An Information-Theoretic Framework For Optimizing Experimental Design To Distinguish Probabilistic Neural Codes
Authors: Po-Chen Kuo
Edgar Y. Walker
Venue: ICLR 2026 Poster (2026)
Keywords: Bayesian brain hypothesis
perceptual decision-making
uncertainty representations
probabilistic neural codes
probabilistic population code
neural sampling code
information theory
experiment optimization
Primary area: applications to neuroscience & cognitive science
TL;DR: Information gap guides optimal experimental designs to differentiate probabilistic neural coding hypotheses by quantifying the expected decoder performance differences

Abstract: The Bayesian brain hypothesis has been a leading theory in understanding perceptual decision-making under uncertainty.
While extensive psychophysical evidence supports the notion of the brain performing Bayesian computations, how uncertainty information is encoded in sensory neural populations remains elusive.
Specifically, two competing hypotheses propose that early sensory populations encode either the likelihood function (exemplified by probabilistic population codes) or the posterior distribution (exemplified by neural sampling codes) over the stimulus, with the key distinction lying in whether stimulus priors would modulate the neural responses.
However, experimentally differentiating these two hypotheses has remained challenging, as it is unclear what task design would effectively distinguish the two. 
In this work, we present an information-theoretic framework for optimizing the task stimulus distribution that would maximally differentiate competing probabilistic neural codes.
To quantify how distinguishable the two probabilistic coding hypotheses are under a given task design, we derive the \textit{information gap}---the expected performance difference when likelihood versus posterior decoders are applied to neural populations---by evaluating the Kullback–Leibler divergence between the true posterior and a task-marginalized surrogate posterior.
Through extensive simulations, we demonstrate that the information gap accurately predicts decoder performance differences across diverse task settings.
Critically, maximizing the information gap yields stimulus distributions that optimally differentiate likelihood and posterior coding hypotheses. 
Our framework enables principled, theory-driven experimental designs with maximal discriminative power to differentiate probabilistic neural codes, advancing our understanding of how neural populations represent and process sensory uncertainty.

Published as a conference paper at ICLR 2026
AN INFORMATION- FRAMEWORK FOR OP- TIMIZING EXPERIMENTAL DESIGN To DISTINGUISH PROBABILISTIC NEURAL CODES
Po-Chen Kuo & Edgar Y. Walker
Department of Neurobiology and Biophysics University of Washington Seattle, WA 98195, USA {pckuo,
edu
ABSTRACT
The Bayesian brain hypothesis has been a leading theory in understanding per- ceptual decision-making under uncertainty. While extensive psychophysical ev- idence supports the notion of the brain performing Bayesian computations, how uncertainty information is encoded in sensory neural populations remains clusive. Specifically,, two competing hypotheses propose that early sensory populations en- code either the likelihood function (exemplified by probabilistic population codes) or the posterior distribution (exemplified by neural sampling codes) over the stim- ulus, with the key distinction lying in whether stimulus priors would modulate the neural responses. However, experimentally differentiating these two hypotheses has remained challenging, as it is unclear what task design would effectively dis- tinguish the two. In this work, we present an information-thooreee framework for optimizing the task stimulus distribution that would maximally differentiate com- peting probabilistic neural codes. To quantify how distinguishable the two prob- abilistic coding hypotheses are under given task design, derive the informa- tion gap-the expected performance difference when likelihood versus posterior decoders applied to neural populations-bby evaluating the Kullback-Leibler divergence between the true posterior and task mass-naagin surrogate poste- rior. Through extensive simulations, we demonstrate that the information gap accurately predicts decoder performance differences across diverse task settings. Critically, maximizing the information gap yields stimulus distributions that op- timally differentiate likelihood and posterior coding hypotheses. Our framework enables principled, theory-driven experimental designs with maximal discrimina- tive power to differentiate probabilistic neural codes, advancing our understanding of how neural populations represent and process sensory uncertainty.
INTRODUCTION AND RELATED WORK
Effective perceptual decision-making requires organisms to process sensory information while accounting for the uncertainty inherent in the noisy and ambiguous sensory observations. The Bayesian brain hypothesis theoretical roots tracing Laplace and von Helmholtz (de Laplace 1820; [Hllmollz) - that the brain maintains internal generative models of the world and performs inference by computing probability distributions over task-relevant latent world states (Kmill & Richards 1996 Knill & Pouget, 2004). This framework has proven successful in explaining various aspects of human and animal perception, from multisensory integration and object recognition to motion perception and sensorimotor learning (Ernst & Banks 2002 Weiss et al. 2002; Kersten et al., 2004 Alais & Burr 2004 Körding & Wolpert) (004).. Extensive be- havioral evidence demonstrates that humans and animals perform near optimally in perceptual tasks that require uncertainty estimation, strongly suggesting that sensory neural populations encode both task-relevant stimulus features and their associated uncertainty (Fiser et al. 2010; Pouget et al.) 2013 Qamar et al., 2013 Ma & Jazayeri 2014). However, the neural implementation of proba- bilistic computation remains actively debated, and how probability distributions are encoded and

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A)
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Two competing hypotheses on how sensory uncertainty information is encoded in early sensory neural populations. A) Likelihood coding hypothesis, exemplified by the probabilis- tic population code (Ma et al., 2006), proposes that early sensory populations encode the likelihood function over the stimulus, with posterior computation deferred to downstream areas. B) Posterior coding hypothesis, exemplified by the neural sampling code (Hoyer & Hyvärinen (002). posits that early sensory populations readily encode the posterior distribution over hidden world state by inte- grating prior knowledge conveyed via feedback connections from higher cortical areas.
Figure 1:
represented the brain is an area of active research (Yang Shadlen 2007 Orbán et al. 2016
Walker et al. 2020 Aitchison et al. [2021] Haefner et al. 2024).
An unresolved question concerns the format of probabilistic representations in sensory processing: Do early sensory populations encode the likelihood function over stimuli, or do they readily repre- sent the posterior distribution that incorporate prior knowledge? The likelihoud coding hypothesis (Fig- [TA] proposes that early sensory populations responding to stimuli (e.g., drifting grating x:) with underlying latent world states (e.g., orientation ()) represent likelihood functions I((0) p(=(0)
(Jazyeri & Movshon 2006 Walker et al. 2020). The classic form of probabilistic population code (Ma et al. 2006) exemplifies this hypothesis, proposing that sensory areas such as the primary visual cortex (V1) represent likelihood functions, accounting for the inherent variability neural popula- tion responses. Previous work has demonstrated that likelihood functions decoded from V1 popu- lation responses are predictive of animals' trial-by-trics choices and reflects uncertainty associated with the sensory stimuli (Beck et al. 2008 Walker et al. 2020).
In contrast, motivated part by the presence of extensive feedback connection from higher cor- tical areas that could convey existing 'prior' information, the posterior coding hypothesis (Fig. IB) posits that sensory populations readily represent posterior distributions over latent world states p(0]z), suggesting that even early sensory areas would incorporate the knowledge of priors to com- pute posterior distributions (Berkes et al., 2011; Festa et al., 2021). The neural sampling code (Hoyer & Hyvärinen 2002) is one illustrative example in this category where neural population is hypothesized to represent a posterior distribution by drawing a "sample" from the distribution and encoding it in its stochastic population responses, suggesting that neural variability naturally reflects
the sampling process from posterior distribution (Orbán et al. 2016: Haefner et al. 2016 Lange & Haefner 2022 Shrinivasan et al. 2023).
The critical distinction between the two probabilistic coding hypotheses lies in whether stimulus priors p(0) would modulate early sensory population responses. While existing studies have demon- strated that specific instantiations of each hypothesis can capture some aspects of observed neural response patterns (Haefner et al., 2016 Shivkumar et al. 2018 Walker et al., 2020), there is yet to be targeted experiment aimed to directly distinguish the predictions from each coding hypoth- esis (Haefner et al., 2024). fundamental challenge lies in identifying experimental designs- specifically, stimulus prior distributions- -hht would maximally differentiate the two coding hy- potheses (Ma & Jazayeri 20144. Since both coding hypotheses can often account for similar neural response patterns under traditional experimental conditions, targeted task designs where their pre- dictions diverge maximally are crucial for distinguishing between likelihood and posterior coding hypotheses (Grabska- Baabska-Barwinske Barwinska et al. 2013 Shivkumar et al., 2018 Lange et al. 2023)
Motivated by research on optimal stimulus design for psychophysics studies (Watson Pelli 1983
Madigan & Williams 1987), electrophysiology experiment (Lewi al. 2006 2011), and efficient
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Figure 2: A decoding approach to differentiating probabilistic neural codes. A) An experimen- tal paradigm consists of two contexts B} with context-specific prior distributions (0). B) Information gap A info the difference in likelihood (blue) and posterior (orange) decoder perfor- mances, can indicate whether the underlying neural population encodes the likelihood function (left) or the posterior distributions (right). C) Deep neural network-based decoders used for decoding the likelihood function (top) or the posterior distribution (bottom) from population responses.
coding (Machens et al., 2005), in this work, we present an information-thhorrae framework for de- signing experiments that optimally differentiate likelihood and posterior coding hypotheses. Our approach quantifies the expected difference in decodable information -hichh we term the infor- mation gap A when applying neural network-basee decoders to extract likelihood or posterior information from sensory neural populations following either coding hypothesis. Specifically, we (1) derive analytic expressions for the information gap under each coding hypothesis, evaluated as the Kullback-Lebllr (KL) divergence between the true posterior and surrogate posterior utiliz- ing Bayes-optimal estimators (Raventós et al. 2023;; (2) validate theoretical predictions through simulations with deep neural network decoders applied to synthetic populations; (3) demonstrate how maximizing the information gap yields stimulus distributions that optimally differentiate the two probabilistic coding hypotheses; and (4) analyze existing neurophysiology datasets to show that conventional single-context experimental design is incapable of adjudicating the two hypotheses, supporting the necessity of the proposed optimized experimental framework.
Our framework provides a principled metric for optimizing experimental designs, establishing the theoretical upper bound on distinguishability between the two probabilistic coding hypotheses for a given task design. By maximizing this metric, we identify stimulus distributions that yield max- imally differential decoder performance- providing rigorous, empirically testable predictions that directly adjudicate between competing theories probabilistic neural representations.
2 INFORMATION GAP
We propose determine whether early sensory populations encode likelihood functions poste- rior distributions by examining how varying stimulus priors affects population responses. Classic orientation discrimination tasks under different contexts naturally involve altered stimulus prior dis- tributions, making them ideal for testing this distinction (Qamar et al.] 2013 Walker et al. 2020) Our experimental paradigm manipulates stimulus prior distributions across two different contexts and examines whether population responses vary according to changes in stimulus statistics (Fig. identical
- a design that would leave likelihood-coding population responses invariant stimulus across contexts while systematically affecting posterior-coding population responses.
decoding framework leveraged to distinguish the probabilistic information content encoded in neural populations. As schematized in Fig. [2,, decoder performance degrades (increase in cross- entropy loss) when attempting to extract mismatched probabilistic content: if a neural population encodes likelihood functions, decoder trained to extract likelihood information should outperform one extracting posterior information, and vice versa for posterior-coding populations. This differ- ential performance between likelihood and posterior decoders thus serves diagnostic tool for identifying the underlying probabilistic representation. Building on recent advances in neural de- coding (Walker al. 2020), employ deep neural network-based decoders that effectively
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extract the encoded information while incorporating the structural assumptions of each probabilisstic coding hypothesis (Fig. 2D).
However, it is unclear what stimulus prior distributions would lead to maximal differentiabiltiy under the two probabilistic coding hypotheses (Fig. [). While intuition suggests using maximally differ- ent context priors, this would limit stimulus overlap across contexts and thus prevent observing how different context priors modulate neural population responses to identical stimuli. This tradeoff- requiring sufficient prior differences to generate distinguishable population responses while main- taining adequate overlap for meaningful comparisons across contexts-cannee be resolved through intuition alone. To address this, we developed an information-theone theoretic framework that quantifies the expected decoder performance difference to systematically optimize experimental designs.
