Truncated proposals for scalable and hassle-free simulation-based inference
Authors: Michael Deistler
Pedro J. Goncalves
Jakob H. Macke
Venue: NeurIPS 2022 Accept (2022)
Keywords: likelihood-free inference
simulation-based inference
Bayesian inference
neural density estimation

Abstract: Simulation-based inference (SBI) solves statistical inverse problems by repeatedly running a stochastic simulator and inferring posterior distributions from model-simulations. To improve simulation efficiency, several inference methods take a sequential approach and iteratively adapt the proposal distributions from which model simulations are generated. However, many of these sequential methods are difficult to use in practice, both because the resulting optimisation problems can be challenging and efficient diagnostic tools are lacking. To overcome these issues, we present Truncated Sequential Neural Posterior Estimation (TSNPE). TSNPE performs sequential inference with truncated proposals, sidestepping the optimisation issues of alternative approaches. In addition, TSNPE allows to efficiently perform coverage tests that can scale to complex models with many parameters. We demonstrate that TSNPE performs on par with previous methods on established benchmark tasks. We then apply TSNPE to two challenging problems from neuroscience and show that TSNPE can successfully obtain the posterior distributions, whereas previous methods fail. Overall, our results demonstrate that TSNPE is an efficient, accurate, and robust inference method that can scale to challenging scientific models.

Truncated proposals for scalable and hassle-free simulation-based inference
Michael Deistler
Pedro J Gonçalves"
University of Tübingen
University of Tübingen
michael, wtt.::dddeeeeeemmmtteeeeeeee wtweaammmttbbbieenemmmm de
pedro redro.gond gww.oooonnaaavnnnnioocmmm weeenitttueiinnnncmm
Jakob H Macke"
University of Tübingen
Max Planck Institute Intelligent Systems
jakob mawkeeeuuittuubigeeeeee
Abstract
Simulation-based inference (SBI) solves statistical inverse problems by repeatedly running stochastic simulator and inferring posterior distributions from model- simulations. To improve simulation efficiency, several inference methods take a sequential approach and iteratively adapt the proposal distributions from which model simulations are generated. However, many of these sequential methods are difficult to use in practice, both because the resulting optimisation problems can be challenging and efficient diagnostic tools are lacking. To overcome these issues, we present Truncated Sequential Neural Posterior Estimation (TSNPE). TSNPE performs sequential inference with truncated proposals, sidestepping the optimi- sation issues of alternative approaches. perform coverage tests that can scale to complex models with many parameters. We demonstrate that TSNPE performs on par with previous methods on established benchmark tasks. We then apply TSNPE to two challenging problems from neuro- science and show that TSNPE can successfully obtain the posterior distributions, whereas previous methods fail. Overall, our results demonstrate that TSNPE is an efficient, accurate, and robust inference method that can scale to challenging scientific models.
addition, TSNPE allows to efficiently
Introduction
Computational models are an important tool to understand physical processes underlying empirically observed phenomena. These models, often implemented as numerical simulators, incorporate mechanistic knowledge about the physical process underlying data generation, and thereby provide an interpretable model of empirical observations. In many cases, several parameters the simulator have to be inferred from data, e.g., with Bayesian inference. However, performing Bayesian inference in these models can be difficult: Running the simulator may be computationally expensive, evaluating the iikelihood-funcinn might be computationally infeasible, and the model might not be differentiable. In order to overcome these limitations, Approximate Bayesian Computation (ABC) methods [Beaumont
al. 2002 2009], synthetic likelihood approaches [Wood] 2010). and neural network-basel methods
le.g.. Papamakarios and Murray. 2016 [Hermans et al. 2020 Thomas et al. 2022 have been developed.
"Equal contribution
36th Conference on Neural Information Processing Systems (NcurIPS 2022).

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A subset of neural network- based methods. known as neural posterior estimation (NPE) Papamakarios and Murray, 2016 Lueckmann et al.
