider a given generic context Denote the set of meanings (also known meaning classes semantic classes) with Mx. The set Mx is partition of the set of S We assume that the set of meanings is finile (although we do not necessarily know its cardinality). Denote with
(x)s Mx
the function that determines the meaning of the sequence s in context x. While fx is typically implemented using calls to an ontailment oracle we abstract away this implementation detail in this paper. For a random sequence S ~ p(-lx), we consider the random variable
(s).
Semantic Entropy The semantic entropy corresponding the context is defined as the Shannon entropy of the random variable
SEx = [[]m
In the remainder of the paper, will occasionally drop the subscript where the dependence the context x is obvious. Semantic entropy is useful for detecting hallucinations because of the following crucial observation (Farquhar et al. 2024).
Higher values of SEx imply the model's response to prompt is more likely to be hallucination.
*Occasionally, the notion of 'context' is used in addition to the prompt, to model the phenomenon that the same prompt can have different meanings in different contexts. To keep notation simple, don't explicitly model contexts. However, if wants to generalize results contexts, it be done considering to be context-prompt tuple. 2We examples borrowed from Parquhar al. (2024).
TTpiccalyy, both X and
the of natural language sequences. However, whether X is immaterial for this paper. 4The term 'partition' used in the mathematical way that sequence always has exactly one meaning.
5te entailment. oracle function that tell if two meanings are distinct.
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The Estimation Problem The aim of this paper is to estimate semantic entropy from finite dataset, based on N calls to the target LLM. For given context x, our samples are represented as a list of indepen- dently generated sequences
$1. SN p(-1x).
For each sequence, we can determine its meaning, obtaining the corresponding list of meanings
m. mN, where m, = fs.)
We are also given the probabilities of generating ...... SN, which we denote with p(s.(x). Note that these probabilitics can be obtained at extra cost when generating sequences from the LLM. Our overall dataset is defined
D ($1, ((s.(()s))......... mn p(sw/x)).
The dataset can in general have repeated elements. It is important to note that we only have the probabilities for the sequences that were actually generated, which might represent very small fraction of all possible sequences. Morcover, an important feature of our problem is that we want to achieve reasonable estimates using as few samples as possible. When generating the dataset, we do have the ability to ask for more data, i.e. increase N until we are satisfied that our estimate of semantic entropy is good enough. We will make this precise in Section
3
A Bayesian Estimator for Semantic Entropy
We now give a sketch of our estimation process. Since we have finite data, our estimate of semantic entropy will be noisy. Adopting the Bayesian philosophy, we construct a random variable h that represents our belief about the value of the semantic entropy SFx, based limited available data. contained in dataset D (we define h formally later in the Section). Since we are motivated by detecting hallucinations, our focus is on measuring the quantities
E [b] and Var [h],
(1)
i.c. the mcan and variance of our Bayesian belief about what the value of the semantic entropy is.
In this Section, we describe a Bayesian process for forming probabilistic belief over h. For presentation purposes, we first derive our estimator under the assumption that that the number of meaning classes is known, i.e. [Mxl = K. Under this assumption, in Section 3.1 we describe the basic variant of the estimator, which only uses the list of meanings my...., my. In Section 3.2 we then extend the estimator to also make use of the probabilitios of the generated sequences ((s] ..,,)))))..(s)')))
p(sN)ssxx x). In Scction 3.3 we remove the assumption that K is known, defining a hierarchical Bayesian system that maintains belief about K. In Section 3.4
we summarize
methodology in the form of pseudo-code.
3.1 Basic Variant of the Estimator
We first summarize the dataset, counting how many times we sampled each meaning. The counter for meaning E M is defined
c- m = j---
(2)
We have c We seek use the information from the counters to get an idea about how the true distribution over meanings looks like. We adopt the Bayesian modeling philosophy, using a belief distribution. Specifically, Bayesian belief about the probability distribution over meanings is modeled as
Bp = Dirichlet(a + CI,..., CK),
(3)
where we used the letter B, to indicate that the probability distribution used as belief and K is the number of meanings. The value a represents prior of the Dirichlet distribution and is hyper-parameter
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of our method Equation 3 has Bayesian interpretation the posterior distribution, when the prior is chosen to be Dorichlet(o,, and the likelihood is categorical. Consider random variable distributed according to Bp:
b ~ Bp.
Here b A is itself a probability distribution, representing the fraction the total probability mass assigned to each meaning. A belief about b induces a belief about its entropy, represented with the random variable
h - H[].
The expectation as per equation 1 can be computed analytically as
H[bpp,, [1+Ka+ -
1 I++Ka= Ka Ee, E
a+c +1), Ka+ E -E
B h
[
Jb
where 2 is the digamma function (see Appendix A.1 for proof and derivation of similar expression for the variance integral).
3.2 An Estimator that Also Uses Sequence Probabilities
An LLM doesn't just give mlaning-classes but also probabilities of the generated continuations. Condi- tioning on this additional information can make our estimates of semantic entropy much better. Recall that the probability of generating is denoted with (x). We can define a constraint bounding the probability of each meaning, writing
constr(b, D)
by sE :ED::EED. [x
Σ
p(s(x)
j=1...KK
Intuitively, equation 4 holdd because the probability of meaning j is at least equal to the sum of probabilities of distinct sequences with that meaning. The bound is not equality because it is possible (and typically the case) that we didn't 't generate all sequences that correspond to this meaning. Probabilistically, the constraint can be interpreted as an event, i.e. something we can condition on. We in fact do that, modifying the estimator to sample from belief conditional on the constraint:
b Bp constr.
This conditioning is in fact key to obtaining good empirical results and is, far our approach. While the conditioning operation allows to leverage all information at our disposal, it also makes the process of computing the expectation in equation 1 more complicated. In practice, the integrals for E[h] and Var[h], which are defined as:
tell, novel to
/ H Hbbptt trunc (b; D)db,
E hi
Var h
H[|b trunc
PB
(b; D)dbb) [h)
will have to be computed approximately using a Monte Carlo method. Here, the density of a truncated Dirichlet random variable is defined as
(b)
if b E constr(b, D),
trunc PB
(b:D)
Incomartb Pe(b)dd pa(b)db
(5)
if b & constr(b, D),
We study the sensitivity of method the choice of Appendix

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where we used the notation (b(b) to denote the Dirichlet PDF. We treat particular choice of the integration algorithm as an implementation detail and defer its discussion to Appendix A.2 Note that, even though we are using a Monte Carlo method, obtaining estimates of semantic entropy is is still relatively cheap. This is because the integration routine is orders of magnitude cheaper than increasing N. In other words, sampling meanings from an LLM is expensive while MC integration has negligible cost.
3.3 Unknown Number of Meanings
Previously, we assumed that we know the number of meanings possible for a given context x, i.e. [Ma = K for a known value of K that can be used to design the estimator. This is not a realistic assumption since the number of possible meanings can be vastly different for each context and is not typically known a priori. We resolve this dilemma in a Bayesian way, representing our Bayesian belief about the number of meanings using a probability distribution.
B& = Discrete((KI, (KM, Am))
(6)
Here, the parameters , AM are relative frequencies of each support size K, The parameters can be computed using a small (separate) training dataset and M is the maximum observed support size.
Our belief about the number of meanings can be used to estimate the entropy in hierarchical way. Specif- ically, given we have observed that there are at least Kmin different meanings, we obtain the probability distribution
K ~ Bx|
| (K > Kmin).
Here, we used bold font for K to denote it is a random variable. We can use probabilities of this discrete distribution to take an expectation of entropy estimates conditioned on particular values of K:
Bh = EK E [h[]],
Var [h] Ex [Var [h] + Vark [ [b[K],
(7)
where the quantities E [b| and Var [b[] can be obtained as described in Section 3.2 In our pseudo-code (see the next Section), equation
is assumed to be implemented using a procedure ACCREGATESUPPORT:
3.4 Algorithm
Stopping Rule Having specified the estimator for the quantities E[h] and Varhh], we still need to specify the stopping rule for determining the right number of samples N. Tt is natural to keep drawing more samples until
Var[h|2 2
(8)
where 7 is desired level of precision. Increasing y indicates we are satisfied with lower-quality semantic entropy estimates, which allows us to use smaller N. Intuitively, this stopping rule reflects the fact that we only want to know the semantic entropy to a certain level of procision. We summarize the idcas introducod in Sections
3.2 and 3.3 in Algorithm T
Benefits of the Bayesian Approach We will see in Section D that our Bayesian estimator works better than other semantic entropy estimators in practice. We now discuss the three theoretical reasons why this is the case. First, our estimator leverages small training set to learn the prior, as in equation 6 Other estimators don't do this because, as they are not Bayesian, they do not have the concept of the prior Second, the Bayesian estimator combines all evidence (the probabilities of generated sequences and the meanings) in an optimal way, based on the Bayes rule. Other estimators don't do this. Third, our methodology gives us a handle on [ar[h] (the variance in the belief about what the true semantic entropy is) allowing us to further improve sample efficiency by using stopping rule in equation S
7 Note that the size of the training does scale with the numher queries to the estimator inference time, hence the cost. becomes entirely amortized.