Experimental paradigm Consider a generative model of sensory observations 0 - where represents noisy sensory observations (e.g. drifting gratings) generated according to the conditional distribution p(z[0) given the hidden world state 0 (e.g. orientation). We consider an experimental paradigm as introduced in Fig. A with two contexts c = ( AA B), each with its associated context frequency p(c) and context-specific prior over the world state p(0|c) (0). We assume that the contexts of the current session are explicitly cued to ensure that subjects adopt the intended context- specific prior without having to infer the context from the stimuli. Once the subjects trained on both contexts and the task performance stabilizes, we probe how early sensory populations represent probabilistic information.
Given neural population response vectors т, our goal is to assess the difference in decoder per- formances between likelihood decoder (r) and posterior decoder gp(r), optimized through minimizing the cross-entropy loss to extract likelihood functions and posterior distributions, respec- tively. We adopt an information-theonm in decoder performance- quantity we termed information gap A inlio for the two coding hypothe- ses under the theoretical limit of optimal decoders of probabilistic information. This quantity thus measures the expected increase in cross-entropy loss incurred when a decoder is forced to extract probabilistic content that is not actually encoded by the population responses. Although any em- pirical decoder would underestimate the true sensory information content, we posit that the derived theoretical limits would serve as reference points in evaluating the effectiveness of a task design in differentiating probabilistic neural codes. Below, we derive the information gap under each of the two probabilistic coding hypotheses.
approach to derive the expected cross-entropy difference
Information gap for likelihood coding hypothesis A Given discretized sensory observations {z,), }. task design specified by (p(c).p((() Ve E (A, and a generative model p(z,(0), the expected difference in cross-entropy loss between optimal likelihood and posterior decoders, or the information gap into for likelihood-code population rt. p(x16), is derived (see Appendix A.I for full derivation):
L = ep(an.e (p |l (0))
|l (p) p(c = A) Σ plz, p(z, (1)))))p(0))
D (0) 11 (p) p(c B)
Σ p (1 (0)}
where (0]2z) is the true posterior given observation x, which is the output of an optimal likelihood decoder. The surrogate posterior (f). which is the output of an optimal posterior decoder on likelibood-coding populations, is given by:
Ip(c (0) + plc B)p p(z,10) (0) Es. fpp(c (0') + p(c = B)p (0)))0(p))))))) p(x,(0)}
Since likelihood codingg populations TL contain no prior information, an optimal posterior decoder trained on such population cannot perfectly decode the posterior distribution. Instead, output of the optimal posterior decoder converges to a Bayes-optimal estimator as determined by marginalization over context distributions p(c) and (0).
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Information gap for posterior coding hypothesis Amp Given discretized sensory observations I E [, a task design specified by (p(c). p(())) Vc E (A. and a generative model p(z, the expected difference in cross- cross-entropy loss between optimal likelihood and posterior decoders, the information gap Apto for a posterior-coding population rp ~ p(0]ær), is derived as (see Appendix
A. full derivation):
Ap inlo = Ep(ene) (PxL( || [T,
Σ (173)7)
xz)) (0)) p(c A) [ ((2x)0))AA(0)
- +DKL(P (0 ll (0) p(c= = B) 100[p(ann p( (012=10)pp110 (01}}
B
(3)
where (0)xz;) is the true posterior given observation z,, which is the output of an optimal posterior decoder. (i,, denotes surrogate posterior which is the posterior distribution associated with the output of an optimal likelihood decoder. The sum in Eq. 3 includes only pairs (2), that satisfy the condition expressed below in Eq. 4 as they are the only observations that would yield non-zero decoder performance difference. These are scenarios where identical population responses Tp (encoding the same posterior across the two contexts e E {A, B}, i.e., TP. & B must map to different likelihood functions (() and p(xx|0), respectively), preventing the optimal likelihood decoder from achieving perfect decoding. Observation pairs that do not satisfy Eq. thus has no contribution to the sum in Eq. 3 The condition is given as:
Ve, p (0]zz) = =p0(00m))) + Vo, pa (0)) 10) x p (0) p( 0)
(4)
With this, the surrogate posterior distributions for the pair (æy, 2k) are given by:
( = 5xx (0)) 3 (0)
Eg (0)
( = z (7
where (ix (0) denotes the output of the optimal likelihood decoder on the posterior-coding popula- tion, approaching a masginaagina,ized, Bayes-optimal estimator of the likelihood functions given by Eq. below. Z L (0) and z L5xx (0)] are normalization constants dependent on E5x (0), defined
as [0)] = Eo p" (0) (e))) (0) and 2 [L) (0)] = Eo RB (0)
The Bpyes-optimal likelihood function estimator E (0) can be determined (up to a multiplicative constant) by solving the following implicit equation using fixed-point iteration (see A.T for detail):
P3 (0 p p7 (0) PA (0)
(0) x
(5)
pf T. (0) pB (())
where oz and of denote the frequencies of each context conditioned on observed neural population
responses of r.j. or vP.k- Let us first define StA == plc = A,E, ( (0))((111)) and se = plc
B) Ee pd (0)p((x)))) Then the context frequencies P3 and p. are given by:
= A/r = rp. vipA = 1S, +S _ = p(c = B/r = vTP.)) = S +S = =
In summary, our information -thooreetcctto framework quantifies the differentiability of the two proba- bilistic coding hypotheses under a given task design by deriving analytic expressions of the informa- tion gap the expected difference in decoder cross-entropy performances- for both likelihood coding hypothesis (Eq. [ and posterior coding hypothesis (Eq. (3). The key insight stems from iden- tifying the formula for the task mask-margnnaliedd, Bayes-optimal estimators when decoding mismatched probabilistic information that is, when decoding the posterior from likelihood-coding population (Eq. 2 or when decoding the likelihood function from a posterior-coding population (Eq. 5)
Below, we empirically validate that information gaps accurately predict decoder performance dif- ferences under given task designs simulated likelihood posterior encoding neural populations across diverse task settings. We then demonstrate how maximizing the information gap enables targeted experimental designs that optimally differentiate the two probabilistic coding hypotheses.
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3 SIMULATION EXPERIMENTS
To validate that the information gap accurately predicts decoder performance differences under both probabilistic coding hypotheses, we conducted comprehensive simulation experiments. We con- structed synthetic likelihood-coding and posterior-coding neural populations, and applied likelihood and posterior decoders on these synthetic populations. These simulations serve two complementary purposes: validating our theoretical framework and providing practical insights into the scaling and convergence behavior of the information gap measure.
Task design: Gaussian context priors We consider Gaussian context priors motivated by clas- sic orientation-basel discrimination experiments (Orbán et al. 2016 Walker et al.) 2020). In this paradigm, subjects perform orientation discrimination task with two contexts E B), with the context for each session sampled randomly, i.e. p(c = A) = p(c = B) = 0.5. Within each ses- sion, the trial-to-trial hidden world state 0 (i.e. orientation) is drawn from context-specifie Gaussian prior distributions (0) = N(u", {))) where je and (a are task-specific parameters. In the simulation, we use identical variances for the two Gaussian priors o = σ.
We simulate noisy sensory observations I by drawing from the conditional distribution defined by a given generative model p(z]()) This stochastic process can be seen as capturing both intrinsic neuronal noise and extrinsic uncertainty in stimulus features. This generative model can be ex- perimentally manipulated by varying stimulus parameters such as contrast. Indeed, lower contrast induces increase in observation variance, reflecting increased sensory uncertainty. In the simulation, p(x[(0) is modeled as Gaussian distributions to reflect Gaussian- orientation tuning curves com- monly observed in V1 neurons and to capture the effect of different contrast levels by systematically varying the standard deviation (Walker et al. 2020).
For simulated population responses, we implemented Poisson neuron models with Gaussian tun- ing curves (Walker et al.] 2020). Likelihood-coding population's 's mean firing rates TL are encoded through Gaussian tuning curves based the sampled sensory observations 2, while posterior- coding population's mean firing rates rp are additionally modulated by the context-specific prior (0), effectively encoding the posterior o (0|zr) x p(x²(0)= (0). In both cases, spike counts were then generated by sampling from Poisson distribution with the given mean firing rate. We addition- ally considered more complex, bio-realistic gain-modulated Poisson neuron model for simulating population responses Goris et al.] (2014). As shown in Fig. _ deep neural networks are trained with cross-entropy loss serve as flexible, powerful decoders of probabilistic distributions, decod- ing either the likelihood function or the posterior from the simulated neural population responses (Walker et al. 2020). See A.3 for full details of the simulation experiments and decoder setups.
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Figure 3: Decoder performance difference on simulated populations converges to the theoret- ical prediction of information gap. A) On simulated neural populations encoding the likelihood function (left, blue) or the posterior distributions (right, orange) responding to high contrast stimuli, the difference between the likelihood and posterior decoder performances converges to the theoret- ical value of information gap (dashed lines) as the total number of trials increases (top, with fixed number of neurons 500), and the total number of neurons in the population increases (bottom, with fixed number of trials = 30k). (shaded areas denote the s.t.d. across 5 random seeds.) B) Same for medium contrast stimuli and C) for low contrast stimuli.

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Scaling and convergence We first examine the scaling and convergence properties of the theo- retical prediction of information gap (see additional analyses and ablation studies in [ and A.7). Fig. _ demonstrates convergence of the empirical decoder performance differences on simulated Poisson neural populations across various stimulus contrast levels. For a given set of task param- eters, decoder performance differences for both simulated conulaaioon-iiellhooooe and posterior-coding (orange) populations-rapiic tion gap (dashed lines) computed via Eq. [ and as the number of trials increases (top) and as the number of neurons increases (bottom). This empirical convergence suggests that the information gap measure derived from our framework accurately predicts the asymptotic decoder performance difference quantifying the effectiveness of a task design.
(blue)
converge to the theoretically derived informa-
Validation across parameter space We next assess the validity of the theoretical prediction of information gap across wide range of simulation settings. Across different levels of stimulus con- trast, at least ten different sets of task parameters are selected to compute the theoretical value of information gap and to simulate likelihood and posterior encoding populations. Fig. 3 systemati- cally compares the thcoretical predictions of information gap and the empirical decoder performance difference across diverse task design parameters, both under the Poisson neural model (top) and un- der the more complex, gain-modulated Poisson neural model (bottom. Goris et al. (2014)). On both types of simulated neural models and across different contrast levels, the comparison reveals markable agreement between the information gap prediction and the empirical decoder performance difference for both likelihood and posterior coding hypotheses.
Notably, information gaps for likelihood-coding populations (Ain exceed those for posterior- coding populations (A by up to an order of magnitude. Our framework provides an intuitive explanation: for likelihood coding hypothesis, every observation contributes to the information gap calculation, whereas for posterior coding hypothesis, only pairs satisfying Eq. 4 contribute to the estimate. This asymmetry suggests that distinguishing posterior-coding populations presents greater experimental challenges, requiring careful task design to achieve sufficient statistical power.
Overall, these simulation results establish that our information-theoreer framework accurately pre- dicts decoder performance differences for neural populations following either probabilistic coding hypothesis, providing a quantitative foundation for designing targeted, theory-driven experiments to differentiate probabilistic neural representations in early sensory areas.
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Figure 4: Information gap accurately predicts decoder performance difference on simulated populations across diverse task settings. A) On simulated Poisson neural populations responding to high (left), medium (middle), and low (right) contrast stimuli, theoretical values of information gap (x-axis) accurately predicts the decoder performance difference on simulated neural populations (y-axis) across multiple task design parameters, for both the likelihood-coding and posterior-coding populations. (Fach color marks one set of task parameters used for both types of simulated popula- tions; Error bars denote the s.t.d. across random seeds.) B) Same for simulated populations using a more complex, bio-realistic gain-modulated Poisson neural model (Goris et al. 2014).
