Bayesian Oracle for bounding information gain in neural encoding models
Authors: Konstantin-Klemens Lurz
Mohammad Bashiri
Edgar Y. Walker
Fabian H. Sinz
Venue: ICLR 2023 poster (2023)
Keywords: information theory
evaluation metrics
Bayesian
Neuroscience
encoding models

Abstract: In recent years, deep learning models have set new standards in predicting neural population responses. Most of these models currently focus on predicting the mean response of each neuron for a given input. However, neural variability around this mean is not just noise and plays a central role in several theories on neural computation. To capture this variability, we need models that predict full response distributions for a given stimulus. However, to measure the quality of such models, commonly used correlation-based metrics are not sufficient as they mainly care about the mean of the response distribution. An interpretable alternative evaluation metric for likelihood-based models is \textit{Information Gain} (IG) which evaluates the likelihood of a model relative to a lower and upper bound. However, while a lower bound is usually easy to obtain, constructing an upper bound turns out to be challenging for neural recordings with relatively low numbers of repeated trials, high (shared) variability, and sparse responses. In this work, we generalize the jack-knife oracle estimator for the mean---commonly used for correlation metrics---to a flexible Bayesian oracle estimator for IG based on posterior predictive distributions. We describe and address the challenges that arise when estimating the lower and upper bounds from small datasets. We then show that our upper bound estimate is data-efficient and robust even in the case of sparse responses and low signal-to-noise ratio. We further provide the derivation of the upper bound estimator for a variety of common distributions including the state-of-the-art zero-inflated mixture models, and relate IG to common mean-based metrics. Finally, we use our approach to evaluate such a mixture model resulting in $90\%$ IG performance.

Published as a conference paper at ICLR 2023
BAYESIAN ORACLE FOR BOUNDING INFORMATION GAIN IN NEURAL ENCODING MODELS
Konstantin- Lurz',, Mohammad Bashiri' Edgar Y. Walker", Fabian H. Sinz
Institute for Bioinformatics and Medical Informatics, University of Tübingen, Germany
2 Department of Physiology, Computational Neuroscience Center, University of Washington, USA Department of Computer Science, University Göttingen, Germany
equal contribution, sinzeess, uni- iingen.
ABSTRACT
In recent years, deep learning models have set new standards in predicting neural population responses. Most of these models currently focus on predicting the mean response of each neuron for given input. However, neural variability around this mean is not just noise and plays a central role in several theories on neural computation. To capture this variability, we need models that predict full response distributions for a given stimulus. However, to measure the quality of such models, commonly used correlation-based metrics are sufficient they mainly care about the mean of the response distribution. An interpretable alternative evaluation metric for nikelihood-based models is Normalized Information Gain (NInGa) which evaluates the likelihood of model relative to a lower and upper bound. However, while lower bound usually easy to obtain, constructing an upper bound turns out to be challenging for neural recordings with relatively low numbers of repeated trials, high (shared) variability, and sparse responses. In this work, we generalize the jack-knife oracle estimator for the mean commonly used for correlation metrics- to a flexible Bayesian oracle estimator for NInGa based on posterior predietive distributions. We describe and address the challenges that arise when estimating the lower and upper bounds from small datasets. We then show that upper bound estimate is data-efficient and robust even in the case of sparse responses and low signal-to-nove upper bound estimator for a variety of common distributions including the state- af-the-art zero- inflltnt mixture models, and relate NInGa to common mean- based metrics. Finally, we use our approach evaluate such a mixture model resulting 90% NInGa performance.
ratio. We further provide the derivation of the
INTRODUCTION
In recent years, systems neuroscience has seen great advancements in building neural encoding models of population activity 24 the conditional mean of the response distribution given a stimulus and are consequently evaluated on mean-based measures such as correlation or fraction of explainable variance explained (FEVE). However, neural responses exhibit a great deal of variability even when the animal is presented with the same stimulus. This variability is not not just noise, but might be a symptom of underlying neural computations. In fact, many normative theories that link first principles to neural response properties, like the Bayesian brain hypothesis [18, neural sampling 12 or probabilistic population codes [17]. make predictions rely on the variability of neural activity around the mean [15 to use neural encoding models as a quantitative underpinning for these theories, models are needed which accurately predict and are evaluated on complete response distributions. While progress has been made at building such models [22]2), it is not clear what upper bound on the performance we expect. However, this question is important as it gives indication how close models to the true system.
21 16
6 23 Most of these models focus estimating
1 5
If we want
In the case of mean- predicting models, correlation based metrics are often used for evaluation [6 g) Correlation is an interpretable measure since it is naturally bounded between -1 and 1. However, for vanilla correlation, it is impossible for any model to achieve correlation of 1 in the presence of

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Point Estimate
Bayesian
Comparison of lower and upper bound likeli- hood estimates (Null vs GS) per neuron. Left: For many neurons, the PE approach yields worse GS than the Null score. The Bayesian method results in the ex- pected outcome of upper bound scores being higher than lower bound scores. Right: Two example neu- 2.13 rons demonstrating where the PE method fails (red) or suc- ceeds (green).