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Algorithm Estimate of Semantic Entropy for a prompt
repeat
f(p()))) LLMSAMPLE()
Sample and store the corresponding probability. Determine the meaning.
= (s)
5: D.append (s, m. p(s(x)
6: Kmin < [m E ED]
7: for (1,..... Kmin] do
8: Cj cii m, = J, m, E D)
P. Use equation
end for
10:
for
K E Support (Bx/ (K> kmin)) do
11:
PB trunc (b; is defined per equation
A truncated Dirichlet PDF.
12:
εK- + NUMEBRICALLL
H bb; D; )bb)
13:
PB (b; D; K)bb) e²x K + NUMERCALLAYYVVTOGRAAEE H H
trune
00TK = 2² (2)
14:
15: end for
16: 2, var
ACGREGATESSUPPORTT. vark)
Apply equation for k E (Kminy .... K Kmax).
17: until var 27
18: return
4 Baselines
4.1
Other Estimators For Semantic Entropy
There existing baselines for measuring semantic entropy, both of which coming from the paper by Farquhar et al. (2024). They are called discrete semantic entropy and semantic entropy in that paper, although we use the term histogram semantic entropy for the former and rescaled semantic entropy for the latter to avoid confusion with the concept semantic entropy itself, which is independent of the estimator used.
Histogram Semantic Entropy This estimator samples fixed number of sequences, computes the mean- ing of each sequence and then computes the entropy of the empirical histogram of the meaning distribution. Tt is computed from the meaning counts cj as Σ, N log S
Rescaled Semantic Entropy This estimator also samples a fixed number of sequences and then computes the meaning each. However, it assigns probabilities to each meaning in a different way. First, one defines the un-normalized probability distribution
((m[x)
Σ
p(s(x).
sE(s sED, f*(s)= m)
This is then normalized
q(m x)
p(m|x)
Σm (mm x
and the semantic entropy is computed as the Shannon entropy of this distribution. In addition, the probabil- ities p(s[x), which normally correspond to the multiplication of the probabilities of each token conditioned on past tokens, are heuristically replaced by the exponent of the mean log probability of cach token, a process known length normalization. We report results for the rescaled estimator both with and without the (heuristic) length normalization.
4.2 Other Baselines for Hallucination Detection
Semantic Entropy is not the only way of detecting hallucinations. We consider two non-entropy baselines.

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P(True) It has been shown that LLMs have surprising introspective abilities, i.e. one can often see if the LLM is hallucinating simply by simply asking it (Kadavath et al. 2022). We use the same procedure for measuring P(True) as was employed by Farquhar et al. (2024).
Sequence Log Likelihood Recently, Aichberger et al. (024) suggested using the log probability of the sequence generated greedily as a predictor of hallucinations, justifying using the notion of zero-one scoring rule (Hofman al. 2024). Unfortunately, this method is not directly compatible with the evaluation protocol of Farquhar et al. 20024) which does not perform greedy generation. In order to stay within the boundaries of that protocol, we instead used the log likelihood of a sequence generated with a low temperature g
5 Prior Work
Hallucination Detection We do not provide a complete survey of hallucinations in Large Language Models, instead referring the reader to the work of Ji et al. (2023). We focus our work on combating 'confabulatory' hallucinations also addressed by Farquhar et (2024). i.c. situations where the LLM is randomly adding spurious facts to its replies. This is the same kind of hallucinations that was considered
by FFllippova 2020 and Maynez ct al. (2020
Semantic Entropy We build on the works that pioneered semantic entropy (Kuhn et al. 2023 Farquhar ct al. 2024) by providing a more statistically efficient estimator. Our work is different from entropy probes [ossen et al. 2024), which attempt to distill a thresholded version of semantic entropy into a classifier, although the ideas can certainly be used in conjunction with each other (similar ideas were also explored by Chen al. (2024)) Our estimators do not attempt to leverage similarity in the meaning clusters (Nikitin ct al. 2024 Qiu & Milkkulaneen 2024). instead focusing on obtaining accurate estimates of vanilla semanttcc entropy for given sample budget as possible.
Bayesian Entropy Estimation Wolpert & Wolf (1994) have provided the foundations for Bayesian estimators of entropy for arbitrary priors, and also pioneered the specialization to the case of the Dirichlet prior. Hausser & Strimmer (2009) have provided an explicit summary of the equivalences between the Dirichlet-Bayesine estimator and various pre-existing entropy estimators, for different values of the Dirichlet prior parameter. [Archer et al. (2014) have provided overview of past work Bayesian entropy estimation, in addition to extending the framework to distributions with countably infinite support
Epistemic Uncertainty in LLMs The distinction between epistemic and aleatoric uncertainty (Gal et al.) 2016 2017; Kendall & Gal, 2017) has been proposed as a useful idea in modeling the behavior of LLMs (Abbasi Yadkori et al.) 2024). In this paper, we do not distinguish between alcatoric and epistemic uncertainty, instead staying in the framework Farrurr et al. (2024) and modeling the combined predictive uncertainty. While an accurate model of epistemic uncertainty would almost certainly lead to improved hallucination detection, we leave such extensions to further work.
Human Perception of Hallucinations Hallucinations are related to how confident LLMs are about their outputs. Recent research (Steyvers et al.) 2025) studies how such self-confidence intrinsic in LLMs relates to how humans perceive it. Our work is largely orthogonal this effort. Indood, we treat the definition of semantic entropy as a given and focus on finding the statistically most efficient way to estimate it.
6 Experiments
6.1 Experimental Setup
Evaluation Methodology Our goal is to measure the quality of semantic entropy estimates quantified with AUROC on hallucination detection tasks. To do so, we follow the methodology from the paper by
8The temperature set 0.1.
We da not use their infinite support framework, instead modeling unknown support using techniques described in Section
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N = 2
LLM
Datasel
LL
P(true)
SE- Bay-Bayes
SE-Histogram
SE-Rescaled
SE-Rescaled (h)
NQ
0.583 + 0.000
0.461 + 0.000
0.723 0.007
0.652 + 0.010
0.654 + 0.013
0.644 + 0.014
SVAMP
0.631 + 0.000
0.469 + 0.000
0.855
0.022
0.748 0.024
0.749 + 0.027
0.759 + 0.025
Llama-2
Squad
0.624 0.000
0.441 0.000
0.735
0.654 0.007
0.659 = 0.017
0.649 0.010
Trivia QA
0.594 + 0.000
0.436 +: 0.000
0.737 = 0.006
0.670 0.012
0.675 + 0.012
0.673 = 0.010
NQ
0.627 = 0.000
0.615 0.000
0.728 0.008
0.640
0.663
0.011
0.649 0.020
SVAMP
0.647 + 0.000
0.414 0.000
0.845 + 0.022 0.761
0.774 0.082
0.771 = 0.028
Llama-3.2
Squad
0.610 + 0.000
0.555 1 0.000
0.664 0.019
0.024
0.642
0.029
0.621 0.027
Trivia QA
0.614 +: 0,000
0.710 + 0.000
0.768 +: 0.004 0.699
0.706 + 0.007
0.708 + 0.008
Llama-3.3-708 Trivia QA
0.556 + 0.000
0.686 + 0.000
0.775 - 0.003
0.653
0.650 + 0.006
0.651 = 0.006
NQ
0.695 + 0.000
0.731 0.000
0.705
0.013
0.647
0.700 0.007
0.654 0.009
SVAMP
0.645 0.000
0.843 + 0.000
0.876 = 0.011
0.793 +
0.815 0.028
0.812 + 0.024
Mistral
Squad
0.698
0.000
0.687
+ 0.000
0.667
0.010
0.618
0.005
0.671 0.017
0.631 0.010
Trivia QA
0.672 1 0.000
0.647 1 0.000
0.682
0.008
0.638 1 0.011
0.645 1 0.013
0.643 1 0.012
N = 5
LLM
Dataset
LL
P(true)
SE- SE-Bayes
SE-Histogram
SE-Rescaled
SE-Rescaled (h)
NQ
0.583 = 0.000 0.461 + 0.000 0.752 + 0.006
0.730 + 0.009
0.701 + 0.012
0.725 + 0.010
SVAMP
0.631 + 0.000 0.469 0.000
0.871 + 0.011
0.849 + 0.012
0.853 0.013
0.858 + 0.013
Llama-2
Squad
0.624 = 0.000
0.441 0.000
0.756 0
0.004
0.711 = 0.012
0.752 = 0.005
Trivia QA
0.594 = 0.000 0.436 = 0.000 0.763 0.009
0.734 0.007
0.737 = 0.006
0.735 + 0.006
NQ
0.627 = 0.000 0.615 0.000 0.760
0.732 + 0.012
0.691 0.005
0.733 + 0.012
SVAMP
0.647 = 0.000 0.000
0.860 + 0.010
0.850 + 0.004
0.864 + 0.009
Llama-3.2
Squad
0.610 + 0.000
1 0.000
0.707 1 0.012
0.667 0.021
0.705 1 0.013
Trivia QA
0.614 +: 0.000 0.710
0.004
0.775
0.763 1 0.005
0.777 + 0.004
Llama-3.3-70B Trivia QA
0.556 = 0.000 0.686 0.000 0.79 1.000
0.750
0.740 + 0.004
0.747 1 0.006
NQ
0.695 = 0.000 0.731 0.000 0.78
0.762 0.007
0.728 0.008
0.756 + 0.007
SVAMP
0.645 = 0.000 0.843 = 0.88 0.019
0. 0.866
0.855 + 0.027
0.878 + 0.022
Mistral
Squad
0.698 = 0.000 0.687
0.000 0.731 #
0.719
0.699 0.008
0.712 = 0.008
Trivia QA 0.672 + 0.000 0.647 1 0.000 0.691 0.005 0.688 1 0.005 0.684 1 0.004
0.690 1 0.005
Table 1: Measured AUROC for a fixed budget of N 2 and N 5 samples per prompt.
