Learning What to Ask and What to Believe: Distilling a Variational Bayesian Agent into Language Models

Baichuan Li; Yanling Chang; Alfredo Garcia

cs.LGOct 31, 2026

Learning What to Ask and What to Believe: Distilling a Variational Bayesian Agent into Language Models

Authors: Baichuan Li, Yanling Chang, Alfredo Garcia

Organizations: Southern Methodist University · University of Virginia

Abstract

Interactive tasks such as eliciting a user's preferences or diagnosing a hidden mechanism require an agent to track its uncertainty over a latent state, ask informative questions and act on the answers. Language models have no explicit mechanism for this, whereas Bayesian information seeking provides one through posterior inference and expected information gain (EIG), and we ask whether it can be distilled into an 8B language model. Our teacher, a lightweight Bayesian agent with a sequential variational belief and EIG query selection, matches or approaches exact and particle references on three task families. We generate synthetic interactions with the teacher on episodes disjoint from the benchmark and fine-tune \Student{} to reproduce its queries and its belief state as a structured scratchpad. On three Multi-Turn Puzzles tasks the student raises Circuit Decoding from 0.00 to 0.93 success, lowers the normalized rank of the recommended film from 0.50 to 0.10 and cuts word guessing from 32 to 3.4 attempts, exceeding the strongest reported language models on the two comparable tasks. Exposing the belief state closes much of the gap where the decision is an implicit computation rather than a read-out of the interaction; in matched teacher contrasts, EIG supervision consistently outperforms random querying, whereas greater inference fidelity does not uniformly improve the distilled student; a 70B language-model teacher does not match the Bayesian teacher in downstream student performance, and the learned skill does not transfer to an unseen task family. Variational Bayesian information seeking can thus be amortized into a language model by transferring the belief state that drives its questions and decisions.

Figures & tables

Panel of Figure 1 — Figure 1: Overview. A Bayesian teacher interacts with synthetic episodes disjoint from the benchmark; its interactions are serialized as actions only or as belief scratchpad plus action and distilled into Llama-3.1-8B-Instruct, which is then evaluated interactively on the complete benchmark test splits. The targets are shown for two turns of one movie-recommendation dialogue of the teacher; the scratchpad adds the <belief> and <scores> lines (blue; formats for all tasks in Appendix D.4 ).

	Movie Recommendation	Circuit Decoding	Word Guess
Latent state $\theta$	weights $\in[0,0.9]^{8}$ (continuous)	$24$ -bit truth table (structured binary)	one of $40$ words (finite symbolic)
Prior $p(\theta)$	uniform on the grid $\{0,0.1,\dots,0.9\}^{8}$	uniform over the tables realizable under the inventory	uniform over the $40$ words
Query $a$	pair of seen films ( $190$ )	(circuit, input) ( $24$ )	vocabulary word ( $40$ )
Observation $y$	Yes / No / No Preference	one bit	$5$ colors ( $\leq 243$ patterns)
Budget $T$	$10$	$18$	$40$ (minimize attempts)
Decision $d$	one of $40$ unseen films	all $24$ bits	the secret word

Table 1: The three tasks and the teacher’s variational family for each (

1{,}000

300

400

test episodes).

Task	Metric	Base (8B)	Student	Teacher	Student	Teacher	Reported LM
			ELBO+EIG		Exact / Particle Bayes+EIG
Circuit	Success@18 $\uparrow$	$0.000$	$\mathbf{0.927}$	$1.000$	$\mathbf{0.840}$	$1.000$	–
Circuit	benchmark score (%) $\uparrow$	$0.7$	$\mathbf{97.4}$	$100.0$	$\mathbf{92.8}$	$100.0$	$39.8$
Movie	Success@10 $\uparrow$	$0.032$	$\mathbf{0.345}$	$0.410$	$\mathbf{0.409}$	$0.562$	–
Movie	benchmark score (%) $\uparrow$	$50.4$	$\mathbf{90.0}$	$92.9$	$\mathbf{93.1}$	$96.5$	$84.3$
Word	mean attempts $\downarrow$	$31.97$	$\mathbf{3.37}$	$2.29$	$\mathbf{3.10}$	$2.29$	–