Figure 1:
GS model log sa-likellihodd
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trial-to-trial fluctuations. Therefore, model correlation often normalized by upper bound oracle estimator [19] [6], which is commonly obtained by computing point estimates of the conditional mean using the responses repeated presentations of the same stimulus. For a likelihood -based metric, a similar normalization to bounded and interpretable scale would be desirable, especially for: 1) Assessing whether model has achieved its "best possible" performance for given dataset, and 2) comparing models that are trained on different datasets, which can exhibit different levels of achievable performance. To this end, one can use Normalized Information Gain (NInGa) [14], which uses an estimate of both upper and lower bound, to put the likelihood between two meaningful values. However, the challenge lies in how these bounds can be obtained for noisy neural responses.
In this work, develop a robust way to estimate such lower and upper bounds for NInGa neuronal responses. We show that point estimate approach for obtaining the upper bound fails and demonstrate that this is caused by the lack of robustness for the estimate of moments beyond the mean. This is especially pronounced when dealing with data that have few samples, sparse responses, and low signal-to-noise ratios which are common characteristics of neural responses. To mitigate this problem, we propose a generalization of the point estimate approach to a full Bayesian treatment using posterior predictive distributions. Our approach yields lower and upper bounds which are proven to be robust to all the above-mentionn expression for the Bayesian estimator for zero-inflated distributions that can be efficiently estimated under very general conditions by solving only single one-dimensional integral on bounded interval. These distributions capture the sparse nature of neural responses, particular for 2-photon recordings, and include state-of- the-art zero-inflated mixture models [22] 2 Using this full-likelihood -bseed metric, we then evaluate zero-inflated mixture model and find that it performs remarkably well at 90% NInGa. Finally we experimentally and mathematically relate NInGa to other common metrics for the performance of neural prediction models which are based on the mean and derive general conditions under which likelihood and correlation metric identify the same predictive function.
complexities in neural data. We derive a general
2 METHODS
2.1
NORMALIZED INFORMATION GAIN AND PROBLEMS WITH POINT ESTIMATES
Information Gain Let p(y|z) denote the distribution of a neuron's 's response y to a stimulus x. In order to evaluate and interpret the modeled distribution p(y|z) use Normalized Information Gain (NInGa) 14 20 which sets the model likelihood on an interpretable scale between an estimated
lower and upper bound:
(log ply x));,a (log po()))
NInGa
(logr-(y x))) (log po (w))) y.
using a Null distribution Po()) and Gold Standard distribution p. (2)z)) This method of computing NInGa can be interpreted as a normalized comparison of lower bounds of mutual information (20] and Appendix E In general, IG is completely flexible in the choice of Null, Gold Standard, and trained model distribution. Therefore, it can be used for model comparison across different distribution families as long the GS and Null model are kept the same. The Null model should reflect basic
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aspects of the response. Here, we choose Null model that does not account for any stimulus-related information, resulting in the marginal distribution of responses Puly). The Gold Standard (GS) model p. (3)lz) the other hand, should be the best possible approximation of the true conditional distribution p(y) [z). Here, we use an oracle model that has more information than the model under evaluation, such as access to repeated presentations of the same stimulus. Importantly, we do this in leave-one-out fashion: given a set of n repeats, the GS parameters of a target repeat i are estimated from - 1 left-out repeats /i. However, as we demonstrate below, estimating robust GS model can be challenging.
Point Estimate (PE) GS model The parameters 0 of the upper bound estimator can be obtained as point estimates (PE) from - left-out repeats:
(B x) p(y:(0;) with 0, f(y
where f function used to obtain the point estimate of 0. In our case, f represents moment matching. For correlation- based performance metries of neural prediction models, a jack-knifed mean estimator over repeated presentations of the same stimulus Elys[z] = - Ey, Eu, 3/j is commonly used as an oracle predictor for the conditional mean to obtain an upper bound on the achievable performance the presence noise [9][6]] While the posterior predictive distribution is generally conditioned stimulus x because it requires responses to the repeated presentations of the same stimulus, for the remaining this manuscript we will drop the conditioning on æ for brevity.
Problems with the PE approach To demonstrate the problems with point estimate GS models, we modeled neural responses with a zero-inflated Log-Normal likelihood and estimated the upper bound using the PE approach (see Appendix [A.] for details data, 22 2 inflated distributions, and Appendix [ for the moment matching derivations). Since the GS model estimates parameters per stimulus. should yield higher likelihood values than the Null model whose parameters are not stimulus-specifie. However, applying the PE approach to neural data, we observed that the Null model outperforms the GS model for the majority of neurons (Fig. [ black points). The reason for this effect is that the PE approach is sensitive to the sparse distribution of the data, which combined with few responses per stimulus results in an overconfident estimation of the GS parameters (see Fig. 1 on the right; Appendix D for a more detailed analysis on where the PE fails).
for details zero-
2.2 BAYESIAN GOLD STANDARD MODEL
Gold Standard Model based on posterior predictive distributions To avoid an overconfident GS model, we add uncertainty to the parameter estimation and estimate the GS model in fully Bayesian fashion via the full posterior predictive distribution:
P. (vlly
0) p(Oly))) do
(2)
likelihood
posterior
Note that the PE approach is special case within this framework for which p(0)y),) = 5(0- f(y,)) collapsed onto delta distribution.