Farquhar et al. (2024) as much as possible, deviating from only by (1) separating out the dataset generation
phase and the entropy estimation phase, (2) varying the sample budget N and (3) removing bugs from the dataset generation code. We defer the detailed discussion of the methodology to Appendix B
LLMs and Source Datasets We investigate the behavior of four LLMs. We use Llama-2-7b-char for comparability with Farquhar et al. (2024). We also evaluate on the much more modern Llama-3. 3.2--BB and on -tttaaa-------------tt---------
Instruct, the large 3.mma--.3-70B--nsstte to as Llama 2, Llama 3, Llama-3.3-70 and Mistral in our figures. We used the TriviaQA (Joshi et al., 2017).
These LLMs are referred
SQUAD (Rajuurkar et al.) 2016), SVAMP (Patel et al. 2021) and NQ (Lee et al. 2019) datasets. Due to computational constraints, we only ran the largest LLM on one dataset (TriviaQA).
Derivative Entropy Estimation Datasets For each combination of LLM and dataset, we generated derivative dataset of 100 LLM generations per prompt for 1000 prompts, which we then used to estimate semantic entropy. We will release these derivative datasets upon acceptance allowing researches without GPU access to work on even better estimators for semantic entropy.
Train and Test Our Bayesian Semantic Entropy estimator requires a training set to estimate the prior on the size of the support of the meaning distribution as in equation 6 We use the first 200 prompts from each derivative dataset the training set and the remaining 800 as the test set.
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Temperature Following the methodology of Farquhar et al. 2024). the N LLM responses generated with temperature 1.0. On the other hand, the LLM response about which we seek to determine if it is a hallucination is generated with temperature 0.1. Because GPU response generation is expensive, we did not tune those temperatures.
6.2 Results
We performed two types of experiment. First, we studied the setting of fixed budget of samples per prompt. Second, we varied the number of samples per prompt, giving harder prompts more data. In both cases, we note that we can support = I because we have access to the probability the generated sequence, which already gives us an imperfect but still useful handle on the entropy (for example, if the probability is close one, we know that the entropy is almost zero).
Fixed Budget Per Prompt Results for N 2 and N = 5 samples per promp D Bold font is applied as follows: the estimator with the best mean performance is put in bold, logether with all the others with overlapping confidence bars. It can be seen that the Bayesian estimator mostly outperformed or tied with other approaches to measuring semantic entropy, with the difference being greater for small N. We can also see that it is difficult to conclude which version of the rescaled estimator better.
To are shown in Table
Main Experiment: Adaptive Budget Per Prompt As described in Section 3.4 the Bayesian frame- work gives us an additional handle on sample complexity in that we can use the variance of the belief about the semantic entropy as proxy for confidence. Results bars for the AUROC estimates in our paper represent 1.96 times the standard error. It can be seen that our Bayesian estimator is nearly Pareto-optimal in the sense that achieve belter AUROC than other ap- proaches to semantic entropy, regardless of the value of N. Note that performance of the adaptive Bayesian estimator for a given N will in general be better than performance for the same fixed value of N. This is because, while the number of prompts still N on average, harder prompts will get more samples (and easier prompts will get fewer). Concerning the non-semantic-entrop baselines, we outperform them for all for Llama and 3, while needing N 2 3 for Mistral. We stress one additional take-away from the experiment: our Bayesian estimator is often competitive even for N = 1. This is completely counterintuitive since the entailment oracle (a crucial component of semantic entropy) is not needed in that case.
shown in Figures
2 and All confidence
Note on Consistency By definition, for large enough sample budget, any consistent estimator will produce the same correct value of semantic entropy, which will give rise to the same value of AUROC. There are two reasons why such convergence does not happen in practice in our experiments. First, not all compared estimators are consistent (for example the rescaled estimator with heuristic sequence probabilities is not). Second, convergence to the exact same value might require a sample budged far in excess of 10.
Alama--33-700. Trivis QA
0.75
AUROC
0.70
0.65
0.60
0.35
0.50
5
Bayesian SE.
Histogram
Rescaled (heuristic)
Rescaled
Log Likelihood
P(True)
Figure 1: Results in the adaptive budget setting (Llama 3.3, 70 billion parameters).
10see Appendix C for results for other values N.
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Llama-2, Trivia QA
Llama-2, NQ
0.75
0.70
AUROC
AUROC
0.65
0.6
0.60
0.55
0.5
0.50
0.45
10
N
Llama-2, SVAMP
Llama-22 Squad
0.8
0.8
11
0.7
AUROC
AUROC
0.7
0.8
0.5
2
3
9
10
Bayesian SF
Histogram
Rescaled (heuristic)
J-en
Rescaled
Log Likelihood
P(True)
Figure 2: Results in the adaptive budget setting (Llama 2).
Llama-34 3.2, Trivia QA
Ilama-33 3.2, NQ
0.75
0.75
0.70
AUROC
0.70
AUHOC
0.65
0.65
0.60
0.60
0.55
0.55
2
6
9
10
7
10
Llama-3 3.2, SVAMP
Llama-3.2 Squad
0.70
AUROO
AUROC
0.65
0.55
0.50
N
Bayesian
Histogram
Rescaled (heuristic)
Rescaled
Log Likelihood
P(Truc)
Figure 3: Results the adaptive budget sctting (Llama 3).
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Mistral, Trivis QA
Mistral, NQ
0.80
0.70
0.75
0.63
0.70
AUROC
AUROC
0.65
0.60
0.60
0.55
0.55
10
Mistral, SVAMP
Mistral, Squad
0.75
0..8
AUROC
AUROC
0.65
0.7
0.60
0.55
0.5
0.50
3
10
Bayesian SE
Histogram
b-+ Roscaled (heuristic)
1-+
Roscaled
Log Likelihood
P(True)
Figure 4: Results in the adaptive budget setting (Mistral).
7 Conclusions, Limitations and Perspectives
We have described a new Bayesian estimator for measuring semantic entropy. The proposed estimator has systematically outperformed other semantic entropy baselines several practical settings. Like any Bayesian method, our approach is constrained by the choice the prior, where our choice can be viewed as limited expressivity. This can be mitigated by using more complex hierarchical priors, for example along the lines proposed by Nemenman al. (2001). We leave this further work.
References
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Lukas Aichberger, Kajetan Schweighofer, and Sepp Hochreiter. Rethinking uncertainty estimation in natural language generation. arXiv preprint arXiv:2 12. 15176, 2024.
Evan Archer, Il Memming Park, and Jonathan W Pillow. Bayesian entropy estimation for countable discrete distributions. The Journal of Machine Learuing Research, 15(1) 2833 2868, 2014.
Narayanaswamy Balakrishnan and Valery B Nevzorov. A primer statistical distributions. John Wiley & Sons, 2004.
Chao Chen, Kai Liu, Ze Chen, Yi Gu, Yue Wu, Mingyuan Tao, Zhihang Fu, and Jieping Ye. INSIDE: LLMs' internal states retain the power of hallucination detection. In The Twelfth International Conference on Learning Representations, 2024. URL httts:://peereeeies 2ttp:///er...eeeeeooroaaaaa22220
William G. Cochran. Sampling techniques. Proceedings of the Edinburgh Mothematical Society, 13(4):342- 343, 1963.