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4 TASK OPTIMIZATION TO DIFFERENTIATE PROBABILISTIC NEURAL CODES
Given the strong agreement between the empirical decoder performance differences and the theo- retical information gap measure, we now demonstrate how to optimize task designs to maximally differentiate the two probabilistic coding hypotheses. The goal systematically explore the task parameter space to identify task parameters that would yield maximum information gap.
4.1 INFORMATION GAP LANDSCAPE FOR GAUSSIAN CONTEXT PRIORS
For tasks with Gaussian context priors, we evaluate the information gap across the two-dimensional task parameter space defined by (1) the distance between the two Gaussian means d | 1l and (2) the shared standard deviation for both Gaussian priors (Fig. The landscapes of information gap across three different contrast levels are shown Fig. 5 for both likelihood-coding populations (top) and posterior-codee
populations (bottom). We first observe that the information gap landscape depends on the stimulus contrast level, suggesting that experimental design should be tailored specific stimulus features such as contrast. In addition, decreasing contrast expands the parameter region yielding substantial information gaps for both probabilistic codes, agreeing with the intuition that prior information becomes more influential when sensory observations alone provide insufficient information for reliable inference.
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Figure 5: Information gap landscapes inform practical task designs that optimally differentiate probabilistic representations neural populations. A) Information gap as function of task parameters (d: separation between context priors, and context prior standard deviations) for both the likelihood coding hypothesis (top) and the posterior coding hypothesis (bottom) when presented with high contrast stimuli. The asterisks identify strategic task designs that achieve the tradeoff where posterior-codie codinng information gap approaches its maximum while likelihood cooiinnddddinggggooo maintains sufficient discriminative signal. B) Same for medium contrast stimuli and C) for low contrast stimuli.
Strategic task design Crucially, for given contrast level, the information gap landscape depends on the underlying probabilistic coding hy hypothesis. revealing a trade-off when optimizing experi- mental design: task parameters that maximize the discriminability for likelihood-codii populations diverge from those optimal for posterior-coding populations. This divergence necessitates a strate- gic selection of parameters that balance discriminative power across both hypotheses. Considering the notable asymmetry information gap magnitudes-with posterior-coding values typically an order of magnitude smaller than likelihood-coding ones, one might prioritize parameters that max- imize cysterior-cdding discriminability while maintaining adequate likelihood codinng sensitivity. The asterisks in Fig. ideettiy such strategic "sweet spots" where cesterior-coding information gap Ajro info approaches its maximum while iiklihood-coding information gap A maintains sufficient discriminative signal. For low contrast stimuli, such optimization occurs with prior separation of
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d = 30" and standard deviation of a z 20". As contrast increases, the optimal task parameters shift toward smaller prior separations and narrower standard deviations.
4.2 INFORMATION GAP LANDSCAPE FOR NON-GAUSSIAN CONTEXT PRIORS
We next explore the feasibility of using other types of stimulus context prior distributions. Specif- ically, we test the effectiveness of heavy-tailed priors such as student's s t-distribution and Cauchy distribution (see AseeAA.9for for analysis on thin-tailed priors). Fig. @hshww the information gap landscape under medium contrast using the student's 's t-distribution (top) or the Cauchy distribution (bottom) as stimulus priors. Compared to Gaussian priors, areas with substantial information gap become more limited. In particular, posterior- information gap is zero almost throughout the entire param- eter space, indicating that heavy-tailed priors are not suitable for distinguishing posterior-coding populations. Our theoretical framework provides explanation: under heavy-tailed priors, there are barely any observation pairs satisfying Eq. 4 that would contribute to the information gap (See A.8 for details). Finally, almost no overlap exists between areas where A info for each coding hy- pothesis is maximized, suggesting that any choice of task parameters optimal for identifying one hypothesis will necessarily sacrifice the effectiveness of identifying the other. Overall, this analysis suggests that heavy-tailed priors are not ideal for differentiating probabilistic coding hypotheses.
In summary, our framework transforms parameter selection from heuristic search principled opti- mization, directly identifying task designs that maximize statistical power for differentiating proba- bilistic neural representations. The resulting information gap landscapes can guide targeted experi- ments toward parameter combinations most likely to yield decisive empirical results.
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Figure 6: Information gap landscape suggests heavy tailed distributions are not ideal stimulus prior distributions for differentiating coding hypotheses. A) Using student's t-distribution with degrees of freedom a function task parameters (separation d and standard deviations σ) for both the likelihood coding hypothesis (middle) and the posterior coding hypothesis (right) shows decreased information gap with minimal overlap compared to task design with Gaussian context priors. B) Same for Cauchy distribution stimulus priors with task parameters separation d and scale 7.
3 as stimulus priors (left), information gap under medium contrast stimuli as
EMPIRICAL RESULTS ON NEUROPHY DATA
Distinguishing the two probabilistic coding hypotheses depends on how population responses change across contexts with different priors, yet existing datasets mostly provide only single fixed stimulus context without manipulation in priors. To demonstrate that existing neurophysiol- ogy datasets with single-context experimental designs cannot adjudicate the two coding hypotheses, we report empirical results on orientation decoding from real neural data using the Allen Brain Observatory Visual Coding Neuropixels Dataset (Siegle al. (021). Under such single-context experimental design with uniform prior, our theory predicts no performance difference between the likelihood and posterior decoders, i.e. A 0. In Fig. 7 performed orientation decoding

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analysis the Allen Visual Coding dataset. The result shows indistinguishable performance be- tween the likelihood and posterior decoders (difference = 0.0024 0.064, p = 0.63), which agrees with our theoretical prediction. In fact, previous decoding work macaque V1 similarly discussed why their experimental design resulted ambiguity in differentiating coding hypotheses due to the lack of multiple context priors (Walker al., (020). This result on empirical data underscores why future experiments incorporating context-dependent prior manipulations will be essential for adjudi- cating the competing probabilistic coding hypotheses and how our proposed targeted experimental paradigm could help optimize the task design for such experiments.
Allen Brain Visual Coding Dataset
Figure 7: Decoding analysis on the Allen Visual Coding datasets (Siegle et al., 2021) shows indistinguishable decoder performance difference. Across 169 sessions with large enough trials (> 300 trials), the decoder cross-entropy performance difference (likelihood decoder- posterior decoder) is 0.0024 + 0.064, which not significantly different from the model prediction of 0 (p = 0.63) (Each dot indicates one session.). This empirical result highlights the necessity of the context-dependes prior manipulation for distinguishing probabilistic coding hypotheses.
Decoder performance dift. (nats) [Likeilhood posterior decoder)
Date: 0.002420.064
value: 0.63
0.00
-0.15
Theoretical prediction
6 DISCUSSION AND CONCLUSIONS
We presented an information-thoon thecrretic framework for optimizing experimental design to address whether early sensory neural populations encode likelihood functions or posterior distributions. We derive analytic expressions for information gap-the the expected decoder performance difference when extracting mismatched probabilistic content. This measure quantífies how effectively experimen- tal design can distinguish between competing probabilistic coding hypotheses, providing precise predictions validated through extensive simulations. Most critically, maximizing the information gap yields principled experimental designs that can optimally discriminate between probabilistic neural codes, enabling decisive experiments to resolve a fundamental debate about Bayesian com- putation in the brain. By developing theoretical framework to quantify how well experiments can distinguish between competing coding hypotheses, this approach demonstrates how computational theory can directly guide experimental neuroscience.
Although our approach follows the popular ideal observer framework for studying perceptual decision-making under uncertainty (Ma et al.] (006] Beck et al. [2008] Walker et al. (022)). it could naturally be extended to incorporate imperfect priors given empirical psychophysical results. The imperfect or biased prior of subject can be inferred by analyzing their psychophysical curves, which can subsequently be used in place of the ground-trume information gap (See the detailed procedure in A.4 and Fig. [10). Furthermore, it should be noted that likelihood and posterior coding are not the only relevant theories to be considered, but they can be regarded as two extremes that differ in the probabilistic quantity encoded by the sensory population. Consequently, mixed or intermediate hypotheses that lie somewhere between the two canonical hypotheses may be proposed, where the neural population encodes some mixture of like- lihood and posterior (for example. Ganguli & Simoncelli (2010)). In A.5 and Fig. [T] we discuss how our framework and optimized tasks can be extended discriminate more nuanced, mixed cod- ing hypotheses. By optimizing task parameters to maximally separate the canonical hypotheses, we simultaneously maximize sensitivity to discriminating more nuanced probabilistic coding theories.
prior to adjust the calculation of the
Scope and limitations To compute information gap, our framework requires reasonable genera- tive models and thus may require prior work establishing neural response properties. In addition, the decoding approach requires sufficient neural population response data for training. Our framework also provides a foundation that can be extended in several directions: 1) The framework extends be- yond orientation- basod stimuli to continuous observations and other types of distributions through numerical methods; and 2) Incorporating more bio-realistic neural models such as noise correlations and nonlinearities would further strengthen predictions.
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Reproducibility statement The source code for computing the information gap, implementing both likelihood and posterior decoders, and running all simulation experiments available at The detailed derivation of information gap is presented in [ The details of simulation experiments including synthetie neural population response generation procedures, deep neural network decoder architectures and hyper- parameters, and training procedures are presented in A.3
https://github.comm
thank Patrick Zhang and Suhas Shrinivasan for their insightful feedback and discussions. We thank Daniel Sitonic for the technical support. We gratefully acknowledge the community support from the University Washington Computational Neuroscience Center and the Allen Institute for Neural Dynamics. We also thank the anonymous reviewers for their valuable and constructive review.
Acknowledgements
REFERENCES
Laurence Aitchison, Jannes Jegminat, Jorge Aurelio Menendez, Jean-Pascal Pfister, Alexandre Pouget, and Peter E Latham. Synaptic plasticity as bayesian inference. Nature neuroscience, 24(4):565-577) 2021.
David Alais and David Burr. The ventriloquist effect results from near-optimal bimodal integration. Current biology, 14(3):257-262, 2004.
Jeffrey M Beck, Wei Ji Ma, Roozbeh Kiani, Tim Hanks, Anne K Churchland, Jamie Roitman, Michael Shadlen, Peter Latham, and Alexandre Pouget. Probabilistic population codes for bayesian decision making, Neuron, 60(6): 142-1112, 2008.
Pietro Berkes, Gergö Orbán, Máté Lengyel, and József Fiser. Spontaneous cortical activity reveals hallmarks of an optimal internal model of the environment. Science, (6013))83338 87, 2011.
Pierre Simon de Laplace. Théorie analytique des probabilités, volume 7. Courcier, 1820.
Mare O Ernst and Martin Banks. Humans integrate visual and haptic information in statistically optimal fashion. Nature, 41 415(6870):429-44 433, 2002.
Dylan Festa, Amir Aschner, Aida Davila, Adam Kohn, and Ruben Coen-Cagli. Neuronal variability reflects probabilistic inference tuned to natural image statistics. Nature communications, 12(1): 3635, 2021.
József Fiser, Pietro Berkes, Gergö Orbán, and Máté Lengyel. Statistically optimal perception and learning: from behavior to neural representations. Trends in cognitive sciences, 14(3):119-130, 2010.
Deep Ganguli and Eero Simoncelli. Implicit encoding of prior probabilities in optimal neural popu- lations. Advances neural information processing systems, 23, 2010.
Deep Ganguli and Eero P Simoncelli. Efficient sensory encoding and bayesian inference with het- erogeneous neural populations. Neural computation, 26(10):2103-2134 2014.
Robbe LT Goris, Anthony Movshon, and Eero P Simoncelli. Partitioning neuronal variability. Nature neuroscience, 17(6):85-8665, 2014.
Agnieszka Grabska-Barwinska. Jeff Beck, Alexandre Pouget, and Peter Latham. Demixing odors- fast inference in olfaction. Advances in Neural Information Processing Systems, 26, 2013.
Ralf M Haefner, Pietro Berkes, and József Fiser. Perceptual decision- probabilistic infer- ence by neural sampling. Neuron, 90(3):649-660, 2016.