Efficient estimation for zero-inflated distributions In general, the integral in equation 2 is intractable and the posterior predictive distribution can only be evaluated using numerical approx- imations. Only for certain choices of likelihood the integral can be solved analytically if the right conjugate prior was chosen.
Here, show that the posterior predictive distribution can be efficiently computed for the class of zero-inflated distributions 22 Fig. 2 principle, such a distribution is a mixture of a delta- distribution at and density of a positive part. In practice, replace the delta-distribution with a uniform distribution in a small interval [0, r) and shift the positive part to the interval [r, oo) so that the mixture components do not overlap
The probability density for a single stimulus x: is then defined as:
p(y|=) (1 q(=)) (u()) + 4(x)-
(x))
(3)
positive distribution on (F,00)
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Figure 2: Graphical model for zero- inflated distribution. stimulus x deter- mines the probability whether neu- rons fires or not and the parameters 81 of the response distribution of the non- zero response distribution. A Bernoulli random variable my determines whether a neuron fires on a particular trial i. If my 1 response is drawn from ()), otherwise from p(y: (6)
q
my
81
80
y/
LA visisn
where the mixing proportion q and the parameters 0₁ are dependent When using a predictive model, q and 0 are typically predicted from the stimulus x using an encoding model. Zero-inflated likelihoods the basis of current state-of-the-art likelihood-based neural encoding models 22 2)
Here we show that the posterior predictive distribution for zero-inflated likelihood boils down to a one dimensional integral over q if the posterior predictive distribution of the positive part is known.
Lemma 1. The posterior predictive distribution of cero-inflated distribution as defined in equation is given by
plu: = (llnlyy) (2(1-) 2) p(ayy)) dq ((lnly))) Sg p(ayx))) dq
(11))
if <
if 2T
where ri and y) denote the set of zero and non-zero responses in Yoi. respectively.
Proof. See Appendix C
This means that the posterior predictive distribution can be computed efficiently for a large class of positive distributions, including the zero-inflated Log-Normal or even zero-inflated Flow models 2] In the next section, we demonstrate that this Bayesian treatment yields a GS model that is more robust against outliers and yields higher likelihoods than the Null model, as expected (Fig. orange points).
EXPERIMENTS
3.1 ANALYSIS OF GOLD STANDARD MODELS
In this section, we investigate why the Bayesian approach outperforms the PE and test its robustness under different numbers of left-out repeats y and different signal-to-noise ratios. For this analysis, used responses from 7672 neurons to 360 stimuli, each repeated 20 times well as simulated data. Details about the datasets are provided in Appendix A. and Appendix A.2 respectively. We base our posterior predictive distribution
zero-inflated Lor-Normal distribution with zero-threshold = exp(-10 10) and parameters 01 = (μ, o²) referring to the first and second moment of the log- transformed positive responses (Appendix Cj. The conjugate prior for the Log-Normal part of the distribution is -oveal-Inverse- Gaamee distribution p(0,) N X, a, 33 in log space whose parameters can be chosen freely. We discuss choices of prior parameters later in section 3.2
Bayesian GS estimates higher order moments better In order to determine which parameters of the likelihood profits most from the Bayesian estimation, we compared GS models where the individual parameters are either estimated via the PE or the Bayesian approach (Fig. 33)). For this we used the largest number of left-out repeats n that the likelihood improves with the Bayesian estimation of μ (orange vs. yellow bar) as well as o² (light blue vs. yellow bar) individually. Consequently, the highest performance achieved when both parameters relative gain in performance is much higher for o² than for the μ4, reflecting a lower robustness of the higher moments compared to the first moment in log space.
19 available in our real neuron dataset. First, we observe
estimated via the Bayesian approach (dark blue vs. yellow bar). Interestingly, the
Bayesian GS is data-efficient Datasets can vary in how many repeats per stimulus they contain. Since metric should be comparable across datasets, NInGa ought to yield consistent estimates for
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Figure 3: Comparison of the point estimate and Bayesian GS model. Comparison of different GS models where the individual parameters are either estimated via the PE or the Bayesian approach. The number of left-out repeats y is 19. Colors likelihood value, i.e. the dark blue bar. b: Similar to a but for different numbers of left-out repeats yu- Left: Simulated data. The likelihood normalized w.r.t. the ground truth likelihood. Right: Neural responses. Likelihood normalized w.r.t. the maximum likelihood value, i.e. the dark blue bar 18 left-out repeats. c: Upper bound likelihood scores for different signal-to-noise ratios and different number of left-out repeats y Left: Point Estimate. Right: Bayesian. Likelihood is normalized w.r.t. the ground truth likelihood. In all panels the likelihood values are averaged over stimuli and neurons, and the eтror bars and shaded areas show SEM over random selections of the left-out repeats.