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Sebastian Farquhar, Jannik Kossen, Lorenz Kuhn, and Yarin Gal. Detecting hallucinations in large language models using semantic entropy. Nature, 630(8017):625--6)) 630, 2024.
Katja Filippova. Controlled hallucinations: Learning to generate laithfully from noisy data. ar preprinl wwXiv:2010.05870 2020.
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Yarin Gal ct al. Uncertainty deep learning. PhD thesis, University of Cambridge, 2016.
Jean Hausser and Korbinian Strimmer. Entropy inference and the james-stein estimator, with application to nonlinear gene association networks. Journal of Machine Learning Research, 10(7), 2009.
Till Hoffmann. Moments of the dirichlet distribution. http:///ww//p/////o/////////////
https://tillahofffaaet htt.:ffaannnggttt...e. 2025-04-28.
2015.
Accessed:
Paul Hofman, Yusuf Sale, and Eyke Hüllermeier. Quantifying aleatoric and epistemic uncertainty with proper scoring rules. arxiv preprint ar arxix:2444.1 12215, 2024.
Ziwei Ji, Naycon Lee, Rita Frieske, Tiezheng Yu, Dan Su, Yan Xu, Etsuko Ishii, Ye Jin Bang, Andrea Madotto, and Pascale Fung. Survey of hallucination in natural language generation. ACM Compul. Surv., 55(12), March 2023. ISSN 0360-0300. doi: 10. 10.1145/3571730. URL https: http:::///oi.rg/1..1111111 1t1///0.114//367733
Mandar Joshi, Eunsol Choi, Daniel Weld, and Luke Zettlemoyer. triviaqa: A Large Scale Distantly Super- vised Challenge Dataset for Reading Comprehension. arXiv e-prints, art arXiv:1705.0355) 2017.
Saurav Kadavath, Tom Conerly, Amanda Askell, Tom Henighan, Dawn Drain, Ethan Perez, Nicholas Schiefer, Zac Hatfield-Dodds, Nova DasSarma, Eli Tran-Johnson, et al. Language models (mostly) know what they know. arXiv preprint arXiv:2207.0222, 2022.
Alex Kendall and Yarin Gal. What uncertainties do need in bayesian deep learning for computer vision? Advances in neural information processing systems, 30, 2017.
Jannik Kossen, Jiatong Han, Muhammed Razzak, Lisa. Schut, Shreshth Malik, and Yarin Gal. Semantic entropy probes: Robust and cheap hallucination detection in llms. arXiv preprint arXiv:2406.154 15927, 2024.
Lorenz Kuhn, Yarin Gal, and Sebastian Farquhar. Semantic uncertainty: Linguistic invariances for uncer- tainty estimation in natural language generation. ar preprint arXiv:2302.0900 09664, 2023.
Kenton Lee, Ming- Wei Chang, and Kristina Toutanova. Latent retrieval for weakly supervised open domain question answering, arXiv preprint XrXix:1966. 003000, 2019.
Joshua Maynez, Shashi Narayan, Bernd Bohnet, and Ryan McDonald. On faithfulness and factuality in abstractive summarization. arXie preprint arXiv:2005. 00666,, 2020.
Tya Nemenman, Fariel Shafee, and William Bialek. Entropy and inference, revisited. Advances in neural
information processing systems, 14, 2001.
Alexander Nikitin, Jannik Kossen, Yarin Gal, and Pekka Marttinen. Kernel language entropy: Fino-grained uncertainty quantification for Ilms from semantic similarities. XrXiv preprint arXiv:245..2000, 2024.
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Arkil Patel, Satwik Bhattamishra, and Navin Goyal. Are nlp models really able to solve simple math word problems? arXiv preprinl XrXi::2103.0770 071911 2021.
Xin Qiu and Risto Mikkullanen. Semantic density: Uncertainty quantification for large language models
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Pranav Rajpurkar, Jian Zhang, Konstantin Lopyrev, and Percy Liang. SQuAD: 100,000+ questions for machine comprehension of text. In Jian Su, Kevin Duh, and Xavier Carreras (eds.), Proceedings of the 2016 Conference on Empirical Methods in Natural Language Processing, pp. 2383 2392, Austin, Texas, November 2016. Association for Computational Linguistics. doi: 10.18653v11/D1-1122 URL https: Saiaaantooooog.oo//D-------
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A Integral Computation
A.1 Mean and Variance of Expected Entropy under the Dirichlet Distribution
Here, we derive the analytical expressions for the mean E[h] and the second moment E(h²] of the entropy
K
h A H = Σ
by log b;.
where the probability vector b (b1...., bx) follows Dirichlet distribution b ~ Dir(a) with parameters K Or (015...,0g)) Let oo Di= 04. The results involving digamma (1)) and trigamma (1)) functions
are based on standard properties of the Dirichlet distribution (Wolpert & Wolf 1994 Hausser & Strimmer 2009).
Mean Entropy [[h] Using the linearity of expectation and the known expectation E(b; log by] for a Dirich- let distribution:
K by log bi = Σ
E[h] E
Ebb, log b;]
i=1
(vl(a,
1) s100 +1)
K
ai
= + + 1)
waa +1)
1=1
K
= <0000 + 1)
(a:
+
ax
This formula corresponds to the one used to compute E[h] in Section 3.1 (after substituting the appropriate parameters + c,).
Second Moment E[h]]
We start by expanding the square of the entropy:
K
2
h2
by log bi
(b.by log by log
Applying the expectation:
K
K
E
(b,b, log by log
K
K
Elb,b, log by log
E (log by 1
Elb,b, log log by]
The calculation requires the expectations E[b (log ) and E[b,b, log by log byl for # j. Before the derivation of these quantities, let derive some useful lemmas. Lemma A.1. Let b ~ Dir(a), where a = (aj...., ag). Let i (I....,K). Then:
a Jas
log b(bp() = v(ao) - v(as)) + logbs.
where vbx)) is the digamma function.
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Proof. First, let us re-write this quantity:
a
8
log lbba((aa))
log (an) - log [(a)) + E (ax
1) log bn
aa,
da,
k=1
Using the chain rule:
dlog [ dao va(o) - = ((0)).
log [(ao)
doo
Therefore:
log log((bba)) vb(00) d dog (as))
+ log
da;
do;
= <v(0)) - -v(a)) + +logb;
Lemma A.2. Let b ~ Dir(a), where a = (ai..... ax). Let i {I....,KK K}. Then:
E[[log b,) (vh(aa)) vi(())) + (v(a.) vaao))
where b(x) is the digammma function.
Proof. It is known that:
Ellog bi] = <b(a)) ((o))-
It follows that:
I ((bla) (v(o) - vaa))) log b,] db vb(au) (aas)) + Elllo by]
If we apply derivative to this quantity, it must be O, since it is a constant:
Do. f p(bla) (b((o)) - vaas + log b.] db = 0
a
For this integral, we apply the Leibniz integral rule for differentiation under the integral sign. Applying the product rule for differentiation under the integral sign get:
(Dpibla)) (vb(w)) - v(ax) + log b.] db + / p(bla) das
(v(ao) - +(a;) + log db = 0
Da,
By using Lemma A.1 and the log-derivative trick, we can substitute
Op Das
p das
Ologe = ((a)
f ((bla) (W(ov) vbaa)) + log 62)² db + / p(bla)) (ao) Ur(0)) db = 0.
where - (an)) sh (ao) dan th (ao), oa (a,) = 1h (ax), and Da, log b, 0.
We can further simplify this expression by using the expected value definition:
((b(ao)) ((ss)) + log b,) th (av) - t/ (a:) 0
(1)
Let C, ((())) vaa.)) EEooo b,]. Term (1) becomes:
[E[( + bgg E(C + 2C, log by 2
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C 20, b,] [[og b,)
Substitute Ellog b:] = -C;
E[oog b,) C² + 22:(2((C0 +
-C Blogg
Substituting this back into the equation derived from differentiation:
(-0f + E(logbb b,) b)) + (a0) (11(4))
[looo, 6, C² - vi((o) + 4/ (a;).
Substitule C = (b(an) - (aa)):
[ b, 6 (v(an) - -baa))) (1 2 (ao).
Rearranging gives the desired result:
E (log b,) (wi((a) (i + (+)()) 2ba0)))²
Lemma A.3. Let b Dir(a:), where (a1,...,, og) Let i,j (I,..., K} and # Then:
Blog b, log by] -1/1 (0o) (v)():) - ((o)) (b(az) - ((o)))
where bbx) is the digamama function and thi (I) dz (x)) is the trigammaa function.
d
Proof. From Lemma A.1 know that, for given E (I,..., K}, the following quantity is 0:
f p(bbaa)) (b(())) (aa) + log bi] db 0.