Ralf M Haefner, Jeff Beck, Cristina Savin, Mehrdad Salmasi, and Xaq Pitkow. How does the brain compute with probabilities? arXiv preprint arXiv:2409.02700 2024.
H von Helmholtz. Versuch einer erweiterten anwendung des fechnerschen gesetzes im farbensystem. 7. Psychol. Physiol. Sinnesorg, 2:1-30, 1891.
11

---

Published as a conference paper at ICLR 2026
Patrik Hoyer and Aapo Hyvärinen. Interpreting neural response variability monte carlo sampling of the posterior. Advances neural information processing systems, 15, 2002.
Mehrdad Jazayeri and J Anthony Movshon. Optimal representation of sensory information by neural populations. Nature neuroscience, 9(5):690-696 2006.
Daniel Kersten, Pascal Mamassian, and Alan Yuille. Object perception as bayesian inference. Annu. Rev. Psychol., 55(1):271-355 2004.
David C Knill and Alexandre Pouget.
The bayesian brain: the role of uncertainty in neural coding and computation. TRENDS in Neurosciences, 27(12):122-71, 2004.
David C Knill and Whitman Richards. Perception as Bayesian inference. Cambridge University Press, 1996.
Konrad P Körding and Daniel M Wolpert. Bayesian integration in sensorimotor learning. Nature, 42776971) 2004.
Richard D Lange and Ralf M Haefner. Task-induced neural covariability signature of approxi- mate bayesian learning and inference. PLoS computational biology, 18(3):e 2022.
Richard D Lange, Sabyasachi Shivkumar, Ankani Chattoraj. and Ralf M Haefner. Bayesian en-
coding and decoding as distinct perspectives on neural coding. Nature neuroscience, 26(12): 2063-2072 2023.
Jeremy Lewi, Robert Butera, and Liam Paninski. Real-time adaptive information- opti- mization of neurophysiology experiments. Advances in Neural Information Processing Systems, 19, 2006.
Jeremy Lewi, David M Schneider, Sarah MN Woolley, and Liam Paninski. Automating the design of informative sequences of sensory stimuli. Journal of computational neuroscience, 30(1):181- 200, 2011.
Wei Ma and Mehrdad Jazayeri. Neural coding of uncertainty and probability. Annual review of neuroscience,,7( 37(1):205-220 2014.
Wei Ji Ma, Jeffrey M Beck, Peter E Latham, and Alexandre Pouget. Bayesian inference with prob- abilistic population codes. Nature neuroscience, 9(1)):432--4438 2006.
Christian K Machens, Tim Gollisch, Olga Kolesnikova, and Andreas VM Herz. Testing the effi-
ciency of sensory coding with optimal stimulus ensembles. Neuron, 47(3) 2005.
Robert Madigan and David Williams. Kikiimum-lkkelhhowe psychometric procedures in two- alternative forced-choice: Evaluation and recommendations. Perception & Psychophysics, 42 (3):240-249, 1987.
Gergö Orbán, Pictro Berkes, József Fiser, and Máté Lengyel. Neural variability and sampling-base probabilistic representations in the visual cortex. Neuron, 92(2):530-543, 2016.
Adam Paszke, Sam Gross, Francisco Massa, Adam Lerer, James Bradbury, Gregory Chanan, Trevor Killeen, Zeming Lin, Natalia Gimelshein, Luca Antiga, et al. Pytorch: An imperative style, high- performance deep learning library. Advances in neural information processing systems, 32, 2019.
Alexandre Pouget, Jeffrey M Beck, Wei Ji Ma, and Peter E Latham. Probabilistic brains: knowns and unknowns. Nature neuroscience, 16(9):1 170-11788 2013.
Ahmad T Qamar, R James Cotton, Ryan G George, Jeffrey M Beck, Eugenia Prezhdo. Allison Lau- dano, Andreas S Tolias, and Wei Ji Ma. Trial-to-trial, uncertainty based adjustment of decision boundaries in visual categorization. Proceedings of the National Academy of Sciences, 110(50): 20332-20337, 2013.
Allan Raventós, Mansheej Paul, Feng Chen, and Surya Ganguli. Pretraining task diversity and the emergence of non-bayesian in-context learning for regression. Advances neural information processing systems, 36:14228. 10fe14228-1444 14246, 2023.
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Daniel B Rubin, Stephen D Van Hooser, and Kenneth D Miller. The stabilized supralinear network: a unifying circuit motif underlying multi-input integration in sensory cortex. Neuron, 85(2):402- 417, 2015.
Sabyasachi Shivkumar, Richard Lange, Ankani Chattoraj, and Ralf Haefner. A probabilistic pop- ulation code based on neural samples. Advances in neural information processing systems, 31, 2018.
Suhas Shrinivasan, Konstantin- Keesman Lurz, Kelli Restivo, George Denfield, Andreas Tolias, Edgar Walker, and Fabian Sinz. Taking the neural sampling code very seriously: A data-driven approach for evaluating generative models the visual system. Advances in Neural Information Processing Systems, 06:21945-211555 2023.
Joshua H Siegle, Xiaoxuan Jia, Séverine Durand, Sam Gale, Corbett Bennett, Nile Graddis, Greg- gory Heller, Tamina K Ramirez, Hannah Choi, Jennifer A Luviano, et al. Survey of spiking in the mouse visual system reveals functional hierarchy. Nature, 592(7852):86-92, 2021.
Edgar Y Walker, R James Cotton, Wei Ji Ma, and Andreas S Tolias. neural basis of probabilistic computation in visual cortex. Nature Neuroscience, 23(1):122- 202..
Andrew B Watson and Denis G Pelli. Quest: A bayesian adaptive psychometric method. Perception & psychophysics, 33(2):1 13-120, 1983.
Yair Weiss, Hero Simoncelli, and Edward Adelson. Motion illusions as optimal percepts. Nature neuroscience, 5(6):598-604, 2002.
Tianming Yang and Michael N Shadlen, Probabilistic reasoning by neurons. Nature, 447(7148): 1075-1080, 2007.
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A TECHNICAL APPENDICES AND SUPPLEMENTARY MATERIAL
A.1 INFORMATION GAP DERIVATION
Consider generative model of sensory observations + s; where is the noisy sensory obser- vation (e.g. a drifting grating stimulus) generated according to the conditional distribution p(r (f), where 0 is the hidden state of the environment (e.g. true orientation of the drifting grating stimulus). Note that the likelihood function is given by [()) = p(r[0) for specific observation æ. Consider an experimental setup where there are two possible stimulus generation contexts: c = (A, B) with their associated context-specifg latent priors p" (0) = p(0cc) and their context frequencies p(c).
Given a sensory observation x and context c, the context-dependent posterior distribution of 0, denoted as (0 ) = p(0|zz c), is given by the Baye's rule:
p'(0)x) p² 20,
(x(x)
(x) (0)
Since the generative process 0 - x is independent of
of (x((0')) p p0(0')
p(z|0) (0)
Ev r p(z(0') (0')
x p(x 0) (0)
For given neural population response vector r, consider two competing probabilistic coding hy- pothesis:
1. Likelihood coding hypothesis: TL p(z)0), where the neural population responses rL is hypothesized to encode the likelihood function of the stimulus p()
2. Posterior coding hypothesis: rp (())))) where the neural population response rp hypothesized to encode the posterior distribution of the hidden state given the stimulus p(0[x).
We consider whether it is possible to differentiate the probabilistic information content encoded in given neural population responses through a decoding approach. Intuitively, if a neural population encoding the likelihood function, then decoder decoding the likelihood function should lead to better performance then a decoder decoding the posterior distribution; vice versa if the neural pop- ulation is encoding the posterior distribution. In other words, decoder performance degrades when trying to decode mismatched probabilistic content, such that the difference in decoder performance when decoding the likelihood function versus decoding the posterior distribution can be used to dif- ferentiate whether given neural population is encoding the likelihood function (likelihood coding hypothesis) or the posterior distribution (posterior coding hypothesis). Below, we formalize this intuition by deriving the expected decoder performance difference.
Consider applying decoder function g which is optimized decode some probabilistic information content from the neural population responses under cross-entropy loss:
g(r) - p(-) where is a decoder function
Note that to establish the expected difference between decoder performances, we assume ideal de- coders in derivations. Empirically, we assume the decoder is expressive enough (e.g. multi-layer perceptron, MLP) and fully trained, and the data is abundant such that the performance of the de- coder would closely approximate that of the ideal decoder.
Adopting an cntermattionthhoreede approach, our goal to derive the expected difference between decoder performances measured in cross-entropy when decoding the likelihood function versus decoding the posterior distribution from given neural population responses r, a quantity that we termed the information gap, info between the decoders under a given experimental design specified by (p(c), Vc E {Á, B) and a generative model p(= Below we will separately derive the information gap for likelihood coding hypothesis, Annoo and the information gap for posterior coding hypothesis, inno respectively. As by by-product, our -heeorettion-hhorrttee analysis framework also allows for deriving the expected decoder performance for each decoder under the limit of perfect decoding measured in cross-entropy.
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A.1.1 INFORMATION GAP FOR LIKELIHOOD CODING HYPOTHESIS INPO
For likelihood coding population, the neural population responses rt. encode the likelihood func- tion of the sensory stimulus, which are not modulated by and hence independent of the context prior.
f(p(z(0)) where f is some neural encoding function.
Note that since the decoders are optimized under cross-entropy loss:
(pp,)) EE q) Hp)) + (p q)
when q" = p H(p,q") minimized.
Decoding performance of a perfect likelihood decoder 9L
Applying a likelihood decoder gr to likelihood coding population TL, we want
(T - p))
Let us assume the observation space can be discretized into E {z;}, and consider the neural population responses associated with each Ny;:
Vz,, c TL; = TL. f(p(x,10)
Since TL, is context-inde innieettiideeeneett, let us denote the likelihood decoder output (r(,)) = Ir (TL.)).
Since ground truth context prior (0) is provided the likelihood decoder gr. as schematized in Fig. 2 with the likelihood decoder output 91 (rL.s) and the corresponding context prior pt (0), the context-dependent decoded posterior distribution L., (f) is given by:
(0) .L. gr. (Pt..) p (0), where Vi. a normalization constant.
The entsopy loss for data samples associated with F,, i.e. (8)z.)) is minimized when:
(0) = 1d (0(xz))
mi.; (7 (0) p(2;(0) (0)
p(x;)
D2 (r1.)) LL p(x; where az, is constant
(6)
That is, after training, the likelihood decoder output 9L (TL.:) will converge to gf (rL.s) x p(x;10) given enough samples.
To get the expected cross-entropy loss across the entire data set, we marginalizing over all i; c, and the expected cross-entropy loss for perfect likelihood decoder can be evaluated as: ( (p" xz), (E
Ep(as,,))
= Ep(r.,)) [ (p)(())) + (0(xa. |l (E)(
= Ep(zi, (Op"(()))))
Σ (p x0)x)) p(z,,c)
Σ ((p plc) Σ (((0))0)00))
Σ H(p)(0)z))) p(c)
Σ p(a:10)p
Σ [((A(())i))) p(c = A)
Σ p(a, ((a,00)p400
H(pB((0)x:)) p(c = B) Σ pl p(x100p=(0))
where the second equality holds because (K || ()) = for a perfect likelihood
decoder as derived above. That is, the expected cross-entropy loss for perfect likelihood decoder should approach the expected posterior entropy as determined by the context frequencies and context prior distributions given by Eq.
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Decoding performance of the best possible posterior decoder 9P
Applying a posterior decoder gp to a likelihood coding population TL, we want:
gp(rr.) - p (0((z)
However, since there is no context information encoded in the population responses TL. the posterior
decoder 9P cannot achieve the same performance as the likelihood decoder in Eq. 7 as there are scenarios where identical inputs (ri.)) are trained to map different outputs (p"(#) (x)) depending on the inaccessible ground-truth context information p0)..