the same in b. Normalized wrt. the max.
different numbers left-out repeats y particular in the regime low n The right panel of Fig. 3D shows this neural data where first observe that the results of Fig. Ba qualitatively consistent for different numbers repeats. The effect of the Bayesian parameter estimation on the likelihood performance, however, is much number of left-out repeats decreases the PE approach suffers much more than the Bayesian and completely fails n (vanishing yellow bar). To test the higher neural responses since the real neural dataset contained maximally 20 repeats. In the left panel of Fig. 3D we explored the differences between the two approaches for up to 30 repeats and observed that the Bayesian treatment consistently yields better likelihood than the PE estimate (yellow bar does not completely converge dark blue bar). The probabilistic treatment of p. however, seems to become less important in the high of the first moment compared to the second moment log space (compare the difference between orange and yellow for high vs. low 1).
pronounced in the low
regime: As the
regime, we simulated
regime than that of o², reflecting the higher robustness
Bayesian GS robust to different SNRs Apart from different numbers repeats, datasets can also vary in terms of signal-to-noiss ratio. We therefore simulated neural data with different underlying means and variances per stimulus, resulting in different SNR values (see Appendix A.2 2 for details). We then tested Bayesian and point estimate GS models on this data (Fig. (] and observed that the Bayesian approach consistently outperforms the PE approach across all SNRs (data for Fig, B3 left panel had an SNR of 0.42).

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Figure 4: Real neural data evaluation of neural encoding model trained on zero-inflated Log- Normal likelihood (ZIL) using the Bayesian upper bound estimate (GS). a: A sub-optimal prior for the GS model results in lower performance than the trained model (TM). Normalized wrt. the max. likelihood value, i.e. the blue bar. b: The GS model outperforms the TM when its prior was optimized. The TM estimates all parameters similarly well. Grey scale colors indicate models from the same distribution as the TM (ZIL) for which all parameters are estimated by the trained model except for the parameters in parentheses. Normalized wrt. the max. likelihood value, i.e. the yellow bar. e: Normalized Information Gain (NInGa) for the TM and the models grey scale models from b. Values on top of bars indicate the likelihood per image and per neuron bits. In all panels the likelihood values are averaged over stimuli and neurons, and the error bars and shaded areas show SEM over 5 random initializations of the TM weights. There are no errorbars on the red, yellow and dark grey bars since they do not involve TM.
3.2 EVALUATING NEURAL ENCODING MODELS VIA NINGA
In Section 3.1
we demonstrated that the Bayesian approach for obtaining an upper bound estimator greatly outperforms the point estimate (PE) approach in several aspects. From now we therefore only use the Bayesian estimator when referring to Gold Standard Model. In this section, we use the lower and upper bound estimates to evaluate neural encoding models via Normalized Information Gain.
To this end, trained encoding model on dataset which is publicly available [16] containing the responses of 5335 neurons in mouse primary visual cortex to 5094 unique natural images. The dataset split into 4472 Hiimulus-rsppone pairs for training, 522 for validation, and 100 for testing. The stimuli in the test set were each repeated 10 times resulting in n = 9 repeats for the GS model be computed on. The neural encoding model and the training are the same as the model provided by Lurz et al. [16). Briefly, the encoding model consists of two parts: (1) A core network which is shared across neurons with four convolutional layers (some of them depth-separablee (9) resulting in 64 feature channels, followed by batch normalization and ELU nonlinearity. And (2) neuron- -sproffic Gaussian readout mechanism [66 that learns the position of the neuron's 's receptive field (RF) and computes a weighted sum of the features at this position along the channel dimension. While Lurz et [66 used Poisson loss to train the models, we chose the negative loglikelihood of
loss function. This means that the model needs to predict three parameters (g. μ4, σ²) instead of single mean firing rate À as in Lurz et al. [16]. The readout thus learns three weight vectors to combine features extracted via the core network, at neuron's is learned RF position. Since the metric that the model will be evaluated on is NInGa, the early-stopping criterion changed from correlation to likelihood. The results of these experiments with neural encoding models are summarized in Fig. and will be explained in detail below.
zero-inflated Log-Normal (ZIL) distribution
Choice of prior is crucial for the GS model Up to this point, we chose the prior couron-independently We chose the parameters of the Hermal-Inverse-Gammee prior based on the average conditional mean Eg. [Ey [yyyyle [=] and average conditional variance [Var,,,y of our dataset. This left with number of neurons samples which fit the prior on, resulting in one identical prior for every neuron. However, we observed that the resulting GS model (yellow bar) was outperformed by the trained encoding model (TM model, blue bar) and did not provide good estimate of

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performance upper bound (Fig. a). To obtain a better upper bound oracle model, we therefore optimized the prior descent for each single neuron. This is analogous to other oracle models that have access more information than the predictive model under evaluation, thus providing an upper bound, Note that this approach results in a more conservative estimate of the model performance (Fig- bb, compare the blue and yellow bars). We also tried other approaches (c.g. MAP estimate) to obtain a better GS model but none of them outperformed the Posterior Predictive GS (see Appendix TT.