Ground
truth
2017, Greenberg et al. 2019]. train neural density estimator on simulated data such that the density estimator directly approximates the posterior. Unlike other methods, NPE does not require any further Markov-chain Monte-Carlo (MCMC) or variational inference (VI) steps. As it provides an amortized approximation of the posterior, which can be used to quickly evaluate and sample the approximate posterior for any observation, NPE allows the application in time-critcal and high-throughput inference scenar-
APT
ios [Gonçalves et al., 2020 Radev et al., 2020 Dax et al. 2021]. and fast
TSNPE
application of diagnostic methods which require postenor samples for many different observations (Cook et al. 2006 Talts et al.
2018 In addition,
unlike methods targeting the likelihood (e.g., neural likelihood estimation, NLE [Papamakarios et al., 2019, summary statistics from data and it Lueckmann et al. 2019)). NPE can learn equivariances the simulations to improve the quality of inference [Dax et al. 2021 2022].
APT vs TSNPE. Top: Prior (gray) and true pos- terior (black), APT matches true posterior within the prior bounds but 'leaks' into region without prior support. TSNPE (ours) matches true posterior.
Figure 1:
If inference is performed for particular observation X.. sampling effi- ciency of NPE can be improved with sequential training schemes: Instead of drawing parameters from the prior distribution, they drawn adaptively from a proposal (c.g.. a posterior estimate obtained with NPE) in order to optimize the posterior accuracy for particular x,, These procedures are called Sequential Neural Posterior Estimation (SNPE) PPapamakarios and Murray, 2016 Lueckmann et al. 2017 Greenberg al. 2019] and have been reported to be more simulation-efficien than training the neural net- work only on parameters sampled from the prior, across set of benchmark
tasks [Lueckmann et al.) [
Despite the potential to improve simulation-efficiene two limitations have impeded more widespread adoption of SNPE by practitioners: First, the sequential scheme of SNPE can be unstable. SNPE requires modification of the loss function compared to NPE, which suffers from issues that can limit its effectiveness on (or even prevent their application to) complex problems (see Sec. 2 Second, several commonly used diagnostic tools for SBI TTatss al. 2018 Miller et al.) 2021 Hermans et al.] [021] rely on performing inference across multiple observations. In SNPE (in contrast to NPE), this requires generating new simulations and network retraining for each observation, which
often prohibits the use of such diagnostic tools [Lueckmann et al., 202 Hermans et al. 2021].
Here, we introduce Truncated Sequential Neural Posterior Estimation (TSNPE) to overcome these limitations. TSNPE follows the SNPE formalism, but uses a proposal which a truncated version of the prior: TSNPE draws simulations from the prior, but rejects them before simulation if they lie outside of the support of the approximate posterior. Thus, the proposal is (within its support) proportional to the prior, which allows us train the neural network with maximum wiwiiimuu--iiklhooooooo in every round and, therefore, sidesteps the instabilities (and hence 'hassle") of previous SNPE methods. Our use of truncated proposals is strongly inspired by Blum and François (2010) and Miller et al.] (020 202T]. who proposed truncated proposals respectively for regression-ade adjusmmentt approaches ABC and for neural ratio estimation (see Discussion). Unlike methods based on likelihood(-ratio) estimation [Mllerr et al., 202T, Hermans et al. 2021. TSNPE allows direct sampling and density evaluation of the approximate posterior, and thus permits computing expected coverage of the full posterior quickly (without MCMC) and at every iteration of the algorithm, thus allowing to diagnose failures of the method even for dite-sninniio parameter spaces (we term this 'simulation-barl based coverage calibration' (SBCC), given its close connection with simulation- based calibration, SBC, Cook al. [2006]. Talts al. (2018).
We show that TSNPE is as efficient as the SNPE method 'Automatic Posterior Transformation' (APT, Greenberg et al. 2019)) several established benchmark problems (Sec. 4. We then demonstrate
that for two chal challenging neuroscience problems, TSNPE-but not APT-can can robustly identify the
posterior distributions (Sec. 4.2
2

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prior
simulated data
density estimator
posterior estimate
truncated proposal
p(0)
X ~ p(x
E log Dos (o((()))
96 xx,)
((0)
Propose parameters for new simulations
Figure 2: Truncated Sequential Neural Posterior Estimation (TSNPE). The method starts by sampling from the prior, running the simulator, and training neural density estimator with maximum- sampled from the prior,
likelihood to approximate the posterior. In subsequent rounds, parameters but rejected if they lie outside of the support of the approximate posterior. With these proposals, the neural density estimator can be trained with maximum maximum--ikellhood likelihood in all rounds.