Let us consider the neural population responses associated with each observation at::
VI;, c: L., LL f(((z;10)
The frequency of a context given the observation of data samples associated with x, is given by:
p(c| = zz)) = plc,
p(x,)
p(c) p(a,(c)
Ee p(c') p(xx(c")
p(c) Eo (0) p(x,10)
Ee p(c') Ee of (0) p(x,
Let us denote
5/ = p(c = A) Σ p (0)p(z:10)
s = plc B) Σ p (0)p(z,1))
Hence, we can define the observation-dependent context frequency for given as:
p¹ah = p(c Alz = x;) = S/A (( + + 5
p.¹h = : B|x= - /5, + S
Now, let denote the posterior decoder output (0) = gr(r..) highlighting that the output can be interpreted directly as the posterior distribution over the hidden state 6, as schematized in Fig. 2 Since the posterior decoder output is agnostic the specific context and the associated prior p under cross-entropy loss, (0,(0) is trained to minimize the expression below:
min (P)
Ep(cla)) (0)) qp.))))
min (P,)(0)
p H( (z) gr. Hh (())))
min
(pppA(0l log p (0) log (0)]
(0)
min (P.(
+ +0 log (0)}
Since P (0 and p (0)z,) are both probability distributions over 0, and pe + o 1, the expression of (0)2z))) + B (0)az)) represents proper probability distribution over 0. Therefore the loss above minimized when:
(0) = pAPn* (0)m:) + P PP
S
p (0)))((2)10)
S
P 1 (0))((z;)))
S + S.f Eor<(0)p(z.11)
SA SB E. 2 (0)p(z,0
lp(c= = A)p B)p p(x
Ip(c= = A)p +(c= B)p (p) 1(r
= (p 1')
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That is, after training, the best possible posterior decoder output for data samples associated with Dis ie. (0), is as if the decoder were to use a surrogate prior:
(0) - p(c= (0) + p(c (0)
(8)
which is the task- mask-marinnalized, Bayes-optimal estimator of the prior distributions over 0 across contexts c {A, Interestingly, this surrogate prior distribution is independent of z;,
Since likelibood-coding populations rL contain no prior information p0))), a posterior decoder trained on such population responses cannot perfectly decode the posterior distribution. Instead, the posterior decoder output converges to a Bayes-optime estimate of context-dependent posteriors determined by the context distributions p(c) and p0). To obtain the expected cross-entropy loss across the entire data set, marginalize over all Ti G, yielding:
Ep(ms,e) (p [z)), (0)] e [ (p" (0)z4) + DxL (p (0)x.) (0))
ep),,.) [M] (p" (0]x,))) ep(,,))) 1 (p (0)x;) ) (0))
CE loss of the perfect likelihood decoder (Eq.
Eptas
(p
(0))
Information gap for likelihood coding population AL
From Eq. 9 let us define Ainoo, the information gap between perfect likelihood decoder (gi) and the best possible posterior decoder (gp) applied on likelihood-coding population, evaluated as the expected difference in cross-entropy loss between the two decoders:
info = Ep(on.o) (F ll (0)(
Σ DKL 11 (b)) p(.,,c)
Fi.c
DKL (v'(0a0) (0)) p(c) Σ p(1
Σ
Σ DKL (0) p(c) Σ p(z;10)p((0)
Σ (x |l (p) p(c = A)
Σ p(zz 1(2)
8
DRL (p (f) (0) p(c B) Σ p(z:10)p
(10)
Eq. I0 provides an analytic expression for the information gap for a likelihood population under task design specified by ()()).((()) and a generative model p(x:()). Per observation æ, the expression evaluates the KI. divergence between the true posterior (f( (r)..) and a surrogate posterior (O), which is the output of the best possible posterior decoder utilizing the mask-narginl Bayes-optimal estimator of the prior distribution (Eq. (). The KL divergence is then marginal- ized across 2; derive the total expected performance difference between likelihood decoders and posterior decoders.
A.1.2 INFORMATION GAP FOR POSTERIOR CODING HYPOTHESIS A INFO
For posterior coding population, the neural population responses rf encode the posterior distribu- tion over given æ under the context c, i.e. (0)xx), and are therefore modulated by and dependent on the context prior (0):
r (f)=)) where is neural encoding function.
Decoding performance of perfect posterior decoder gr
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Applying posterior decoder 9P posterior-coding population Tp, want
gP(Tr) - (f12z)
As before, let us assume the observation space can be discretized into æ E {2t). and consider the neural population responses associated with each Ię:
Vx e: rp.; (((=)))
We denote the output of posterior decoder (0) == gr(rp,), which is context-dependent rē. depends the context As the output of the posterior decoder (f, can be directly
interpreted as the posterior distribution (schematized in Fig. ( the cross-entropy loss for data samples associated with 22, C, i.e. H (y (0lax)) fr, (0)), is minimized when:
gr. (0) p" (0/z,)
That is, after training, the posterior decoder output gr(rf,) will converge to (0) = p²(0 provided sufficient training samples available.
To ohtain the expected cross-entropy loss across the entire data set, we marginalize over all 24 yielding:
ep (p' (0)] pp (O(((())))) + (ea (f))
E E (0(=(());;
Σ ((p)(0)2;)) p(,)))
Σ ((0((()2))) p(c) Σ p(2a)0)p
H(p p(c) Σ [p(E:(0))((0)
Σ ((px(() p(c A) Σ 1(z
22
H(p (fl=,) p(c= B) Σ p(2,10)p
(11)
where the second equality holds because DKL (p xx) || 4r, (0) = 0 for a perfect posterior decoder as derived above. Hence, the expected cross entropy loss for perfect posterior decoder on a posterior coding population should approach the expected posterior entropy as determined by the context frequencies and context prior distribution as given by Eq. [1] Note Eq. [1]is is the as the expected cross-entropy loss for perfect likelihood decoder on a likelihood coding population as derived previously in Eq. 7
Decoding performance of the best possible likelihood decoder 9L
Applying a likelihood decoder gr to a posterior coding population Fp. we want gL(rr) - p(x(0)
In contrast to the mismatched decoding scenario of applying posterior decoder to likelihood- coding population where the posterior decoder cannot perfectly decode the posterior distributions from population responses for any observation x;, application of a likelihood decoder to posterior- coding population requires more intricate considerations-wv reason below that only some v; would cause the likelihood decoder to fail to perfectly decode the likelihood function from posterior- coding population. We first reiterate that the posterior population responses T are context depen- dent, which means that for the same 2,, the neural responses rf,; are different across the two con- texts. Hence, from the perspective of likelihood decoder, for each I; the inputs (neural responses rf,) are different across contexts, but the target output (p(x;(0)) is the same. Because the ground- truth context priors p (0) explicitly provided the likelihood decoder, this scenario "pressures"
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the decoder to learn many-to-one mapping, which is generally achievable for sufficiently power- ful likelihood decoder (Fig. 2C).
To identify the condition in which the likelihood decoder would fail perfectly decode the like- lihood function from posterior coding population responses, recall that when applying a posterior decoder likelihood-coding population, the main reason why the posterior decoder cannot be per- fect is that it is forced to map identical inputs (FL,)) into multiple distinct target outputs (p" (f]x.)). In other words, the decoder cannot be perfect because is trying learn one-to-many mapping. Given this insight, for the scenario of applying a likelihood decoder on a posterior-coding popula- tion, we identify the condition under which likelihood decoders are forced to map identical inputs to distinct target outputs. Consider the set of pairs X = {(z), zz)}. where each pair (Iz, Zx) satisfies:
rpj R P. (010,)) = p (0)0x), vo (can be measured in terms of KL divergence) + (0) 10, x x (0)) p(zx10), vo
B
(12)
That is, we consider the condition re. r where the inputs (rp, or vp..) to the likeli- hood decoder 9r. are (approximately) the same but the target output differs based on the context (p(æ,)0) (0) p(xx(()). Under the assumption of ideal decoders, the set of pairs in X {(rg. 2x)} are the only scenarios where is impossible for an ideal likelihood decoder to be perfect. these scenarios, identical inputs (population responses encoding the same posterior distributions) need to decoded into different outputs (distinct likelihood functions), which is not achievable by any functional decoder, regardless of training sample size expressive of parametrization.
With the insight that only observations in the set of pairs X = (zy, .x) where Eq. 12 is satisfied will cause the likelihood decoder to fail to perfectly decode the likelihood function, let now derive the expected likelihood decoder output for each pair. Firstly, consider the frequency a context given an observation of neural responses associated with rp. r
plc A, = pp. PP.k)
p(c AIr TP.
V VTP.))
p(r = Tp. V P.kk
p(c A) Eop
p(c A) Eo PA (0)((=;(0) + p B) -o EarB (B
Similarly, we have:
p(c= B) En (0)p(z))) (0) p(c= = Br= = rps V vp.) p(c= A) (0)p(2)) + plc = B) (()ppaz)
Let us denote
= plc = A) Σ p (0)p((2,10)
s plc B) Σ p (0))((2))))
Define the observation--deeeddent context frequency for observing data samples coming from rp.
B or T.P.
0 == = A/r = rØ V vTpA)= = S5 + B = p(c B|r = rp V (TR)) = 5 /S,, 5
Now, let us denote the context-independemt likelihood decoder output as (0) = gr = Tp. V "..)). The context-dependent posterior distribution given the corresponding context prior p0( is
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given by:
()(O)Ga(0) (0)
(0) Eer ((0)))((0))
(0)
(0)0)(u(0) (0(0))n(0)
(0) (0) = EΦ ("(()))((()) 2E16((0)
where (0)] and (C are normalization constants dependent on (0), defined
(0²()=(0)] = Σ p (0)Gxa00)
[c)0 = Σ (0)C()))))
Under cross-entropy loss, we want (f) (and hence its associated posteriors ( and (0)) to minimize:
min (0)
(0))
E.)))
(0) PEp (f|zx) log (0)
min (0)
min (0x
{
Σ ps
log pd (0)p(zx))))
p p0(O)(Iz,)0)
P" (0)0)
Σ Rr ('(0))(22)1))
1
(0))
p
log
(13)
Eo (0()')((x))))))
Define
P3 A p (0)p(22,10) Σ. ((0)(((1))))) = p(c- = A)p S S
13(0) = 13
((0))((z)(0)
p(c = B)p B (0)p((2)))))))
"e (0) = (00 = Ph Ee ((()p((Z))))
S/²² S
Note
Σ (o) plc A) Ee E0 P3
S/A¹ + 5
Σ (E p(c= B) Σ o B (0 p(ax)p -P
S/A3 + S
The cross-entropy loss term in Eq, 13 can rewritten as:
(()))(0)) = Σ p (0) (logp^(0) log (0) - log (711
" (0) [oog> (0) + log , (0) - log 2 [Cxx (c)x
1 (0) log A (0) + / log log (0)
Σ (Ms (0) + 1 (0)) log L (0)
+
Σ n² (0)] log 1 (x) - Σ ue (0)
log 2 (011
(14)
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we can that L(af) L(l), Va 0, as the normalization factors cancel out the multiliccative effect. Therefore E" that minimizes L determined up to a multiplicatiie constant, agreeing with our intuition that the output of a likelihood decoder should be only determined up to a multiplicative constant as in Eq. 6
Note from Eq. 14
The above minimization happens the critical point (xx(() where (5 (0)
0, V0, with L defined aL
in Eq. 14
Before proceeding to find the minimum for this variational calculus problem, let us first evaluate:
(l) [0)] (0)x(0) (0) Z [0) = (l) (E (0)
a
Σ p (0))00)(00) -P'()) (=)00))00))==>000
0'
0
To find the minimum, let us take the derivative of I. with respect to lyx (0) and set it to zero:
0 (L))
(lzx(0)
θ (8)
14 log + (2)()) log (logp((8) (0)
Σ [uf (0) 2 (0)) log (x)
Σ n (0) log 2f - 1 MR (0) log 2 (001
(0) 12
Eo
(0) (0)
3
(0)
0lgx(() (0)
(l)
1 (0) + (0)
(00)
( (0) p(0}
Pk
(0)
(
Therefore the minimization happens when (determined up multiplicative constant):
(0) + + ()((0)
(0) x
2 (0) 07 (0)) T
() P php" (8) (K
(15)
(0) x (()
Eq. 15 gives an implicit expression for (0), since both [(]( and 2f105k] depend on (0).