Now, we differentiate this quantity with respect to aj, where j #
/ p(bia) (b(ao)) - vaas)) log by] db = 0. Dag
a
For this integral, we can apply the Leibniz integral rule for differentiation under the integral sign. Applying the product rule for differentiation under the integral sign we get:
( (omiblaa) (eb(aw) - v(as)) + log be] db + / p(bla) a day [b(av) v(as) + b;g db
da,
Due to Lemma A.1 and the log-derivative trick, we can substitute
Op Blogg (#)) +oog by): do;
Bas
f ((bla) bla,) + log (va(a)) - vb(ss) + l;g db / ((bla) db = 0,
where a dny ((ao)) Un (ar),
a Ba, ((a)) 0 since j + i, and a Baz log by 0.
We can re-write the previous quantity by using the expected value definition:
B((((aa)) (aa) log (V)()) vb(a,) log b:)] th (ax) = 0
(1)
(2)
Let C, ((0) ((as)) EEloo bi] and C2 (ao) v(az)) -Ellog by]. Term (1) becomes:
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B[(C, + (C logb;)) = BiC,C, + C, logg + C3 log log b
C,C, + C,Ellog b,] Cl b.] Ellog b, log by]
CC Cj + Ellog by log by]
-CC, - + Ellog b, log by] = - C.C, + Ellog by log by].
Substituting this back into the equation derived from differentiation:
(-C.C, + Elllog b, log by]) + 1/1 (0x) =0
Ellog b, log by} = - -1
Substitute C: = vb(())) (a;) and Cj- v((0)) v(az):
Bloog by log by] (w(as) +)(a)) ((a))) 4// (ao)-
Rearranging gives the desired result:
Ellog b, log (10( + (1((a)) - (H( vb(ao) for i * j.
Lemma A.4. For b ~ Dir(x),
B b.. f(b]] Elb, ... ...l. [ell''
(1,..., . K} are n indices, b' ~ Dir(a Ex= es,), and ex is the k-th standard basis vector. where in in
in E
Proof. First, let us re-write the term by bj on the counts o₁:
b b
bi
b
j=l
k=1
where is the count of the indices equals to k: cs = Ei= 1(i, = k). Now, we can substitute this into the expected value:
K II k
K
Eib,, f(b)] = B(a) / f(b)
b
[I kmi
b
804-1
db
f(b)
+
-1 db
B(x)) SK
Define af = as + c and a' = (of...., ox). Let us rewrite the expression by multiplying and dividing by the normalization constant B(a') for the Dir(a') distribution:
Eb,, ...,,.. = B(a') [ ((b)
K
0 db
B(a) SK
B(a')
1=1
B(a')
B(a) SK [(b)p(bla")db,
where p(bla') is the PDF of the Dir(a') distribution.
By substituting the known fact that B(a')
B(a]
Eb,, ... b... (Balakrishnan Nevzorov
2004 Hoffmann
2015) we get the desired result.
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Now, let us apply Lemma A.4 for the i term:
B (logb,) = TBbb). E E"[(lob"]] b?)
where b" Dir(x 2e.). Now apply Lemma A.2
E" [log b (vi(a, 2) Via + +2)) + (vb(a; + + 44a0 2))²
Since bb
a:(a++1) (o we get:
b (logb,)
an(ag + 1) { (1ilaa + + (0 (ap - 2)) + (v(a; + - 44(00 + +)))²}
ai(a; +1)
Applying Lemma A.4 for the + j term:
Blb:bj log by log by] B[b,by] E"[lo Uf log bi]
where b" ~ Dir(( e; + cz). Now, apply Lemma A.3
E" b" log 6/ (ao + 2) + (vla, + 1) s 2)) (v(aa + 1) 10a0 + 2))
avaj Since E[b,by] oo(ao+l) for i # j, we get:
Eib,b, log b, logb,] an (a +
a,a
(-1h (00 + 2) (v(a; + s(as 2)) (v(ag + + 10a0 + +2))}
Combining these terms yields the final expression for E[h²]:
K
Ehh = aolas + 1)
Σ
02 + 1) (wi(a; + - 10 + 2) + (1b(a, + - vsa0 + 2))
Σ a,a (an + + (wias + 1) - s(an + 2))((( + 1) - v6a + 2))}
Variance [ar[h] The variance of the entropy under the Dirichlet distribution can then be computed using the standard formula:
Var [h] E - K[h]²
using the analytical expressions for the first and sccond moments derived above.
A.2 Mean and Variance of Expected Entropy under the Truncated Dirichlet Distribution
Recall from Section
3.2 that define the truncated Dirichlet distribution
PB (bbb) ((D)
if b E constr(b, D),
PB
trunc (b;D)
(9)
otherwise,
where pa(b) the (untruncated) Dirichlet density with parameters c, and
Z(D) 1
(B db constr(b,
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is the (unknown) normalizing constant. Our goal is to compute the expected value and the variance of the entropy. To this end, we want. to compute expectations of the form
PB trunc E [[ = /HIbJ
H[]
PB trune (b;
pb (b; D) db.
In the following, we focus only on the expected value of the entropy (the integral on the left), since the other integral will follow an analogous reasoning.
D). Therefore, we cannot ap- proximate the integral via simple Monte-Carlo approach. naive solution would be standard importance sampling (IS) (Cochran 1963; Owen 2013). This approach consists of selecting a proposal distribution q(b) with full support over the sample space, from which we know how to sample. Then, we apply some algebra to the original integral as follows:
prunn The first problem we face that we cannot directly sample from pB (b;
[[H[b]]
|H[bb T (b:
(): db =
[H]b]
PB
trunc (b; D)
(b)) db Bbby
pb (b; D) ((b)
[[[]]
((b)
As proposal distribution, could simply pick the uniform distribution over the truncated simplex, trun- cated according to constr(b, D).
Now, we transformed the original expectation into an expectation with respect to a distribution from which we know how to sample. Therefore, we can apply Monte-Carle to get an unbiased estimator. First, we sample m samples from y. Then, we approximate the integral as follows:
i
trune (b,; D) 1 m
PB
(b.,) H(b.).
(10)
However, this estimator still requires knowledge of Z(D) to compute pl D) for the importance weights, which we do not know in general. For this reason, in this paper we will use the self-mormaiiiation technique.
Self-normalized importance sampling. We can circumvent the explicit computation of Z(D) by using self-normalized importance sampling (Owen 2013 Swaminathan & Joachims) 2015). Self-normalized IS is similar to standard 1S, but instead of dividing the sum by m, use the sum of the importance weights:
Pa trune (b;
((b.)
Since the expected value of this quantity is m, this results in a biased--uu--coniittent estimator of the integral at hand (Swaminathan & Joachims 2015 and in practice it has been found out to often provide better estimation than simple IS.
After some algebraic simplifications, the integral approximation computed as follows:
AsN = Σ Pa b,)
pa DB(b))H(bb
(11)
This simplified version follows from the following facts:
(bb
we always sample from the truncated simplex, hence p trune (b; D) Z(D)
the normalizing constant cancels out;
the proposal distribution constant and cancels out.
Now, this be computed because sample from the truncated simplex and evaluate Dirichlet PDFs.
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B Detailed Description of the Evaluation Methodology
We took the evaluation methodology of Farquhar et al. (2024) as a starting point and only modified it in ways which were necessary to adapt to our-use case. We describe the evaluation methodology in full this section.
Sto-Stage Architecture The computation stage that does inference in the LLM (which takes over a week on a single A100 80GB) is separated from the stage that estimates semantic entropy (which only uses the CPU, taking on the order of 12 minutes). This is important so we don't have to repeat expensive GPU inference many times when changing details of the entropy estimation. Our code for the LLM inference stage is based on the code by Farquhar et al. (2024), while the code for the second stage new We used the following quantization settings: 8 bit for Llama-333-70B, 16 bit for Mistral, 32 bit for Llama-3.2 and Llama-2.
LLM Inference Stage Conservatively, we performed LLM inference for N = 100 times for each prompt. We use the temperature 1 for all generations. We also generate one extra answer with temperature 0.1, about which we aim to decide whether or not is a hallucination. The source code for the LLM inference stage is based on the code by Farquhar et al. (2024). We found two bugs in the code, which we fixed. The first bug meant that the exact same LLM response (identical string) could be (rarely) assigned to different meaning classes, due to the imperfections of the DeBERTa entailment oracle. The second bug caused sum of probabilitics of generated sequences to occasionally exceed one (it was caused by not storing the probabilities of special lokens ending the LLM response).
Entropy Estimation Stage We implemented the entropy estimation stage as a single Jupyter notebook. In the second stage, when we require a dataset for a smaller value of N, we subsample (with replacement). The notebook can be used to regenerate all the tables and figures in the paper.