Table 2: Main result on the complete test splits: base model, the belief-scratchpad students of the ELBO+EIG teacher and of the exact or particle reference, their teachers, and the strongest language model reported by the benchmark (Gemini 2.5 Pro; none for Word Guess: the normalization is unpublished and its two turn counts disagree). Benchmark score: circuit-wise accuracy (circuit) and

100\times(1-\text{normalized rank})

(movie). All student–base McNemar or Wilcoxon

p<10^{-60}

Panel of Figure 2 — Figure 2: Action-only against belief-scratchpad distillation. (a) Movie Recommendation: success of the action-only students under their own and their teacher’s questions, with their own and a Bayes-optimal read-out, against the belief-scratchpad students and the teachers. (b) Word Guess: success within $k$ attempts for the base model and the two ELBO+EIG students. Labels in (a) give questions / read-out.

	Circuit, Success@18 $\uparrow$		Movie, Success@10 $\uparrow$		Word, attempts $\downarrow$
Teacher	A / B	B $-$ A	A / B	B $-$ A	A / B	B $-$ A
ELBO+EIG	$0.920$ / $0.927$	$+0.007$ $[-0.037,+0.050]$	$0.085$ / $0.345$	$+0.260$ $[+0.228,+0.292]$	$19.2$ / $3.4$	$-15.8$ $[-17.5,-14.0]$
ELBO+Random	$0.500$ / $0.590$	$+0.090$ $[+0.010,+0.170]$	$0.105$ / $0.176$	$+0.071$ $[+0.044,+0.098]$	$15.2$ / $3.1$	$-12.1$ $[-13.4,-10.8]$
ELBO+Uncertainty	–	–	$0.064$ / $0.194$	$+0.130$ $[+0.103,+0.157]$	–	–
Exact / Particle+EIG	$0.643$ / $0.840$	$+0.197$ $[+0.133,+0.257]$	$0.079$ / $0.409$	$+0.330$ $[+0.296,+0.362]$	$15.4$ / $3.1$	$-12.3$ $[-13.8,-10.8]$

Table 3: Action-only (A) against belief-scratchpad (B) students of the same teacher on identical test episodes: score under each format and paired difference B

-

A with

95\%

bootstrap CI.

Contrast	Circuit	Movie	Word
Query rule: random $\to$ EIG	$+0.420$ $[+0.350,+0.487]$	$+0.169$ $[+0.137,+0.202]$	$+0.205$ ( $103/21$ )
Query rule: uncertainty $\to$ EIG	identical rules	$+0.151$ $[+0.117,+0.186]$	–
Inference: variational $\to$ exact or particle	$-0.277$ $[-0.333,-0.217]$	$+0.064$ $[+0.029,+0.098]$	$0.000$ ( $23/23$ )
Teacher type: Bayesian $\to$ LLM teacher	$-0.920$ $[-0.950,-0.887]$	$-0.033$ $[-0.055,-0.011]$	–

Table 4: Teacher-component contrasts: paired differences between students on identical test episodes (second condition minus first) with

95\%

bootstrap CIs; word: Success@3 with discordant counts. Circuit uses action-only students, movie and word belief-scratchpad students (the LLM teacher row is action-only; belief-scratchpad circuit contrasts in Appendix F.1 ).

Appendix figures & tables5 assets

Supplementary material from the paper’s appendix.

Appendix

Condition	Circuit Decoding	Movie Recommendation	Word Guess
Base model	yes	yes	yes
ELBO+EIG student	A, B	A, B	A, B
ELBO+Random student	A, B	A, B	A, B
ELBO+Uncertainty student	(= EIG)	A, B	–
Exact / Particle Bayes+EIG student	A, B	A, B	A, B
LLM teacher student	A	A	–

Appendix

Table 5: Conditions evaluated on the complete test splits. A = action-only targets, B = belief-scratchpad targets. Uncertainty sampling coincides with EIG on Circuit Decoding; the LLM teacher has no belief and therefore no scratchpad condition.