The equation can be solved using fixed-point iteration starting with some initial guess for 4 (0) >
0. For instance:
Initialize ()) x 1
for t 1,
compute z, 2,,100) 2,(00)p(())
(0(0)
Σ (0())((
z BB. [0000((0)
Σ El (0)p (0)
(0 + B (0 m.
update ((++)) jk
(0) オ कل (()))) (0)
Stop when ( (0) converges (up to multiplicative constant).
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That is, given by Eq. 15 after training, the best possible likelihood decoder output for data samples associated with TP.j and Fp. 13 is as the likelihood decoder were to divide surrogate
posterior that is weighted sum of ground-truth posteriors pt 1r (0|zz) p (0/zz) by surrogate
prior that is a weighted sum ground-trume trunc priors 2 (0) + 1 p
(0).
The posterior of the best possible likelihood decoder output gi = i (0) given the corresponding context prior for rp, and Pp. is evaluated as:
(0) (1)0))
15
(0)
(0) Zx [9g
Hence, to obtain the expected cross-entropy loss across the entire data set, we marginalize over all 3y, G, and the total cross-entropy loss for the best possible likelihood decoder can be expressed as:
E cp [H (p (f, ep( e) [H (0)=.)) + Dx
(p' x.) (p
(0)]
Ep(z,,)) [M (0)(0) ) + Ep(r,)) DKL (0) (0)x;) (0))
CE loss for the perfect posterior decoder (Eq,
+ Ep(m..))
(p (0))
(16)
Information gap for posterior coding population Amioo
From equation 16 let us define Ajaoo, the information gap for a posterior coding population between a perfect posterior decoder (g7)) and the best possible likelihood decoder (yī,), as the expected dif-
ference in the cross-entropy loss of the two decoders:
inn
E Eptenee (v 1l (O))
Σ (p (0) p(a,, c), since only x; X ((1,, terms nonzero
Σ DKL ()) p(z,,c)
xxxxcc
E (x (0) p(c) Σ p( (((:0)((())
z,Exic
ΣΣ (D (0lx.) of (0)) p(c) Σ p(a 10)
2,EX
= Σ (p |l ghg (0) p(c = A) Σ p( 10))))00000)
(Dx (f) p(c = B)
Σ p( p(xxx0))))))))) p(
(17)
Eq. 17 provides an analytic expression for the information gap for posterior-ooding population under a task design specified by (p(c), (#(c). and generative model p(z (0). Per pair of observations (Iz, IR),, we evaluate the KL divergence between the true posterior (p (0)x;) or p (0)xx))) and surrogate posterior (0) or qf..t. which the posterior distribution associated with the output of the best possible likelihood decoder utilizing the task-mare marggnaaiiiddd Bayes-optimal estimators as given by Eq. 15 The KI. divergence is then marginalized across the pairs (æz, ( to derive the total expected performance difference between likelihood decoders and posterior decoders.
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A.2 SCHEMATICS FOR TASK DESIGN TRADEOFF
What task design would maximize info Ap
8 (Orientation)
Figure Design tradeoff.
A.3 DETAILS OF SIMULATION EXPERIMENT
A.3.1 SIMULATED NEURAL POPULATIONS
We consider tasks with Gaussian context priors motivated by classic orientation discrimination psy- chophysical tasks (Orbán et al., 2016 Walker et al., 2020). In this task, subjects perform an orienta- tion discrimination under two contexts e E [A. B), with the context for each session sampled ran- domly, i.e. p(c = A) p(c= = B) = 0.5. Within each session, the trial-to-trial hidden world state (i.e. orientation) is drawn from context-specific Gaussian prior distributions (0) = N(,,, (2) where μ and (a) are task-specific parameters. In the simulation, we consider E { {-90,, 90%) }, and use identical variances for the two Gaussian priors o o mental design is fully specified by the tuple of task parameters (n^ nr ",,,), .)) Furthermore, foregoing cardinal orientation consideration, the circular symmetry of orientations suggests that only the sep- aration between the two means d = 11 f would meaningfully impact perception. Given this, always center the two means around zero, meaning 1 - and م. 1dd. We systematically vary (d, o to cover the task spectrum of Gaussian context priors in the simulation studies.
Consequently, the experi-
Noisy sensory observations erative model p(z[0). This stochastic process can be seen capturing both intrinsic neuronal noise and uncertainty in the extrinsie stimulus features. This generative model can be experimentally ma- nipulated through stimulus parameters such as contrast, where lower contrast induces increased ob- servation variance, reflecting increased sensory uncertainty. the simulation, p(x(6) is modeled as Gaussian distributions to reflect Gaussian orientation tuning curves commonly found among simple VI neurons. We model the effect of different contrast levels by systematically varying the standard deviation of the generative model dbts (Walker et al. (0200). To this end, standard deviations dbbs of 8, 15, and 25 are chosen to model the generative model under high, medium, and low contrast levels, respectively. Finally, each trial, the hidden world state 0 is drawn from (0) N(4", (())) and then the observation is drawn from the conditional distribution p(x = N(0, 2
drawn from the conditional distribution defined by the given gen-
For simulated population responses, first implement Poisson neuron models with Gaussian tun- ing curves and Poisson variability (Walker et al. 2020). population of neurons indexed by 1, rang- ing from 5-500 neurons, was constructed with Gaussian tuning curves (Ot,, (0t, with their means 0, tiling up the orientation space and their standard deviations being Tobs- For likelihood-codin populations, the mean firing rate of each neuron each trial, after observation x is sampled, is
determined by the probability density of its Gaussian tuning curve, i.e. f(x)
V2Ro2
scaled with fixed constant of 30 to approximate the typical range of neuron firing rates observed experimentally (Walker et al., (020). For posterior-coding populations, the mean firing rate of each neuron is further multiplied by the context-specific prior (0), thus effectively encoding the pos- terior (0)z) x p(x(0) of pf) in their mean firing rates. For both populations, trial-to-triall spike counts then generated by sampling from Poisson distribution with the specified mean firing rates.
For some simulation experiments, we additionally implemented a more complex, gain- gain-modulated Poisson neuron model for simulating population responses (Goris al. 2014). The modulaeddl Poisson neuron model has been proposed to account for the supra-Poisson variability commonly observed experimentally among VI neurons. In this model, the mean firing rate of the neuron is
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the product of two terms: 1) the original rate determined by the Gaussian tuning curve model, and 2) a bimulls--ndepenndntt gain factor G. Goris et al. (2014) proposed and validated on VI neural data that this stimulus- independent gain factor G can be effectively modeled as following gamma stimulus-indppenden
distribution with a mean of one and variance of og- Based on their results, we choose a biologically realistic value of øG 0.5 in our simulation. Therefore, on trial-to-trial basis, after an original rate is determined according to the procedure in the previous paragraph for the likelihood-coding or posterior-coding population, random gain factor is then sampled from the gamma distribution and multiplied with the original rate to get the mean firing rate for the gain-modulated Poisson neuron model. Similarly, spike counts are then generated from the mean firing rates with Poisson variability.
A.3.2 EXAMPLE LIKELIHOOD AND POSTERIOR
Example likelihood and posterior
Prior_A
Probabitty dentyy 1N
0.04
Prior B
Lkaahood
0.03
Posterior_A A
Pocterfar _
0.02
Orientation..)
Figure 9; Example of the ground truth likelihood, priors, and posteriors.
A.3.3 PROBABILISTIC INFORMATION DECODER
As described in Fig. 2c deep neural networks parametrized by multi-layered perceptrons are trained with cross-entropy loss serve as flexible, powerful decoders to decode either the likelihood func- tion or the posterior distribution from simulated neural population responses (Walker et al.) [2020). We use fully-connected, deep neural networks with two hidden layers, with 300 and 200 units in the first and second layer, respectively. All hidden units are rectified linear units, and dropout rates of 0.5 are used for both layers. The imput dimension to the first layer is the number neurons in the simulated population, ranging from 5-500. The output layer is fully connected readout with no nonlinearity and a dimension of the number of possible hidden states. In our simulation, we consider orientation @ (-909, 90%) and discretize them into one degree bins, leading to a total number of possible hidden states of 181. To facilitate numerical stability, the decoded probability quantity is operating in the log space. The posterior decoder output is treated as the log-posterior, which is directly optimized to minimize the cross-entropy loss. The likelihood decoder output is treated as the log-likel,oood, which is then integrated with the ground truth log-prior to arrive the final output that is optimized to minimize the cross-entropy loss. To encourage smoothness of the decoded probability distributions, an T,2 regularizer on the log-posteriors filtered with a Laplacian filter of the form h = 00.25, 0.5, 0.25) is added to the cross-entropp term, proposed in (Walker ctt al. 2020). We use (0.8, 0.2) for train-validation split for training the decoders A held-out test set used final evaluation and all results in the paper are on the test test. Early stop with patience of 10 and minimal change of 2e-6 validation set cross-entropy loss is adopted to prevent overfitting. All models were constructed and trained using the Pytorch framework (Paszke al. 2019).
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A.4 INCORPORATING BIASED PRIOR FROM BEHAVIOR DATA
Although our main analysis focuses on the theoretical decoding limit using the optimal priors, our framework naturally accommodates model mismatch or biased priors by incorporating behavioral (psychometric) measurements. The procedure is detailed in Fig. TO
Analyze the psychometric curve: In perceptual tasks (e-g., orientation discrimination tasks), deviations in the subject's psychometric curve (correct rate as function of stim- ulus orientation) from the ideal obscrver reveal model mismatch or biased priors.
Infer the subject's model mismatch/ biased prior: Features (such leftward shifts or increased slope/variance) in the psychometric function can be mapped to corresponding biases or increased uncertainty in the subject's internal mismatched prior.
Compute the information gap using the inferred prior: The inferred biased prior can then be used directly in our information gap calculation, yielding predictions that account for the subject's model mismatch and more accurately reflect expected empirical decoder differences.
Ideal Psychometric
Ideal
Ground butth paychematticc carve
Greand petor
Percerbaga (44
Ideal Information
Predict
Compute
Gap
Orannttion
Compare
Experimentally Psychometric e.g. left shifted and widened
Inferred bissed negative bias and Increased varlance
Oroandiuth prizi
Farceraage (N4)
Experimentally
Inferred
Infer
Compute Information
Gap
Obee
Orantaatinn
Grawiattin
Figure 10: The information gap computation can incorporate behavior data by estimating the sub- ject's biased prior from its psychometric curve.
A.5 EXTENDING OUR FRAMEWORK TO MIXED CODING HYPOTHESIS
In this section we discuss a more nuanced probabilistic coding hypothesis and how our proposed framework could be extended to identify or falsify it. As example of mixed coding hypothesis, Ganguli & Simoncelli 2010 2014) proposed combing heterogeneous tuning curves- -wichh embed aspects of the prior with spiking variability that reflects the likelihood, yielding hybrid code in which sensory responses carry both likelihood information and structurally instantiated prior. This example can be categorized as a mixed or intermediate hypothesis, in between the canonical pure likelihood and pure posterior coding hypothesis. Our framework can naturally accommodate such mixed coding hypothesis by evaluating how each decoder performs under mismatched information. As shown in Fig. IT] since now both the likelihood and posterior decoders can recover the correct This rero-info-gap signature distinct and does not arise under optimized task designs for either pure likelihood- pure posterior- coding populations, which produce reliably nonzero and separable values. As a result, optimizing the task to maximally separate the two canonical hypotheses simultaneously maximizes sensitivity to departures from them. A mixed code that yields A info under the same optimized design becomes cleanly identifiable as neither pure likelihood nor pure posterior. Thus this discussion illustrates how our method could generalize beyond the two extreme hypotheses and provides a principled tool for distinguishing both pure and mixed coding schemes.
distributions, our theory predicts an information gap A info
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More broadly, do not claim that likelihood and posterior coding the only relevant theories in the literature, but they represent the two major families of theories that differ in what probabilistic quantity is encoded. Our contribution is to provide principled methodology for experimentally distinguishing such theories. By optimizing task parameters to maximally separate these canoni- cal extremes, we simultaneously maximize sensitivity to discriminating more nuanced probabilistic coding theories like the mixed coding hypothesis.