Variable Budget Our biggest deviation from the methodolggy of Farquhar et al. 2024) is that we consider
different choices of (the number of samples emitted by the LLM), where Farquhar et al. (2024) only considered the case of N = 10.
Label Computation Even with supervised dataset, determining if model hallucinates is not trivial because the LLM can phase the response in an arbitrary way. Following the work of Farquhar et al. (2024) the label determining if model hallueinates, used for AUROC computation, was obtained by computing the F1 metric and thresholding it at 0.5.
Confidence Bars All confidence bars for the AUROC estimates in our paper represent 1.96 times the standard error. Confidence bars are generated by resampling (with replacement) the dataset for given value of N. In our tables, the bold font is applied as follows: the estimator with the best mean performance is put in bold, together with all the others with overlapping confidence bars.
C Additional Experimental Results
Below, we provide full experimental results (measured AUROC values) for a fixed budget choice (the number of samples from the LLM) of N 10).
We will release the source code both stages upon acceptance.
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N
LLM
Dataset
LL
P(true)
SE-Ban Ba-es
SE-Histogram
SE-Rescaled
SE-Rescaled (h)
NQ
0.583 + 0.000
0.461 + 0.000
0.669 0.013 0.500 0.000
0.500 0.000
0.500 + 0.000
SVAMP
0.631 + 0.000
+
0.796 + 0.057
0.500 + 0.000
0.500 + 0.000
0.500 + 0.000
Llama-2
Squad
0.624 0.000
0.441 0.000
0.701 = 0.009
0.500
0.000
0.500 + 0.000
0.500 + 0.000
Trivia QA
1.094
0.436
0.688 0.012
0.500
0.000
0.500 = 0.000
0.500 = 0.000
NQ
0.627 0.000
0.615 0.000
0.683 = 0.012 0,500
+
0.500 - 0.000
0.500 = 0.000
SVAMP
= 0.647
0.414
0.812 0.012
0.500
0.500 + 0.000
0.500 = 0.000
Llama-3.2
Squad
0.610 0.000
0.555 + 0.000
0.617 + 0.021
0.500
0.000
0.500 + 0.000
0.500 + 0.000
Trivia QA
0.614 0.000
0.710 1 0.000
0.731
0.008
0.500
0.500 0.000
0.500 + 0.000
Tlama-3.3-70B Trivia QA
0.556
0.000
0.686
0.000
0.768
0.003
1.500
0.500 + 0.000
0.500 + 0.000
NQ
0.695 0.000
0.731 0.000
0.635 + 0.008
0.500
0.500 + 0.000
0.500 + 0.000
SVAMP
0.645 0.000
0.843
0.844
0.038
0.500
0.500 + 0.000
0.500 + 0.000
Mistral
Squad
0.698 0.000
0.687
0.628 + 0.010
0.500
0.500 = 0.000
0.500 + 0.000
Trivia QA 0.67
0.647 0.000
0.633 0.003
0.500
0.500 + 0.000
0.500 + 0.000
N
2
LLM
Dataset
LL
P(true)
SE
Bayes
SE-Histggram
SE-Rescaled
SE-Rescaled (h)
NQ
0.583 1 0.000
0.461 1 0.000
0.723 0.007 0.652 1 0.010
0.654 1 0.013
0.644 1: 0.014
SVAMP
0.631 + 0.000
0.469 0.000
0.855
0.022
0.748 0.024
0.749 + 0.027
0.759 0.025
Llama-2
Squad
0.624 0.000
0.441
0.000
0.735 + 0.007
0.654 0.012
0.639 + 0.017
0.649 + 0.010
Trivia QA
0.000
0.436 0.000
0.737 0.006
0.670
0.012
0.675 0.012
0.673 0.010
NQ
0.627 0.000
0.615
0.728 0.008
0.640
0.013
0.663 0.017
0.649 + 0.020
SVAMP
0.647 0.000
0.000
0.845 = 0.022 0.761
0.032
0.774 0.080
0.771 = 0.028
Llama-3.2
Squad
0.610 0.000
0.555
0.000
0.664
0.019
0.642
0.029
0.621 0.027
Trivia QA
0.614 0.000
0.710 1 0.000
0.768 1 0.004
0.699
1.000
0.706
+ 0.007
0.708 + 0.008
Llama-3.3-700 Trivia QA
0.556 + 0.000
0.686 + 0.000
0.775
0.003
0.653
0.650 - 0.006
0.651 0.006
NQ
0.695 0.000
0.731 + 0.000
0.705 + 0.013
0.647
0.007
0.654 0.009
SVAMP
0.645 0.000
0.843 + 0.000
0.876
0.011
0.793
0.815 0.028
0.812 0.024
Mistral
Squad
0.698 0.000
0.687
1 0.000
0.667
0.010
0.618
0.671 0.017
0.631 0.010
Trivia QA
0.672 0.000
0.647 + 0.000
0. 682 0.008
0.638
0.011
0.645 + 0.013
0.643 + 0.012
LLM
Dataset
LL
P(true)
SE- Ba-es
SE-Histogram 0.707 + 0.009
SE-Rescaled
SE- SE-Rescaled (1)
NQ
0.583 + 0.000
0.461 + 0.000
0.746 + 0.006
0.678 + 0.012
0.700 0.011
SVAMP
0.631 + 0.000
0.862 + 0.020
+.022
0.804 = 0.031
0.811 = 0.031
Llama-2
Squad
0.624 0.000
0.441
0.000
0.710 0
0.679 = 0.019
0.708 0.012
Trivia QA
0.594 0.000
0.436 = 0.000
0.753 : 0.004
0.706 = 0.009
0.710 = 0.007
0.709 = 0.009
NQ
0.627 + 0.000
0.615
0.000
0.745 0.008
0.697 = 0.008
0.672 0.012
0.699 0.008
SVAMP
0.647 0.000
0.414 0.000
0.843 0.013
0.801 + 0.019
0.800 - 0.021
0.811 0.021
Llama-3.2
Squad
0.610 0.000
0.555
0.000
0.686 1 0.021
0.659
0.016
0.649
0.017
0.654
0.014
Trivia QA
0.614
0.710
= 0.000
0.781 1: 0.008
0.738 : 0.006
0.729 = 0.006
0.746 0.007
Llama-3.33 3.B-70B Trivia QA
0.556 0.000
0.686
= 0.000
00.811
0.702 = 0.004
0.706 = 0.003
NQ
0.695 0.000
0.731 1: 0.000
0.743
11743.0.000 0.009
0.718
1 0.010
0.716
0.009
0.710 0.010
SVAMP
0.645 + 0.000
0.843 = 0.000
0.887
0.858
0.017
0.851
= 0.021
0.870 0.022
Mistral
Squad
0.698 0.000
0.687
0.000
0.010
0.018
0.698
0.014
0.685 0.020
Trivia QA
0.672 1: 0.000
0.647 1 0.000
0.091
0.678 0.010
0.676 - 0.008
0.684 0.010
21

---

Published in Transactions on Machine Learning Research (09/2025)
N =
4 Bay-B
LLM
Dataset
LL
P(true)
SE-
SE- SE-Histogram
SE-Rescaled
SE-Rescaled (h)
NQ
0.583 + 0.000
0.461 0.000 0.754 + 0.002
0.729 + 0.007
0.695 0.014
0.720 + 0.008
SVAMP
0.631 + 0.000 0.469 + 0.000 0.899
+ 0.013
0.866 + 0.010
0.870 + 0.015
0.872 + 0.011
Llama-2
Squad
0.624 = 0.000
0.441 = 0.000 0.771
0.742 + 0.006
0.700 0.008
0.734 = 0.004
Trivia QA
0.594 = 0.000 0.436 0.000 0.767
0.735 + 0.007
0.733 0.007
0.734 = 0.007
NQ
0.627 = 0.000 0.615 = 0.000 0.759
0.722 + 0.006
0.684 = 0.015
0.723 + 0.007
SVAMP
0.647 = 0.000 0.414
0.86
0.843 + 0.005
0.837 0.010
0.850 + 0.003
Llama-322
Squad
0.610 = 0.000 0.555
0.000
0.686
0.014
0.649 + 0.019
0.676 0.018
Trivia QA
0.614 + 0.000 0.000
0.78
0.759 1 0.009
0.746 0.005
0.764 1 0.009
Llama-3.3-70B Trivia QA 0.556 + 0.000 0.686 + 0.000 01 0.793 0.002
0.739 + 0.002
0.734 + 0.003
0.738 + 0.002
NQ
0.695 + 0.000 0.731 0.000 0.762 0.012
0.740 + 0.008
0.730 + 0.006
0.737 + 0.013
SVAMP
0.645 = 0.000 0.843
0.000
0.880 + 0.865 + 0.017 0.851
0.019
0.877 + 0.019
Mistral
Squad
0.698 = 0.000 0.687
= 0.000 0.720 + 0.007
0.705 + 0.011
0.700 + 0.014
0.705 = 0.013
Trivia QA 0.672 = 0.000 0.647
0.000
0.692 + 0.009
0. 0.682 + 0.016 0.678
0.013
0.687 = 0.012
N 5
LLM
Dataset
LL
P(true)
SE- Bayes
SE-Histogram
SE-Rescales
RE-Rescaled (h)
NQ
0.583 1 0.000 0.461
0.000 0.752 1 0.006 0.730 1 0.009
0.701 0.012
0.725 1 0.010
SVAMP
0.631 0.000 0.469
0.000
0.871 0.011
0.849 + 0.012
0.853 0.013
0.858 + 0.013
Llama-2
Squad
0.624 + 0.000 0.441 + 0.000
0.774
0.756 0.004
0.711 + 0.012
0.752 + 0.005
Trivia QA
0.594 = 0.000 0.436 0.000
0.763 0.009
0.734
0.737 0.006
0.735 0.006
NQ
0.627 + 0.000 0.615
0.000
0.760 0.008
0.691 + 0.005
0.733 + 0.012
SVAMP
0.647 = 0.000
0.070
0.860