2 Background
In Neural Posterior Estimation (NPE), parameters are sampled from the prior p(0) and simulated (i.e., sampled from p(x ()). Then, neural density estimator (f)xx) (in our case normalizing flow), with learnable parameters Φ, is trained to minimize the loss:
min L = min E Exxp(x(ee log L x)],
which is minimized and only if, for sufficiently expressive density estimator, 4o (0)x) (())) for all x E supp(p(x)) [Paige and Wood 2016 Papamakarios and Murray 2016). Throughout this study, we refer to training with this loss function as maximun-likelhhod training, although the neural density estimator targets the posterior directly.
Sequential Neural Posterior Estimation (SNPE) aims to infer the posterior distribution p(0[x.) for particular observation x., SNPE initially performs NPE and, thereby, obtains an initial estimate of the posterior distribution. then samples parameters from proposal p(0), which is often chosen to be the previously obtained estimate of the posterior p(0) ( (f) and retrains the neural density estimator (Papamakarios and Murray 2016 This procedure can be repeated for several rounds.
Importantly, if parameters O are sampled from the proposal p(0) rather than from the prior p(0), the estimator qe0 x) that minimizes the maximun-likelihou loss function longer converges
to the true posterior. If one used the Manximum-likelihon loss on data sampled from p(0). i.c.,
L (g))5(0) Ex-p(x10] log 9 (f|x)), then L would be minimized by 94 ( p ( 6())
p(0)
which the true posterior. Multiple schemes have been developed overcome this [Papamakarios and Murray 2016 Lueckmann et al., 2017]. The most recent of these methods, Automatic Posterior
Transformation (APT, or SNPE-C, in atomic version) [Greenberg al., 2019 Durkan al. 2020
employs a loss that aims to classify the parameter set that generated particular data point among other parameter sets (details in Appendix Sec. 6.5))
While APT has been reported to significantly outperform previous methods, several studies have also described cases in which the approach exhibits performance issues: Both the original APT paper (Greenberg et al., 2019) and [Durkan et al. [020 reported that APT can show 'leakage" of posterior mass outside of bounded priors. We demonstrate this issue on simple I-dimensional simulator with bounded prior (Fig. Appendix Fig. Zp. The posterior estimated by APT is only required to match the true posterior density within the support of the prior (details in Appendix Sec. 6.3). Thus, after five rounds of APT, while the approximate posterior matches the true posterior within the bounds of the prior, substantial fraction of posterior mass lies in regions with zero prior probability. In simple models, approximate posterior samples that lie outside of the prior bounds can be efficiently rejected. However, in models with high numbers parameters, the rejection rate can become so large that drawing posterior samples which lie inside of the prior bounds is prohibitive. For example, Glöcklen et al. [022] reported rejection rate of more than 99. 99.999%% model with 31 parameters, thus requiring approximately one minute to draw single posterior sample from within the prior bounds.

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Algorithm 1: TSNPE
Inputs: prior p()), observation X., simulations per round N, number of rounds R, that defines the highest-probability region (HPR,)
Outputs: Approximate posterior (ø-
Initialize: Proposal p(0) p(0), dataset X
for r E [1, R| do
for
do do (())
0
simulate X; p(x
add (0;, x:) to X
argming - x,xxx log (0,
x, J,x
Compute expected coverage(ū(()).u p(0) x p(0) I
// see Alg // see Alg.
We overcome these limitations by using 'truncated' proposal distributions. This allows train with maxxiiuum-lkkelihoon at every round, thereby sidestepping issues of previous SNPE methods.
3 Methodology
3.1 Truncated proposals for SNPE
Given particular observation X suggest to restrict the proposals p(0) to be proportional to the prior p(0) at least in the e gighest-prbbabiity-reee. (HPR,, the smallest region that contains 1 e of the mass) of p(0)x.), i.e.