Likellhood-coding Population
Posterior-coding Population
Mixed-coding Population
Cross
L
info
Ap info
info
mixed -0
Entropy Loss
Likellhood
Posterior
Likelihood
Posterior
Likellhood
Postertor
Decoder type
Figure 11: Under mixed coding hypothesis, the information gap becomes zero.
A.6 EFFECT OF NOISE AND FIRING RATE
As shown in Fig. decreasing firing rates increasing noise slows the convergence of empirical decoder performance differences. More trials are needed for the empirical decoder performance difference approach the theoretical information gap. However, with sufficient data, the decoder performance differences ultimately converge to the same theoretical value. This reflects the expected effect of reduced signal-to-noise decoding becomes harder, but the underlying difference in decodable information is unchanged. Thus, while low SNR increases data requirements, the theoretical information gap remains the correct predictor of the asymptotic difference between the two hypotheses.
A) Changing firing rate Likelihood coding
B) Changing noise level
Devistion of decoder pert dift
from thcoretical value (nats)
Posterior coding
Devistion decoder dift.
Likelihood coding
Posterior coding
Decresoed
0.3
Decressdd
0.05
incrosed
0.09
Tncressed
0.30
0.2
0.00
theoretical
0.1
0.03
0.03
0.0
0.00
100
200
300
100
2000
300
100
200
300
100
200
300
Number of trials (10?)
Number of trials (107)
Number trials (10P)
Number of trials (10")
Figure 12: Effect of firing rate and noise level.
A.7 FACTORS AFFECTING CONVERGENCE SPEED
We conducted ablation studies to examine the factors that determine how quickly empirical perfor- mance converges to the theoretical value of information gap. Our main simulations assume that neural tuning curves tile the full orientation space, consistent with standard VI models Rubin al., 2015 In Fig. I3A, when the population is randomly sampled without full coverage of the entire on- entation space, since no decoder can recover information about orientations lacking tuned neurons, we found that convergence with respect to neuron count becomes substantially slower. In addition, in Fig. 3 the neuron-scaling experiment uses 30k trials so that decoders quickly approach the the- oretical limit. In Fig. 133, we performed ablation with fewer trials (3k trials) and observed that convergence is again slower because the decoder cannot reliably estimate the encoded distributions from limited data. In practice, the above factors can be mitigated by modern neurophysiology pop- ulation recordings that provide large number of trials with hundreds to thousands of simultancously recorded neurons that cover full range of orientation space.
modils/Rubin
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Finally, to demonstrate that our result is robust the level of discretization of the orientation vari- able, we repeated the convergence analysis with higher-resolutin orientation bins (0.259 instead of 1 as in the main results), and obtained indistinguishable results. This confirms that the accuracy of information gap and its empirical convergence are robust to binning and reflect the underlying decodable information rather than numerical artifacts.
A) Random sampling
B) Less trials
Devialion of deccder cift. from Theorotical value (nats)
Likelihood coding
Posterior coding
Devialiio of deccddr pert. citt.
from thecretiical vol (nats)
Likelihood coding
Posterior coding
Driy not ihill mouerage) Fundomsempping sempling
Driynol dMll munnage)) Bandon sumpling
Original inis to 196 Loa blak (mrebe3k)
Original pran... 300 Lew snuh la trideedtt
0.4
Number of units
Number of units
Number units
Number units
C) Bin size
Deviation decader part aite froer theoretical value (ners)
Likellhood coding
Posterior coding
Original (be_set=11
Original (8P_5000-1
Finer birning 28in, 0.2
Fiier binning 1iin s0t-0 250
8.5
Number units
Number units
Figure 13: Examine factors affecting convergence speed. A) Effect of orientation coverage. B)
Effect trial numbers. C) Effect bin size.
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A.8 DETAILED RESULTS ON HEAVY-TAILED PRIORS
Below we first provide the full results of information gap landscape across various contrast levels for heavy-tailed context priors including student's t-distribution (Fig. [4] and Cauchy distribution (Fig. 15). Note that for t-distribution we report the results using degrees of freedom v = 3. When the t-distribution reduces to standard Gaussian distribution, and when distribution becomes the Cauchy distribution. We then provide an intuitive example explaining why the information gap for posterior coding hypothesis is dramatically lower under heavy-tailed context priors compared to Gaussian context priors. The main reason is that under Gaussian generative models, when integrated with heavy-tailed priors, the posteriors tend to become asymmetric (as opposed to Gaussian priors where the posteriors are still symmetric Gaussian), thus limiting the number pairs (zj. that could confuse the likelihood decoder.
0 the
A.8.1 INFORMATION GAP LANDSCAPE
High contrast
B) contrast
20
30
Infor makiaaiin Gai (mats)
Stendard devieiiion
23
20
Likellhood Coding
25
15
15
10
30
40 50
60
10
00
90
10.
30
60
50
50
80,
90
1.0
20
30
50
60 70
80
90
Sedteral aa aanan (011")
Separalion (007)
Separation (ol*))
Standard dev devetton tote"
g
0.00
Informatton Gap (nats)
8
70
70
Posterior Coding
5 x
10 23 30 40
50 60 70 80 90
10 20 30 40 50 60 70 30
30
50
20
30 40
50 70
50 90
Seperation (d)
Separasion (01
Separation (20%)
Figure 14: Information gap landscapes when using student's t-distribution with degrees of freedom 3 as context priors. A) Information gap as function of task parameters (d: separa- tion between context priors, and a: context prior standard deviations) for both the likelihood coding hypothesis (top) and the posterior coding hypothesis (bottom) when presented with high contrast stimuli. B) Same for medium contrast stimuli and C) for low contrast stimuli.
High contrast stimuli
B) Medium contrast stimuli
C) Low contrast stimuli
Information Geo (mal))
(
12 20
25
Likellhood Coding
0.2
0.2
20
30
10
0
50 NO
40
80
0.0
Scale kpw?:
20
Posterior Coding
15
00:
10
0.02
30 70
30 60 80
30 30 Separation 0.00
Separation (d)?:
Figure 15: Information gap landscapes when using Cauchy distribution as context priors. A) Information gap function of task parameters (d: separation between context priors, and 7: con- text prior scales) for both the likelihood coding hypothesis (top) and the posterior coding hypothesis (bottom) when presented with high contrast stimuli. B) Same for medium and C) low contrast stim- uli.
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A.8.2 AN EXAMPLE EXPLAINING WHY AP IS DRAMATICALLY DECREASED UNDER HEAVY-TAILES CONTEXT PRIORS
Gaussian context priors
Student's t context priors
Cauchy context priors
Pobaabltyy deraly (waa)
0.04
6,03
0.03-
6.02
0.02
0.03
0.01
0.01
0.00
0.00
0.00
-75
-50
-25
25
-75
-50
25
-75
-50
Orientation
Orientation
Orientation
p²(0)
(e(@)
pix,(e
p²(8(x)
Figure 16: Heavy tailed context priors, when integrated with Gaussian likelihood function, lead to asymmetric posterior distributions, limiting the pairs of identical posteriors satisfying Eq. 12 that would cause imperfect likelihood decoders on posterior-coding populations. Across task designs with Gaussian context priors (left), student's 's context priors with // (middle), and Cauchy context priors (right), the context priors (0) and p (0) shown in dashed blue and red lines, respectively, Note they all share identical standard deviation or scale parameters to facilitate comparison. One example pair of (x), (18 27%) that satisfies Eq. 12 under Gaussian context priors shown here, with the associated likelihood functions p(z, (0) and p(2x1() plotted in solid gray and black lines, respectively. The posterior distributions under each context priors, (0(o)))) and p (Ra) are shown as solid blue and red lines, respectively. Under Gaussian context priors (left), the two posteriors are equal to each other, i.e. (0(02,)) = =p"(0]=x) hence the two lines overlap. However, as the context priors become increasingly heavy-tailed as under student's t distribution (middle) and Cauchy distributions (right), the two posteriors become more and more asymmetric, leading non-identical posterior distributions that no longer satisfy Eq.
This
example demonstrates why there are much less pairs (Iy, 236) that would satisfy Eq. 12 accounting for the observation that the information gap of posterior-coding population is dramatically decreased under heavy-tailed context priors
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A.9 RESULTS ON THIN-TAILED PRIORS
To provide further examples on non-Gaussian context priors, we examined thin-tailed distributions stimulus prior distributions. We reported additional analyses using canonical thin-tailed gener- alized normal distributions with 3 2) in Fig. 17 The information gap landscape shows that thin-tailed priors similarly lead near-0 posterior-coding information gaps across task parameter space. Our framework provides similar explanation: under thin-tailed context priors, the resulting posteriors become highly asymmetric across contexts (Fig. 18). reducing the feasible set of (z,, pairs that can satisfy Eq. 12 thereby shrinking the posterior-coding information gap, which mirrors the failure mode observed with heavy-tailed priors.
What about uniform priors? As shown in Fig. 18 uniform prior induces no context-dependent modulation of the posterior. Hence, likelihood- and posterior-coding populations become nearly indistinguishall, causing the information gap to collapse for both hypotheses.
Prior distribution
Likelihood coding
Posterior coding
Generalized Normal
Stancard deviation (all")
Informai mammaaon Gap (nats
0.5
25
20 15
0.4
0.04
Information
0.03
to
10
0.02
30 20
30
40 50
63 70 80 90
0.00
0 (Orientation)))
Separation (d)
Figure 17: Information gap landscapes when using generalized normal distribution as context priors. A) Information gap as a function of task parameters (d: separation between context priors, and context prior standard deviations) for both the likelihood coding hypothesis (middle) and the posterior coding hypothesis (right).
Gaussian context priors
Generalized Normal context priors
Unifomm context priors
Prosolty derdity (8.)
0.04
0.03
0.52
00:
0.01
0.00
0.00
0.00
-55
25
25
50
-50
-25
25
-75
75
Orientation
Oriantation
Oriontation
p²(0)
(G
p(x(0)
p(x,(0)
|110]x]
p=10 xz)
Figure 18: Thin tailed context priors, when integrated with Gaussian likelihood function. lead to asymmetric posterior distributions, limiting the pairs of identical posteriors satisfying Eq. [2 that would cause imperfect likelihood decoders on posterior-coding populations.