-: 0.010
0.850 0.004
0.864 + 0.009
Llama-3.2
Squad
0.610 0.000 0.555
0.000
0.710
0.707
0.667 = 0.021
0.705 = 0.013
Trivia QA
0.614 +: 0.000 0.710 1: 0.000 0.792 1 0.004
1.775
0.763 0.005
0.777 + 0.004
Llama-3.3-700 Trivia QA 0.556 = 0.000 0.686 0.000 0.793 0.005
0.750
0.740 0.004
0.747 0.006
NQ
0.695 = 0.000 0.731 = 0.000 0.780 0.006 0.762
0.728 + 0.008
0.756 + 0.007
SVAMP
0.645 + 0.000 0.843
0.000 0.880 + 0.019 0.866 + 0.021
0.855 0.027
0.878 + 0.022
Mistral
Squad
0.698 = 0.000 0.687 0.000
0.731 + 0.008 0.719 1 0.010
0.699
0.008
0.712 = 0.008
Trivia QA 0.672 = 0.000 0.647
0.000
0. 691
+ 0.005 0 6.688 + 0.005
0.684 + 0.004
0. 0.690 + 0.005
LLM
Dataset
LL
P(true)
SE- Sa-Bayes
SE-Histogram
SE-Rescaled
SE-Rescaled (b)
NQ
0.583 + 0.000
0.461 + 0.000
0.755 + 0.005
0.739 + 0.010
0.702 + 0.008
0.733 + 0.010
SVAMP
0.631 = 0.000 0.469 + 0.000 0.892 + 0.013
0.876 : 0.018
0.876 0.017
0.880 + 0.016
Llama-2
Squad
0.624 = 0.000 0.441 0.000
0.776
0.761
0.725 0.006
0.756 = 0.009
Trivia QA
0.594 = 0.000 0.436 = 0.000 0.771
0.743 0.009
0.747 = 0.008
0.745 = 0.009
NQ
0.627 = 0.000 0.615
0.000
0.005
0.741 0.004
0.690 0.012
0.738 0.006
SVAMP
0.647 + 0.000 0.414
0.000
0.009 0.844 0.007
0.833 + 0.010
0.844 + 0.011
Llama-3.2
Squad
0.610 = 0.000 0.555 - 0.000 0.716
0.718
0.671 0.012
0.715 1 0.009
Trivia QA
0.614 = 0.000 0.710 1 0. 0.005
0.783 0.008
0.771 + 0.008
0.784 + 0.008
Llama-3.3 3.B-70B Trivia QA 0.556 = 0.000 0.686 0.000 798 0.003
0.759
0.751 = 0.004
0.756 0.007
NQ
0.695 +: 0.000 0.731 0.000
0.740 1 0.008
0.756 0.008
SVAMP
0.645 = 0.000 0.843 + 0.000 0.888 0.020 0.8.0002000888888888 0.878
0.860 + 0.018
0.885 + 0.018
Mistral
Squad
0.698 = 0.000 0.687 0.000 0.726 0.004 0.721
0.700 0.02
0.011
0.719 = 0.002
Trivia QA 0.672 + 0.000 0.647 1 000000000000.1110000001000000110
0.00
0.697 1 0.011
22

---

Published in Transactions on Machine Learning Research (09/2025)
N =
LLM
Dataset
LL
P(true)
SE- Ba-Bs
SE-Histogram
SE-Rescaled
SE-Rescaled (h)
NQ
0.583 + 0.000
0.461 0.000 0.760 + 0.002
0.744 0.004
0.712 0.004
0.738 + 0.004
SVAMP
0.631 + 0.000 0.469 + 0.000 0.895 + 0.016
0.875 + 0.018
0.875 + 0.017
0.880 + 0.017
Llama-2
Squad
0.624 = 0.000
0.441 = 0.000
1.788100000
0.767 0.004
0.718 0.007
0.763 + 0.005
Trivia QA
0.594 = 0.000 0.436 0.000 0.767
0.050
0.752 0.006
0.751 = 0.006
NQ
0.627 = 0.000 0.615 = 0.000 0.770
0.050
0.697 = 0.009
0.747 = 0.006
SVAMP
0.647 = 0.000 0.414 0.000 0.862
0.847
0.858 0.020
0.847 = 0.020
Llama-322
Squad
0.610 = 0.000 0.555 0.000
0.003
0.725
0.006
0.667 + 0.008
0.719 + 0.006
Trivia QA
0.614 + 0.000 0.000 0.7095000.7755
0.793
0.782
4 0.004
0.774 0.008
0.783 1 0.003
Llama-3.3-70B Trivia QA 0.556 + 0.000 0.686
0.79 0.004
+
0.765
0.006
0.756 + 0.005
0.763 + 0.007
NQ
0.695 + 0.000 0.731 0.000 0.782 + 0.003
0.770 0.005
0.736 + 0.005
0.764 + 0.005
SVAMP
0.645 = 0.000 0.843
0.000
0
0.903 4 0.001
0.890 0.015
0.910 + 0.004
Mistral
Squad
0.698 = 0.000 0.687
= 0.000
0. # 0.008
0.735
0.008
0.702 + 0.010
0.728 = 0.007
Trivia QA 0.672 = 0.000 0.647
0.000
0.697 + 0.011
0. 0.702 + 0.007
0.696 0.009
0.702 0.009
N =
LLM
Dataset
LL
P(true)
SE- Bayes
HEsHoisog
SE-
BE-Rescaled (h)
NQ
0.583 1 0.000 0.461
0.000 0.761 1 0.005
0.749 1 0.006
0.722 1 0.010
0.742 1 0.007
SVAMP
0.631 0.000 0.469
0.000 0
0.888 + 0.007
0.882 0.010
0.889 0.007
Llama-2
Squad
0.624 + 0.000 0.441 + 0.000
0.784
0.778 + 0.006
0.734 + 0.015
0.775 + 0.008
Trivia QA
0.594 = 0.000 0.436 0.000 1.778100000
0.761 0.007
0.762 0.007
0.760 0.007
NQ
0.627 + 0.000 0.615 0.000 0.773 0.005
0.005
0.753 + 0.005
0.704 + 0.004
0.748 + 0.006
SVAMP
0.647 = 0.000
0.011
0.868 +: 0.013
0.862 - 0.012
0.868 + 0.011
Llama-3.2
Squad
0.610 0.000 0.555 = 0.000 0.724
7.724
0.668 = 0.014
0.720 0.009
Trivia QA
0.614 +: 0.000 0.710 1: 0.000 0.797 0.001
0.785 1 0.002
0.778 + 0.004
0.787 1 0.002
Llama-3.3-700 Trivia QA 0.556 = 0.000 0.686 0.000 0.798 0.005
1.772
0.764 0.005
0.770 0.003
NQ
0.695 = 0.000 0.731 = 0.000 0.791 1 0.004
0.784
0.742 + 0.005
0.775 + 0.005
SVAMP
0.645 + 0.000 0.843
0.000
0 0.014 0.888 1 0.015
0.882 0.012
0.893 + 0.016
Mistral
Squad
0.698 = 0.000 0.687 0.000
0.744 0.005 0.745
0.005
0.710 0.004
0.737 + 0.005
Trivia QA 0.672 = 0.000 0.647
0.000
0 699
+ 0.003 0. 702 + 0.006
0.701 0.010
0. 707 + 0.008
LLM
Dataset
LL
P(true)
SE- Bayes
SE-Histogram
SE-Rescaled
SE-Rescaled (b)
NQ
0.583 + 0.000
0.461 + 0.000 0.759 + 0.005
0.749 + 0.008
0.723 + 0.008
0.714 + 0.009
SVAMP
0.631 = 0.000 0.469 + 0.000
0 0.005
0.877 0.007
0.879 0.006
0.880 = 0.007
Llama-2
Squad
0.624 = 0.000 0.441 0.000
+
0.785 0.010
0.738 0.010
0.780 = 0.011
Trivia QA
0.594 = 0.000 0.436 = 0.000 0.700
0.756 0.008
0.758 0.009
0.755 = 0.006
NQ
0.627 = 0.000 0.615
0.000 0.770
0.753 0.006
0.708 0.007
0.747 0.005
SVAMP
0.647 + 0.000 0.414
0.000
0.876
0.880 + 0.012
0.875 + 0.010
Llama-3.2
Squad
0.610 = 0.000 0.555 = 0.000 0.732
0.733
0.674 0.011 0.010
0.727 1 0.012
Trivia QA
0.614 = 0.000
0.000 0.800 0.000
0.793
1.006
0.006
0.788 + 0.007
0.794 + 0.005
Llama-3.33 3.B-70B Trivia QA 0.556 = 0.000 0.686 0.000 0.0.00000.7944 0.79
0.770
0.762 0.006
0.769 + 0.005
NQ
0.695 +: 0.000 0.731 0.000
0.785
0.752 0.008
0.777 1 0.001
SVAMP
0.645 = 0.000 0.843 + 0.000 0.903
0.895
0.877 + 0.015
0.896 + 0.009
Mistral
Squad
0.698 = 0.000 0.687 0.000 0.749
0.717 0.007
0.008
0.739 0.008
Trivia QA 0.672 + 0.000 1: 0.00000000...0000000000001000000 0.70
0.702
0.013
0.704 1 0.013
23

---

Published in Transactions on Machine Learning Research (09/2025)
N
10
LLM
Dataset
LL
P(true)
SE- SE-Bayes
SE-Histogram
SE-Rescaled
SE-Rescaled (h)
NQ
0.583 + 0.000
0.461 0.000
0.763 + 0.003 0.753 + 0.002
0.731 0.007
0.745 0.002
SVAMP
0.631 + 0.000 0.469 + 0.000 0.896 + 0.007
0.886 + 0.006
0.891 + 0.007
0.890 + 0.010
Llama-2
Squad
0.624 = 0.000
0.441 = 0.000
0.003
0.782 + 0.006
0.735 0.005
0.776 + 0.005
Trivia QA
0.594 = 0.000 0.436 0.000 0.776 0.003
0.761 0.004
0.765 0.004
0.760 1 0.004
NQ
0.627 = 0.000 0.615 - 0.000 0.777 0.777 0.004
0.761
0,711 = 0.007
0.756 + 0.005
SVAMP
0.647 = 0.000 0.414 0.000 0.863
0.861
0.856 0.010
0.859 + 0.012
Llama-3.2
Squad
0.610 = 0.000 0.555
0.736
0.686 + 0.008
0.729 + 0.008
Trivia QA
0.614 + 0.000 0.710 0.000 0.801
0.80
0.793
0.791 0.004
0.793 0.002
Tlama-3.3-70B Trivia QA 0.556 + 0.000 0.686 + 0.000 01.98 0.79 0.004
0.777
0.771 + 0.003
0.777 + 0.003
NQ
0.695 + 0.000 0.731 0.000 0.796 +
0.790
0.004
0.758 + 0.006
0.779 + 0.004
SVAMP
0.645 = 0.000 0.843
0.000
0.012
0.916
0.885 + 0.015
0.919 + 0.010
Mistral
Squad
0.698 = 0.000 0.687 = 0.000 0.752