Task	Metric	ELBO+EIG	ELBO+Rand.	ELBO+Unc.	Exact / Part.
Circuit	Success@18 $\uparrow$	$\mathbf{1.00}$	$0.70$	–	$\mathbf{1.00}$
	turns to identify $\downarrow$	$11.0$	$17.1$	–	$11.0$
	EIG-opt. / regret (bits)	$0.98$ / $0.001$	$0.30$ / $0.33$	–	$1.00$ / $0$
Movie	Success@10 $\uparrow$	$0.410$	$0.373$	$0.428$	$\mathbf{0.562}$
Movie	benchmark score (%) $\uparrow$	$92.9$	$91.0$	$93.8$	$\mathbf{96.5}$
Word	mean attempts $\downarrow$	$\mathbf{2.29}$	$3.09$	–	$\mathbf{2.29}$

Appendix

Table 6: Teacher validation on the complete released test splits (

300

1{,}000

400

episodes; five paired seeds on Circuit Decoding and for ELBO+Random on Word Guess, one run on Movie Recommendation). EIG-optimal rate and regret are measured under the exact posterior. ELBO+Uncertainty asks the pair whose predicted answer is closest to even odds (Eq. 3 , Appendix C.5 ) and is run only on Movie Recommendation, where it differs from EIG.

Figure 3: Inference dynamics of the teachers. (a) Circuit Decoding: mean entropy of the belief after each query on the synthetic training episodes (dotted line: the

18

-query budget). (b) One variational update on a training episode: distance of the ELBO from its analytic optimum over the

80

Adam steps, for the first three updates. (c) Movie Recommendation: cosine between the posterior mean and the true weights after each question (synthetic training episodes). (d) Word Guess: mean belief entropy before each attempt on the complete test split (episodes still running; the two EIG teachers coincide).

Figure 4: Validation loss per epoch (synthetic validation split, assistant tokens) of every distilled student, one panel per task. Solid lines with filled markers: action-only targets; dashed lines with open markers: belief-scratchpad targets. Validation loss measures how predictable a teacher’s interactions are, not how good they are: the LLM teacher students have the lowest loss and the worst benchmark scores, the action-only exact-inference circuit student has twice the loss of the corresponding variational student, and the word action-only students’ losses are more than an order of magnitude above the scratchpad students’, whose targets contain the deterministic bookkeeping. Losses are not comparable across target formats within a task (different supervised tokens). The checkpoint with the lowest validation loss is the one evaluated.

Panel of Figure 5 — Figure 5: Students inherit their teacher’s information-seeking policy on Circuit Decoding and only partly on Movie Recommendation and Word Guess: the teacher’s EIG-optimal rate on the training episodes against its student’s on the complete test split, one panel per task, one labeled point per teacher (belief-scratchpad students; the LLM teacher’s students are action-only).

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Language models (LMs) are increasingly used for tasks that require reasoning about hidden variables from a few observations, for which Bayesian inference is the normatively correct solution. While supervised fine-tuning of an LM on the outputs of an optimal

\textit{Bayesian}

model leads to near-Bayesian behavior, standard supervised fine-tuning (SFT) on the true answers for the task falls short of it. But behavior alone does not tell us

\textit{why}

tuning on a

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or an

\textit{oracle}

(true answers) signal differs: whether the resulting LM represents Bayesian beliefs, acts on them, or turns them into a choice the way Bayes' rule does. To compare them, we formulate increasingly demanding requirements for an LM to count as a Bayesian decision maker, spanning its behavior, representations, and computations, and test them on a flight recommendation task. The Bayes-trained LM acts Bayesian, encodes quantities of Bayes' rule in its middle layers, and uses the encoded belief for the recommendation to a certain extent. The oracle-trained LM differs from it both in the beliefs it holds and whether it reads beliefs out into recommendations. Exchanging beliefs between the LMs transfers a part of the Bayesian advantage. Bayes fine-tuning thus installs usable Bayesian beliefs in an LM for reasoning under uncertainty in a way standard SFT on oracle answers cannot, highlighting the advantage of nuanced supervision.

Abstract

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