directly on the sum of leave-one-our GS likelihoods via gradient
Encoding model captures all parameters similarly well In order to investigate which parameters of the response distribution the encoding model predicts well, we conducted an experiment similar to the one shown in Fig. 3 a: We compared the likelihood of the trained model (TM, blue) to cases where we matched one or all of its three parameters (g. p. o²) to the GS model (see Fig. 4 b blue bar vs. grey bars). While we could match the parameter q directly, we used moment matching to obtain the parameters of the non-zero part of the trained model (Log-Normal distribution) from the non-zero part of the GS model (Log-Student-t). We observed that the likelihood of the trained model improved when each of the three parameters were matched with the GS model individually (three lighter grey bars vs. blue bar), where q yielded the highest increase. Matching all three parameters, however, did not result in the same performance as the GS model itself (darkest grey vs. yellow). Since all parameters of the ZIL model were fitted on the GS model, the reason must be the difference in distributional shape of the positive part: Log-Normal for ZII. vs. Log- -ag-Student-t for the GS model.
Encoding model performance is at 90% NlnGa The final NInGa value of the trained model using the Null model and the GS model with optimized prior hyper-paramedd can be seen in Fig. It performs remarkably well at 90% NInGa (blue bar), which corresponds to a likelihood of 7.00 bits per image and neuron (printed value above the bar). The effect of the parameter matching (grey bars) is more emphasized and suggests that the largest performance gain can be achieved in future models by improving the prediction of the parameter We performed additional analyses on multiple datasets show how NInGa facilitates model comparison across different datasets (see Appendix J.
4 RELATION BETWEEN NINGA AND OTHER METRICS
Neural predicting models have so far been mostly evaluated using metrics such as fraction of explainable variance explained (FEVE) 77. correlation, and fraction oracle [6). While we cannot expect that there is a one-to-one relationship between NInGa and these metrics, as NInGa is sensitive to the entire response distribution whereas the commonly used metrics mostly focus on the mean response of a neuron for a given stimulus, we can nevertheless relate NInGa to these metric under certain assumptions.
this section we provide a summary of the relationships to these metrics. Details and proofs can be found in Appendices F and G
Generally, we will demonstrate the relations in two steps: First we show that NInGa linearly depends on the expected Kullback-Leible (KL) divergence between the true distribution p(y|zz) and the model distribution p(y) (r). Then we derive how this KL divergence relates to other metrics.
The NInGa described in Eq. can be re- re-written in terms of KL divergences between estimated/fitted distributions and the true conditional distribution p(ylx) (see Appendix foor the detailed derivation):
NInGa is a linear function of (Dxl)p(().)
(Dx kl I ( (lloolloo)))))))))))))))
NInGa
(4)
(Dez z. [p(v[=) (Po (v))), (Dxz (x(|=)) p- (p (v[z)))>
This shows that the Normalized Information Gain is a linear function of (Dxl (p xx) (+(()=))
with negative slope. It is maximized when average Kullback-Leibler divergence between the true distribution and the model distribution is minimized.
NInGa vs. FEVE Using this fact, we now show that NlnGa is linear in another commonly used metric, fraction of explainable variance explained (FEVE), for a Gaussian likelihood. For a fixed model noise variance d? the KL-divergence (Dxr. [p (()l=) is linear function of FEVE:
of
(DRulp00)))) f(a.) ! FEVE) x
(5)
6?
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Figure 5: Per neuron comparison of NInGa with correlation and FEVE. The data comes from the TM Fig. 4 (one of the models from different initializations). a: Correlation vs. NInGa. b: FEVE vs. NInGa. c: The neurons displayed in panel and b were selected based on whether their monal-to-ootal-variine ratio (STR) was above 0.25. The blue lines depict linear regressions with 95% confidence intervals.
and o3 is the signal variance. Since (Dxx)(yy)))ll@(/e))) and NInGa are also linearly related, then NInGa too is linear function of FEVE. Note that when the estimated noise variance is the same the true noise variance, then Eq. becomes:
where f(ō.) log 8.