30


Reviews:
[{"review_id": "eYDEceF2ys", "reviewer": "Reviewer_UTZ6", "summary": "This paper suggests a framework, i.e. testing KL Divergence of different decoders, to determine which hypothesis is true about the perceptual coding in the brain, likelihood or posterior. Tested on simulated data on gratings with different orientations and contrast levels, it illustrates the potential use of the method for real experiments.", "questions": "- Do the simulation results hold when tested on the actual neural data (e.g on Allen data set mentioned above)?\n- How does the firing rate and noise affect the results?\n- How does the approach handle arousal, running, or other phenomena that affect the neural code in the visual cortex?\n- Decoding approaches especially by powerful methods like deep neural networks are not trustable to many neuroscientists, as the presence of information is different from using the information. For example, one might give all the neural activity in the retina to a deep neural network and infer high-order features such as objects, just like how they can do it on raw images directly. This does not mean that the retina perform object recognition. how does this affect the feasibility of the presented approach?", "limitations": "", "rating": 4, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 1, "strengths": "The paper is well written, specifying the problem, process, and limitations clearly. It also gives a good coverage of the literature.", "weaknesses": "Using KL-divergence and decoding performance is pretty common in modelling in neuroscience, so I don't think using it for a specific problem could be considered a huge contribution. Furthermore, the paper lacks empirical results. There are multiple public data sets that actually contain gratings with different orientation and contrasts, for example Allen Brain observatory visual coding Neuropixel data set( https://portal.brain-map.org/circuits-behavior/visual-coding-neuropixels). So I think testing the practicality of the approach is feasible for the authors. While it is kind of acknowledged in the paper, the method relies on a lot of assumptions, and most importantly optimality of the neual code. There are multiple studies showing that the code is not perfectly optimal, including those referenced in the paper, e.g Haefner, et al, Neuron 2016 in which they explained a Bayesian approach that models the bias toward early observations. Overall, many studies on perception has shown that the neural activity is too complicated to be easily interpreted by models (and that's why we don't know what exactly going on as the authors mentioned too). I can't see how this approach could give a practical solution to that.", "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": "DkIYYOTsIG", "reviewer": "Reviewer_1gnw", "summary": "The authors develop a framework for designing a pair of stimulus ensembles that are optimized to differentiate two models of probabilistic sensory representation for Bayes-optimal observers: the \"likelihood\" model, and the \"posterior\" model.  They show through a number of simulations that the method can successfully distinguish the two models.", "questions": "See above.", "limitations": "", "rating": 2, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "* Optimal experimental design is an important and under-studied area.\n\n* The problem posed, of developing an analysis to compare two standard models proposed for Bayesian inference in the brain, and using this to derive optimal stimulus ensembles for experimental determination of a \"winner\", is relevant and I'm not aware of any published solutions.", "weaknesses": "* The development is, in my view, over-formalized given the intended use and implementation.  Why is it not sufficient to seek two stimulus ensembles that produce the largest discrepancy in estimates produced by the two models?  And how do the results depend on the design and training of the deep neural network decoders that are used to obtain likeilihoods or posteriors from the neural population?\n\n* The paper is built on the direct comparison of two models (likelihood encoding vs. posterior encoding), but there are other proposals in the literature as to how prior probabilities might be embedded in neural systems.  In particular, Ganguli & Simoncelli (2010/2014) propose that inhomogeneous populations of tuning curves encodes the prior, the spiking responses form a likelihood, and a weighted nonlinear decoder can approximate Bayesian estimates.  There's reasonable experimental support for these components, but it's not clear how this formulation would be categorized in the context of the authors' proposed experimental test.\n\n* The authors' formulation relies on a highly simplified description of how an organism (human or otherwise) adapts to a sensory environment. In particular, they assume the posterior observer is aware of the true prior, and is able to use it properly for inference, which seems inconsistent with the large literature on adaptation.  Perceptually, adaptation to an ensemble leads to changes in perceived value (biases) as well as changes in discriminability (variance), but the changes reported in the literature are diverse (especially changes in discriminability), and highly dependent on the perceptual attribute being tested, the adaptation stimuli, the duration of exposure, etc.  In fact,  perceptual adaptation effects have been described as \"anti-Bayesian\", because adapting to one stimulus (or a distribution concentrated in one region of the the stimulus space) leads to perceptual repulsion rather than attraction toward this modified \"prior\". Physiologically, the most commonly observed effect of adaptation is a reduction in gain, but many reports describe shifts or distortions (e.g., flank suppression) of tuning curves.  Again, these effects are quite variable, and depend on details of adapting stimulus ensemble, duration, etc.   Recent work by Tring et. al. (2024) provides a compelling explanation for observed adaptation effects in terms of coding efficiency, but this seems at odds with either the \"likelihood\"  or \"posterior\" hypothesis of the current paper.  So my concern is that experimental design proposed by the authors will lead to inconclusive results, because adaptive changes in perceptual systems are substantial (so not aligned with the \"likeilihood\" model), but are also inconsistent with a change in prior.  \n\n* In describing prior literature, the paper should make mention of work on optimal stimulus design for estimating perceptual discrimination thresholds [eg: Quest (watson/pelli 1983), Pest (Medigan 1987), Information-theoretic stimulus optimization for neurophysiology (Lewi etal 2006/2009), Quest+ (Watson 2017)]. Perhaps more relevant, work by Machens et al 2005 adaptively sought a stimulus ensemble for which a neural population (grasshopper auditory receptors) provided an optimally efficient code.  Not the same as Bayesian decoding, but analogous in spirit.", "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": "hJm2UmDJ95", "reviewer": "Reviewer_EZQq", "summary": "Different theories have been proposed for the neural representations of early visual areas during perceptual decision making tasks. Some proposed that neural population activity represents the likelihood function over the stimulus variable (likelihood theory), while others argued that neural activity represents samples from the posterior distribution of the stimulus variables (posterior theory). So far, it has been difficult to adjudicate between these theories, or falsify both. This theoretical paper proposed a information-theoretical approach toward better ways to distinguish these theories experimentally, although no experimental results were given or analyzed in this paper. Specially, the authors proposed a metric called information gap, that captures the performance difference between the likelihood decoder v.s. posterior decoder. They were able to analytically derive the information gap under certain conditions. Numerical experiments were used to validate these results. They further used this approach to analyze some simple perceptual decision making tasks and made suggestion about how to better design experiments to test the different theories mentioned above.", "questions": "In fig. 2c, top row, why would one still incorporate the priors given it is a likelihood decoder? Also, were ground truth prior assumed there? \n\nLine 249, what does the “task-marginalized” estimator mean? Please unpack the idea. \n\nWhat was the ground truth posterior/likelihood in the simulation studies? Were neural noise considered in addition to the stimulus noise?\n\nFrom Fig. 6, it appears that the optimal experimental design depends on the noise in the system. Practically how can we determine the noise level before designing the experiments to test the theories?\n\nFig. 6, what are the units on the x- and y-axis?\n\nThe authors show the using heavy-tailed priors would not help better distinguish different models. Would using distribution with thinner tail than Gaussian help? What about uniform priors?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "This paper seeks to address a problem that has been difficult to make progress on in the field.  The proposed approach is sound. \n\nThe proposed analytical approach and the results are new, although they only seem to work under some relatively simple task conditions. \n\nThe numerical simulations in the paper were thorough and served to validate the analytical results. \n\nThe approach may provide some useful guide for the design of future experiments to testing the likelihood theory v..s posterior theory of the neural responses (although there may be some questions of practically how to find the optimal design as the optimal design seems to depend on some parameters, e..g, noise in the system, that might not be easily measured in the experiments).", "weaknesses": "— While the authors clearly spent efforts to articulate the theoretical settings and to justify the importance of the problem, I felt there was still some ambiguity. For example, the two theories outlined were the theory of representing likelihood v..s sampling-based code. However, here sampling seems to be an additional assumption. It would be cleaner if the comparison was simply a presentation of likelihood v.s. a representation the posterior. One can also consider a theory of sampling from the normalized likelihood (which would be equivalent to posterior assuming a uniform prior). It was unclear wether sampling is essential in the model comparison. \n\n— the paper only consider prior imposed during the task. It seemed that the results were based on the assumption that the task prior can be learned perfectly. Yet, this assumption may be too strong. \n\n-- What about the long-term priors (or natural priors) the observers already learn? Can the theoretical framework also apply to these? Or would these complicate things?\n\n\n-- My most substantial concern is that the authors only considered a specific path for a relatively narrow setting (although under these settings, the results seem to be solid). I am not sure if the approach would indeed be fruitful or whether there might be easier and more fruitful approach for adjudicating between the different theories. In particular, from Section 4, I am under the impression that practically it would still be difficult to distinguish these theories (Correct me if I have the wrong impression).  I wonder if there would be more fruitful path that would also be easier. For example, instead of using a static setting, how about considering a learning paradigm instead? Suppose we start with a baseline condition, and measure the neural responses. Then we teach the observers to learn a different prior. Under the likelihood theory,, the neural representation in V1 should not change. In contrast, based on the posterior theory, the neural representation in V1 should change according to the prior learned.  So the question is, what is the advantage of the approach proposed in the current paper compared to such an alternative approach?\n\n-- The paper would be stronger if some empirical results from experiments can be shown. Even a re-analysis some previously reported experimental data to demonstrate the advantage of the current method would be useful.", "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": "0W1Feg0FX2", "reviewer": "Reviewer_JDGu", "summary": "The paper puts forth a mathematical formalism for experimentally distinguishing between encoding of likelihood or of posterior at the level of neural populations via decoding, provides an information gap metric for the discriminability of the two, and optimizes in the space of tasks for designs which most clearly separate between the two hypotheses. This paves the way for a more precise experimental validation  of various bayesian accounts of perception and associated neural implementation in the brain. \n\nThe basic setup assumes a family of tasks where the animal needs to estimate and report a one-dimensional latent variable theta conveyed by a sensory stimulus, in two (known to the animal) contexts corresponding to different theta distributions. given the responses of a population of sensory neurons r, from which different probabilistic information (according to an experimentally matched ideal observer) is decoded via flexible deep learning models. Assumptions about the different models are incorporated in the network's architecture following Walker et al 2020. The information gap between them is the KL between the two possible encoded quantities (using discretized random variables for easy computation). The question then becomes how to optimize the distance and degree of overlap for priors in the two contexts to maximize discriminability", "questions": "How does the animal's own added sensory uncertainty get incorporated in the procedure ? can the (unavoidable) model mismatch bias the conclusions in any way or are there formal guarantees that it will not? \n\nIn particular, what if the local neural population encodes beliefs about related random variables but not the exact parametrization of the experiment? I am mentioning this because successful neural sampling (posterior-based) probabilistic accounts of V1 responses use a high dimensional distributed representation (think probabilistic versions of factor analysis or gaussian scale mixtures) related to orientation but not beliefs about theta per se\n\ni got a little confused about terminology, e.g. \"true posterior ...of a ideal likelihood decoder\"?  i think what is meant is an optimal decoder of likelihood or posterior beliefs from neural responses \n\nAs defined, the information gap seems to come from the ability of the agent to correctly know or infer contex, some more discussion on that would be helpful\n\nPlease explain the need for hypothesis specific information gaps, what assumptions it makes about context unknown r, and what it implies for experimental design: do you assume data exists for both a single/cued context version and an inferred/uncued context version of the same task? The elements of the math on the bounds of decoder crossentropy performance described in A1 are individually more comprehensible but the big picture logic and assumptions remain to be inferred between the lines \n\nEq3: i did not manage to follow the logic of restricting the sum to special subpairs since it seems incompatible to the traditional KL definition\n\nWhy not optimize the KL difference between the signal predicted by the two theories in a more direct way? There may be a good reason for the extra complexity but it's not coming across from the text\n\ngiven concerns about data limitations, is there a way to take into account co-occuring behavioral responses as a way to improve precision or at least to address model mismatch?\n\nHow does the number of neurons enter the information gap expressions ? It seems counterintuitive that one can achieve the assymptotic precision for representing a continuous variable with 10 neurons... does that depend on the precision of the x discretization in a substantial way?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "Interesting question, given precise mathematical form.\n\nNovelty and significance: both likelihood and posterior hypotheses make qualitatively similar predictions in traditional experimental designs which has hampered the ability to experimentally adjudicate between probabilistic coding theories. New ways to design experiments for maximal discriminability would circumvent this problem. To my knowledge no formal framing for such experimental design exists to date. \n\nthe idea that crossentropy-based decoders trained on ground truth stimulus identity are linked to ideal observer posterior uncertainty for the same data distribution seems generally useful\n\nNumerical validation of theoretical results in the limit of enough data", "weaknesses": "While i think the general framework is sensible and fruitful, I have concerns about the details of the execution. \n\nAlthough the writing overall is overall pretty clear, the key text with the description of the information gap is difficult to parse and ambiguous. the logic of the construction is not clearly enough explained in that part of the text.  \nThe appendix explains it better that the goal is to link crossentropy of different decoders to ideal observer posteriors arguing that a mismatched decoder will achieve overall less crossentropy performance but the main text cannot really be understood on its own \n\nBut the weakness is that the punchline is rather weak: the results do not seem to provide much insight into how one could design better experiments beyond giving ranges of possible d' parameters that look good c.f. the theory. It is not clear if the (not negligible) experimental data collected in this setup is outside that regime so that experimentalists would know to do something differently.", "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)