((0) x p(0) 1lae
with HPR, (p(0 xx)) S M. Thus, p(0) is a 'truncated' proposal. The key insight is that, when using such proposal and one can train q (0|x) with maximum likelihood:
min C = min Ee- Bxxp(xxo)] |- log go x)],
and 9 (0)xx) will still converge to p(0)x.) (Proof in Appendix Sec. 6.2)
We estimate M the HPR, the approximate posterior M HPR, (go (go Mexximum-likelihon loss employed to train g (0 (0 x) is support-covering, the HPR, of (0(0))x, x))x)) tends to cover the HPR, of p(0 x.) Bishop and Nasrabadi 2006
(0|x))) Since the
In order to obtain the HPR, of 9s (0)x.), we define a threshold t on the approximate posterior density normalizing flow (0)0)x) which allows for closed-form density
do (/(0)x, x,). To do so, evaluation and fast sampling. We then approximate the HPR, of q x.) as
HPR (0) (0))))) R3 1 (0).)))))
We chose T as the e-quattiee of approximate posterior densities of samples from Tø ( (f)x.), and evaluated TSNPE for 10-3, 10-4, and 10-5 Values of e > 0 yield proposal prior which has smaller support than the current estimate of the posterior, e.g., using e 10 10-3 neglects 0.1% % of mass from the wwwoooimatt-pssterriie support. Thus, this approach leads to errors in posterior estimation, e.g.. to mnder-covered" posteriors (Appendix Sec. 6.10 However, empirically, the error induced by this truncation is negligible, we will demonstrate on several benchmark tasks. We note that TSNPE can trained on data pooled from all rounds (Appendix Sec. 6.2). TSNPE is summarized in Alg.
(Fgg. 2
3.2
Sampling from the truncated proposal
To generate training data for subsequent rounds, we have to draw samples from the truncated proposal
5 and here we explored rejection sampling and sampling importance resampling (SIR) [Rubin] 1988 For rejection sampling, we sample the prior 0 p(0) and accept samples only if their ( probability under the approximate posterior 96 (0(x) x) is above threshold

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Figure 3: Diagnostic tool. (a) Parameter 0" (green) lies within the 1-a confidence region (gray) of the estimated posterior. (b) log(p(0"|<:)) is above the I-a quantile of posterior samples. (c) 1-a versus empirical coverage. averaged over 0"
Empirioa a Cooeen 1
underconf.
overconf.
(0g()(0(x))
Confidence level
This strategy samples from the truncated proposal exactly, but can fail if the rejection rate be- comes too high. To deal with these situations, used SIR. For each sample from the truncated proposal, SIR draws K samples from the approximate posterior, computes weights Wi=1...K = hetl:///.//////ll.
normalizes w, such that they sum to one, draws from categorical dis- tribution with weights ~ Categorical (w;). and selects the posterior sample with index s. SIR requires fixed sampling budget of K posterior samples per sample from the truncated proposal and returns exact samples from the truncated proposal for K - oo. Too low values of K lead to too narrow proposals and posterior approximations. When run for number of rounds, this behaviour reinforces itself and can lead to divergence of TSNPE (Appendix Fig. I3). We, thus, chose a high value 1024. In our experiments, we did not observe poor SIR performance, but we emphasise the importance of using tools to diagnose potential failures of TSNPE (see below) SIR (e.g. by inspecting the effective sample size, Appendix Sec. 6.12p. When STR fails, methods such as nested sampling, adaptive multi-level splitting. sequential Monte-Carlo sampling could be viable alterna- tives (Skilling 2004 Cérou and Guyader 2007 Doucet et al. 2001]. We discuss computational costs of rejection sampling and SIR in Appendix Sec.6.11
3.3 Coverage diagnostic
In order for the estimated posterior L (0(x)) to converge to p(O]x.), TSNPE requires (upp(p(0)x.)) C (u.. i.e., the estimated posterior must be broader than the true posterior (proof in Appendix Sec. 6.2). In order to diagnose whether the posterior is, on average,
sufficiently broad, perform expected coverage tests as proposed in Dalmasso al. 2020 Miller
ctt al. [2021]. Hermans et al. [021].