-
= - (1-FFVEE) x SNR
See Appendix for detailed derivations. If the estimated noise variance o² is not fixed, as is the general case, this trend will still be true but the linear relationship between the NInGa and FEVE is more noisy. In Fig. 5b, we show that by evaluating the trained model from Fig. 4 on NInGa and FEVE per neuron. Note that we only display neurons whose signal-to-total-varaae STR - o oo / (o? + of) exceeds threshold which we set to 0.25 (see Fig. 5c).
ratio
NInGa vs. correlation Another commonly used metric is the correlation p(pz: Mz) between the model prediction P. = (0) , and trial averaged responses μl = (y) Assuming a Gaussian likelihood, like in the relation to FEVE, we show that for a fixed model noise and signal variance (Pkl/p(yla (=)==(())))]] is linear function of the correlation between Hl and paz
(DKuI))vl=)[))<>>)))) f(ō.) + (1+
0² 28, p( pliessHe) Me)) a,
(6)
o²
6%
As before, the model noise variance matches the true noise variance, o? = we have:
(DKu(p())=)]))>>))>) 1 a²
20, p(p wz)) SNR
2
o,
Assuming further that the model's signal variance matches the true signal variance, 03 = 0 we get:
(D (
p(pn, Mx)) x SNR
is worth noting that the assumptions for having a linear relationship with KI. divergence are stronger in the case of correlation than that of the FEVE. Specifically, there is additional dependence the signal variance in the case of correlation. This makes sense as FEVE is inherently sensitive to signal variance while correlation is not (see Appendix G for the detailed derivation). As in the case of FEVE, we empirically show (Fig- a) the relation between NInGa and correlation
our trained model from Fig. 4 While the general linear trend can observed, this relation is very noisy, as expected. Note again, that we only show neurons that pass the threshold of ST R 2 0.25.
Does a high likelihood always correspond to a high correlation? Correlation only reflects good estimation of the mean of distribution (see Appendix H). NlnGa, on the other hand is likelihood- based metric and as such depends on the correct estimation of all parameters of the likelihood. So does high likelihood correspond to a high correlation? The answer is: it depends.
Correlation is high the conditional mean is accurately estimated. However, since the mean of the distribution is in general not directly parameter of the likelihood it is not necessarily estimated
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theoretical max entropy)
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training epoch
Figure 6: Comparison of log-likelihood and correlation of model which optimized via maximum likelihood estimation. Left: The model distribution is from the exponential family and has the mean as a sufficient statistic. The optima of the two metrics coincide but an improvement in likelihood does not necessarily mean improvement in correlation. Data distribution: Normal. Model distribution: Gamma. Right: The model distribution does not fulfill these criteria. No similar behavior of likelihood and correlation can be guaranteed. If initialized with optimal correlation, the correlation after training has decreased (dotted line). Data distribution: Gamma. Model distribution: Chi-squared.
accurately even if optimal likelihood is achieved. Therefore, there is in general guarantee that model that outperforms another model on likelihood also outperforms this other model on correlation. However, if the distribution is 1) member of the exponential family and 2) has the mean sufficient statistic [10, an optimal likelihood also implies an optimal correlation. A distribution that fulfills these two criteria guarantees that for optimal likelihood, the mean is optimally estimated and with it the correlation. We demonstrate this effect using simulated data (see Appendix A.2 in Fig. ii the left panel: A model of such a distribution is being optimized on toy data via maximum likelihood estimation and at the end of training the optima of likelihood and correlation coincide. Note however that during training, i.e. for non-optimal likelihood, a training step which improves likelihood does not necessarily yield an improvement of correlation (compare decreasing blue line vs. increasing orange line). If the distribution does not fulfill the second criterion, i.c. if the mean is not a sufficient statistic, no relation between the correlation and likelihood can be guaranteed, not even at the optimum. This can be seen in Fig. 6 in the right panel: An improvement likelihood (orange) does not guarantee improvement in correlation (blue). If the model was initialized such that the correlation is optimal (dotted lines), the maximum likelihood estimation will even result in a model that has lower correlation than had the time of initialization (dotted blue line epoch 1200).
CONCLUSION
In this work, discussed the challenges in obtaining the lower and upper bound estimates for likelihood- measures like Normalized Information Gain (NInGa) in order to put the performance on an interpretable scale. We introduced robust way of obtaining such an upper bound by using the Bayesian framework of posterior predictive distributions. Equipped with this metric, we showed that current neural encoding models are able to predict full response distributions to up to 90% NInGa and we examined which parameters of the distribution the model still fails at predicting. We also gave detailed derivation for obtaining upper bound estimates for the state-of-the -fee----t-e-rrr- art family of distributions, the zero-inflated distributions. Finally, we related likelihood-basee metrics like NInGa to other metries which are commonly used Computational Neuroscience like correlation and FEVE.
There are, of course, also some limitations to the current work. While we were able to fix the catastrophic failure naive PE estimates (Fig. Tp. our estimates of NInGa or the GS model are not perfect some neurons yield NlnGa > (Fig. 5 which is not mathematically impossible but would ideally not happen. Better NInGa or GS estimators will mitigate this issue. Finally, our approach is for single neuron likelihoods. However, the neural variation is structured instance through brain states. Since the general approach of NInGa still works in that situation, our framework is flexible enough for a robust GS model of full populations to be derived in future work.
population level, for

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Published as a conference paper at ICLR 2023
ACKNOWLEDGMENTI
We thank all reviewers for their constructive and thoughtful feedback.