0.749
0.714 0.002
0.741 + 0.006
Trivia QA 0.672 = 0.000 0.647 0.000
0.697
0.
7.70
0.697
0.008
0.702 + 0.009
D Hyperparameter Sensitivity
We examined the sensitivity of method to the hyperparameter a, finding it to be insensitive. The plots below show performance for Llama 2 the Trivia QA dataset for {0.1, 0.5, 1.0]. A value of 0.5 was used for all other experiments.
Lloma-2, Trivia QA
Llama-2 Trivia QA
LAmma-2 Trivin QAA
1
0.7
0.7
0.2
1
AUBOC
ATMOC
0.6
0.6
0.0
10
10
10
N
a 0.5
1.0
Bayesian SE
Histogram
Rescaled (heuristic)
Rescaled
Log Likelihood
P(Truc)
24


Reviews:
[{"review_id": "UnhYVRqqED", "reviewer": "Reviewer_JXMa", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "### Strengths\n\n**Comparisons**. Authors have made sufficient efforts in comparisons against other types of entropies and justify the efficacy of bayesian approach.  \n**Overall Results**. The experimental results well justify their claims and validate the efficacy of Bayesian approach for hallucination detection.\n\n### Weaknesses\n\n**Chosen models**. I notice that authors selects smaller size models to validate their approach (e.g., Llama-3.2-3B-Instruct, Mistral-Small-24B-Instruct-2501, whose activating parameters are small). It is frequent that conclusion made on small models cannot be generalized to larger models. Can authors add more models to further validate the bayesian approach?", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper addresses **hallucination detection**, a setup that requires identification of a model when it generates factually incorrect responses. One existing representative way to do this is to estimate the semantic entropy (aka. uncertainty) to detect hallucinations. On top of existing semantic entropy estimation, this paper proposes to use Bayesian approach, which performs better on hallucination detection in a more effective and efficient way. To validate their approach, they compare their design against baseline on multiple language models (LLaMA, Mistral) and showcase good improvements over existing approaches.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "Please see weakness 1.", "broader_impact_concerns": "The paper does not include a Broader Impact Statement section.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "DQQBKxrkaQ", "reviewer": "Reviewer_jKjg", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "The paper is well written and makes a meaningful and significant contribution. The experiments are well thought out and convincingly show that the method works. It would make this paper stronger if there could be some explanation for why it works better at least against some of the baselines. There is little explanation given for even an intuition for why these results should hold and a supporting experiment.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper introduces an method for approximating semantic entropy using posterior inference on a meaning distribution. This distribution implies a distribution on the entropy of the prompts and can be approximated well using substantially less generated data from an LLM. The results compare favourably with baselines from the original paper as well as other approaches used for detecting hallucinations in LLMs.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "It would be nice to explain why the approximation more accurately predicts a hallucination than the more expensive estimate of semantic entropy. Why do they not converge as N is made larger?", "broader_impact_concerns": "I have no ethical concerns about this work.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "Jil4hIXaVt", "reviewer": "Reviewer_iPCg", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths: \n- Relatively clearly written. Formulations seem correct. \n\n\nWeaknesses: \n- The results are not surprising as Bayesian estimation tends to outperform maximum likelihood style solutions in cases where there is small number of samples. Hence, it is not clear what the novelty of this paper is other than applying the above idea to the particular problem of LLM hallucination detection.\n- The paper is also not very self contained in the sense that the paper refers to (Farquhar et al., 2024) for some critical details such as the relationship between LLM hallucinations and the entropy. It could be useful to explain it at least in a few sentences.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "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": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper suggests an improved version of the semantic entropy estimation for LLM outputs. Based on an earlier study, the authors claim that hallucinations can be detected based on the entropy of the generated responses in the semantic domain by mapping responses into meaning. This paper replaces the large scale sampling and histogram based approach of the referred paper by a Bayesian estimation approach. The experiments show the estimated entropy values and the area under ROC curve for hallucination detection on a few question answering tasks using Llama3-2b model. Based on the figures in Fig. 1 and and Fig. 2 especially in the case where the estimation process uses small number of generations (N < 4), the Bayesian estimation technique proposed in the paper provides better estimates. However, when N is large the difference between different entropy estimation methods gets smaller.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "1. The paper refers to (Farquhar et al., 2024) for some critical details such as the relationship between LLM hallucinations and the entropy. It could be useful to explain it at least in a few sentences. \n\n2. The experimental setup can also be explained in more details\n\n3. In Table 1, do the numbers represent the expected value of the entropy? If so, instead of just writing \"Experimental results \" or saying performance, please clarify the Table 1 caption and explain why the higher mean entropy means a better estimator?  \n\n4. Some analysis on the length of the LLM responses and whether and how they affect the entropy estimation can be discussed.", "broader_impact_concerns": "There is not an explicit section on ethical implications. The paper tries to detect LLM hallucinations and hence can be useful for eliminating some of the potential harms of LLMs.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Decision: Accept (poster)