As described in Dalmasso et al. [020], Rozet et al. [021 and illustrated in Fig. the coverage of
the approximate posterior can be computed as
a
2
xx) (0(00""))
where 0* is sampled from the truncated proposal and x" is the corresponding simulator output. In order to approximate this integral, one has to either evaluate the approximate posterior on grid [Dalmasso al. 2020, Hermans et al. 2021] or apply Monte-Carle average which includes repeatedly sampling (and evaluating) the (unnormalized) approximate posterior [Miller et al. 2021 Rozet et al. 202T]. The first option does not scale to high-dimensional spaces whereas the second computationally In contrast, the expensive for methods estimating likelihood
and, thus, require MCMC. can be sampled from and evaluated in closed-form, lcading to computationally efficient and scalable diagnostic which can be run after every training round.
Expected coverage can be computed as an average of the coverage across multiple pairs (0" x*) Miller et al. 2021 Hermans et al. 2021 and should match the confidence level for all confidence levels a:) E 00, (Fig. Be). We term this procedure of computing the empirical coverage "simulation- based coverage calibration' (SBCC), due to its close connection with SBC [Cook et al. 2006 Talts et al., 2018 (identical under certain conditions, Appendix Sec. Sec. [6.6). For TSNPE, is important that the empirical expected coverage matches the confidence level for high confidence levels (i.e., for small a), since overconfidence in these regions would indicate that ground-truth parameters 0" are falsely excluded from the HPR SBCC is summarized in Appendix Alg. 2
5

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NPE
APT
TSNPE
Acceptance rate
GT samples in support
Gauasian Linear
10"
10" - 10"
1.0
0.9
8 0.6
0.8
0.7
0.5
0.9
SLCP
1.0
C227
0.9
SIR
10
1.0
0.9
0.8
0.5
0.9
Two Moons
1.0
1.0
Lotka Volterra
0.9
C291
100
0.5
0.9
Bernuuli GLM
1.0
1.0
C2ST
0.8
0.9
0.6
0.5
0.9
10
10
10
5
900
2.3
4
5
910
Round
Round
Number of simulations
Figure 4: Performance on six benchmark tasks. Left three columns: classifier two-sample test accuracy (C2ST) of NPE (left), APT (middle), and TSNPE (right) for three simulation budgets. Forth column: Fraction of prior samples within the approximate-poster
HPR, in each round for each simulation budget. Fifth column: Fraction of true-posterior samples within the approximate-postenne
HPR, TSNPE with 10-4 and rejection sampling from truncated proposal.
Results
We evaluated TSNPE on several benchmark tasks and on two complex problems from neuroscience. found that TSNPE performs as well as APT on the benchmark tasks and that robust to choices of e. In addition, we found that, in contrast with APT, TSNPE can successfully infer the posterior distribution for complex models with large numbers parameters.
4.1 Performance on benchmark tasks
We compared TSNPE with NPE and APT on six benchmark tasks for which samples from the ground-truth posterior are available (see Appendix Sec, 6.9 for tasks) [Lueckmann al.) 2021]. We quantified the performance with classifier two-sample test (C2ST), for which 0.5 indicates that approximate posterior identical the ground-truth posterior, whereas 1.0 implies that distributions can be completely separated by classifier. Overall, APT and TSNPE perform similarly well and both outperform NPE (Fig. left three columns). two of the six tasks (Gaussian Linear and SLCP), APT has slightly better performance than TSNPE, whereas two other tasks (SIR

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Experimental data
b
d
TSNPE posterior predictive
0.004
0.6
Wilth
0.000
0.0
AB/PD Bus
AB/PD 9Kta
Figure 5: Pyloric network inference. (a) Data (Haddad and Marder 2021 (b) APT approximate
posterior (ID-marginal) exhibits leakage (red: prior bounds). (c) APT approximate posterior when forcing the density estimator into constrained space. The spike the upper prior bound is odds with previously published posterior distributions [Gonçalves et al.) [0200 Deistler et al. 2021 Glöckler
at. 2022 and produces poor predictive samples (Appendix Fig. (d) TSNPE posterior predictive
sample matches summary statistics of the experimental data.
and Lotka- Voltera), TSNPE outperforms APT. Overall, TSNPE and APT perform similarly well, demonstrating that TSNPE is competitive with previous methods on benchmark tasks.