Konstantin- Klemenss Lurz is funded by the German Federal Ministry of Education and Research through the Tübingen AI Center (FKZ: 011S 01IS18039A). Mohammad Bashiri supported by the International Max Planck Research School for Intelligent Systems. Fabian H. Sinz is supported by the Carl-Zeiss-Stiftnng and acknowledges the support of the DFG Cluster of Excellence "Machine Learning- New Perspectives for Science", EXC 2064/1, project number 390727645. This work was supported by the Intelligence Advanced Research Projects Activity (IARPA) via Department of Interior/Interior Business Center (Dol/IBC) contract number D D16PC00003. The U.S. Government authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright annotation thereon. Disclaimer: The views and conclusions contained herein those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed implied, of IARPA, DoI/IBC, or the U.S. Government.
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11


Reviews:
[{"review_id": "uEAt4JUQRE", "reviewer": "Reviewer_1J5x", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "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": "", "main_review": "", "summary_of_the_paper": "The author's propose a new metric for evaluating / comparing neural encoding models. They call this metric the Information Gain (IG):\n\n$\n\\frac{ \\langle \\log \\hat{p}(y \\mid x) \\rangle  -  \\langle{\\log p_0(y)} }{ \\langle \\log p_*(y \\mid x) \\rangle  - \\langle \\log p_0(y) \\rangle }\n$\n\nI would instead suggest calling this something like *Normalized Information Gain* as previous papers (e.g. Kummerer et al, 2015) call the numerator the information gain (and in that case it has the right units of bits/nats).\nThe IG is a very well-known metric used to compare different models.\nPersonally, I would prefer to use the term log Bayes factor or log-likelihood ratio, but this is just a matter of taste/background.\n\nMuch of the paper is spent trying to find a good \"Gold Standard\" model $p_{\\star}(y \\mid x)$, which is meant to upper bound the performance of any model. The authors use a reasonable Empirical Bayes methodology here. One concern is that they find encoding models that outperform this \"Gold Standard.\" This is not too much of a problem in my view, but the authors may try the following simple trick/suggestion &mdash; I would try taking a having $p_{\\star}(\\cdot)$ be a weighted average of the current gold standard model $p(y_i \\mid \\mathbf{y}\\_{\\backslash i} , x_i )$ and the *global* distribution over all images (not just those with the same image input as trial $i$). The idea is that one has an extremely large number of trials so you could get a very accurate kernel density estimate of $p(y)$ and shrink your under-powered parametric estimator of $p_*(\\cdot)$ slightly towards this (biased, stimulus-independent) of  the neural response distribution. I'm not sure this will work but it may be worth trying.\n\nThe authors conclude by relating IG to existing metrics within the field like correlation and fraction of explainable variance.", "summary_of_the_review": "I have to admit this paper makes me somewhat depressed. To me it is blatantly obvious that information gain is a superior measure of performance relative to existing conventions. It doesn't feel like we should require a whole paper to set the field straight. But sadly I think this may be a necessary story for people to hear. Thus, I do see merit in this paper.\n\nI am so far proposing a borderline reject to this paper because of the missing details and unclear notation in areas outlined above. However, I'm open to raising my score if the authors revise the paper to provide those missing details.", "strength_and_weaknesses": "The main strength of this paper is that it identifies a rather serious deficiency within the current state of the field. Having pointing out this deficiency, it presents a very well-known and rigorous solution.\n\nThe main weakness of this paper is that it focuses more-or-less entirely on conceptual first principles. To be sure, these first principles are really important for us to get right, but I would have liked to see more empirical demonstrations that compare multiple encoding models and revisit old results (e.g. Yamins & Dicarlo-style regressions between deep nets trained not on encoding task, but on different image tasks). A skeptical reader may come away unconvinced without a deeper empirical demonstration.\n\nAnother weakness of this paper, in my view, is that it spends a long time trying to identify a \"Gold Standard\" upper bound model, which isn't really necessary for us to compare and rank different encoding models. Thus, a lot of the paper ends up being somewhat uninteresting to me. Further, while it would be nice to have an upper bound on model performance, this isn't easily done. The authors find out in Figure 5a that their encoding model actually out-competes their gold standard, and they then amend their gold standard to be a \"true upper bound\" in Figure 5b. However, I challenge the claim that this is a true upper bound -- the gold standard model is a parametric model (albiet a pretty good one!) and there is nothing saying someone else couldn't come along with something else to beat it. This is especially true since the gold standard model as currently formulated doesn't share any information across different image types / categories. Presumably this is why it is possible for the deep net encoding model to outperform the gold standard. Overall, I would suggest that the author's revise the paper to modify their language --- i.e. treat the gold standard as a useful benchmark or point of comparison, but not a end-all-be-all upper bound.\n\nA final weakness of this paper is the clarity of the notation and deep net experiments, described below. I believe however that this weakness is addressable in review.", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "oF-0yGgDdsH", "reviewer": "Reviewer_ALKJ", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "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": "", "main_review": "", "summary_of_the_paper": "The authors proposed a method to obtain upper bound for IG. They showed that this method is efficient and robust using simulated data. They derived the estimator on zero-inflated distribution family. Finally they showed that the encoding model achieved high IG performance on a real dataset.", "summary_of_the_review": "This paper is clearly written and the discussed question is important. But the paper is lack of more real examples and comparisons. The authors only looked at one real dataset and didn't compare with other methods. I have proposed some detailed questions and weakness in the above section.", "strength_and_weaknesses": "Strength:\nThe paper is clearly written.\nThe authors are discussing about an important question: how to evaluate a neural encoding model. \nThe authors have shown both theoretical derivations and empirical data studies.\n\nWeakness and questions:\nThe authors mainly test on 1 real dataset, just wonder if the main result (e.g. 90% IG) still holds for other datasets?\nThe mixture model achieves 90% IG on the real data example. Is this good enough? Can IG be used to compare encoding models with different distribution family?\nThere are also other common ways to measure the goodness of a neural encoding model, e.g. decoding performance. I wonder if the authors have done comparisons with other measures? Is IG consistent with other measures?", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "2: The contributions are only marginally significant or novel.", "empirical_novelty_and_significance": "2: The contributions are only marginally significant or novel."}, "extra_text": {}}, {"review_id": "irgCEsP3f0", "reviewer": "Reviewer_rHRG", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "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": "", "main_review": "", "summary_of_the_paper": "This paper identifies an issue in the Information Gain (IG) evaluation metric for neuronal population response prediction, whereby the upper bound 'oracle' likelihood estimate using a Point Estimate (PE) approach (i.e. using a point estimate for the parameters of some simple parameterised distribution like a zero-inflated LogNormal) is often lower than the corresponding lower bound 'null' model estimate. This issue is particularly apparent in settings of few samples, sparse responses or low signal-to-noise ratios (SNRs). The paper proposes to improve the PE approach by instead using a full Bayesian treatment to obtain the posterior predictive distribution for the 'oracle' by marginalising out the parameters with respect to their posterior distribution. \n\nThe paper demonstrates how to efficiently estimate the Bayesian 'oracle' with zero-inflated distributions (which are state-of-the-art ways to model neuronal population responses) and then demonstrates that the Bayesian oracle outperforms its PE counterpart, across different SNRs and sample sizes for the leave-one-out calculations (Figure 3). In section 3.2, the paper uses its proposed IG metric (with Bayesian oracle) to evaluate a trained neural encoding model, highlighting the importance of prior choice, and finally the compares IG and other metrics that have been used to evaluate neuronal population response in section 4.\n\n", "summary_of_the_review": "I am recommending weak accept as I think the paper identifies and answers a clear problem, and that the proposed approach appears to be effective in practice. I'd like to see more development as to the reason why the proposed approach works (in 'weaknesses', along with other suggestions which will hopefully improve the paper).", "strength_and_weaknesses": "Strengths:\n1. The problem the paper is tackling is largely well motivated and seems to be of importance in neural encoding models (the reviewer is not an expert in this area)\n2. The proposed Bayesian oracle solution is simple and appears quite effective empirically at reducing the issues faced by the PE oracle.\n\nWeaknesses:\n1. The reviewer thinks some aspects of the approach could be developed more thoroughly: for example, does we need the full Bayesian posterior predictive in order to obtain better upper bounds or would say using a maximum-a-posteriori PE suffice in ameliorating the issues associated to current PEs. Equivalently, are current PE oracles underperforming because they are PEs, or is it because perhaps that they are overfit to the observed repeated responses (and could be improved with regularisation)? The paragraph above section 2.2 highlighting the issues with the PE approach did not read convincingly and the suggested experiment would help if I have understood correctly.\n2. Likewise, it would be interesting to see *which neurons* are the ones who are being helped by the proposed approaches in the paper: for example, one could plot the difference between the PE and Bayesian oracle as a function of the sparsity of the neuron response q, which would be interesting to see.\n3. The Bayesian oracle does still seem to have a (small) proportion of neurons for which the null model outperforms the GS model (in figure 1). Is there a reason for this/can we identify which neurons these are?\n4. The final section 4 seems a bit orthogonal to the rest of the paper, which is focused on improving IG. It would be good to develop why we care about IG relative to other metrics more though, but this should come earlier in the paper. \n5. Could the authors provide some more motivation for what benefits does improving the upper bound for the IG evaluation metric provide in practice, e.g. can we say something about neuron encoding models that we previously couldn't with the PE upper bound?", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"correctness": "3: Some of the paper’s claims have minor issues. A few statements are not well-supported, or require small changes to be made correct.", "technical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work.", "empirical_novelty_and_significance": "3: The contributions are significant and somewhat new. Aspects of the contributions exist in prior work."}, "extra_text": {}}]

Author rebuttal:

Decision: Accept: poster