In order to get insights into the improved performance of TSNPE as compared to NPE, we computed the fraction of prior samples that lie within the HPR, of the approximate posterior (Fig. 4 fourth column). In tasks with broad posteriors and few simulations, the HPR, is almost as wide as the prior and thus the performances of NPE and TSNPE are similar (e.g., SLCP with 1 simulations). In other tasks and with more simulations. the HPR, is much narrower than the prior, leading to an improvement simulation efficiency (e.g., Lotka- Lotka-Volterra Volterra with 100k simulations).
Finally, we evaluated whether the HPR, of the approximate posterior contains the support of the true posterior (Fig. q fifth column). We computed the fraction of true-posterror samples within the HPR, of the approximate posterior. For most tasks, fewer than 0. 1% of samples were excluded, and the rate of erroneously rejected samples decreased more simulations with 1k and 10k simulations, many ground-truth samples were rejected and TSNPE performed poorly, but NPE and APT also failed to solve the task. Thus, while truncated proposals can potentially induce posterior biases, these only have a negligible effect on the performance of TSNPE. We note that TSNPE performance is qualitatively unaffected by the choice of e < 10-4 and proposal sampling scheme (Appendix Fig. Fig. Fig 10p. Applying truncated proposals to APT leads to equally good or worse performance than 'standard APT, depending on the task (Appendix Fig.
used. In the Lotka- Volterra task
4.2 Efficient and robust inference in two complex neuroscience problems
Next, we evaluate the performance of TSNPE on two challenging neuroscience problems, where the competitive advantage TSNPE is fully realized.
Pyloric network We applied TSNPE to a challenging real-world simulator from neuroscience: The pyloric network of the stomatogastric ganglion in the crab Cancer Borealis (Prinz et al. 2003 2004
The model has 31 parameters and simulates 3 voltage traces that we reduce to 18 summary statistics.
The prior distribution is uniform within previously described parameter ranges (Prinz all 2004
Gonçalves et al. 2020). We identify the posterior distribution given experimentally observed data
Haddad and Marder 202T (Fig- () with APT and TSNPE (13 rounds, 30k simulations per round).
When applying APT "out of the box' (from 'sbi' toolbox
et al. 2020 ). the rate samples within the prior bounds was 0.02% after the second round and
of approximate-posteere 0.0000% after the third round (Fig. (5)), which rendered a fourth round too computationally expensive.
We attempted to overcome these issues by appending transformation T to the density estimator q (0(0)x x) such that its support constrained match the support of the prior. In practice, we used a sigmoid transformation. While the resulting approximate posterior exhibited no leakage, this setup revealed another problem when running APT: In transformed (i.e., unbounded) space, the density estimator 4 (0)x) can put significant mass in regions outside of the training data. When forced into constrained space, these 'leaking" regions lead to spikes the bounds of the parameter space (Fig. 5p. further details in Appendix Sec. 6.5 illustration, additional tests and full posterior in Appendix
7

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Morphology
b
Obscrvation
Four predictives for Stop
Step
Step
Step
bAP
dend.
bAP dend.
Saparatte nets
e
Posterior distribution
f
Three Posterior predictives for Step
0.0
0.04
Posterior predictive
Empirical coverage
Ensemble
0.0
0.04
Step
Step
Step
0.0
0.001
0
soma
bAP dand.
bAP dend.
0.0
4.0
Confidence level
8
Figure 6: TSNPE on LSPC. (a) Cell morphology. (b) Observation. (c) Four prior samples for the
Step protocol. (d) Coverage for and 10 neural nets. (e) Posterior. True parameter in red. (f) Three posterior predictives for the Step 1 protocol. (g) One posterior predictive for all protocols.
Figs.
16 and
18
These spikes are at odds with previously published posterior distributions
[Gonçalves al.
2020
Deistler et al., 2021| Glöckler al. 2022 and samples from these parameter
regions do not produce good simulations (Appendix Fig. 19p. This demonstrates that leakage occurs APT even when the density estimator is forced into constrained space and that these issues lead to an incorrect posterior approximation well to poor predictive samples.
We applied TSNPE to this task for rounds without any issue.
The resulting posterior produces samples that closely match the observed data (Fig. 511 more samples in Appendix Fig. [5 posterior
distribution across all 31 parameters in Appendix Fig. 20p. The obtained posterior similar to
previously published posteriors [Gonçalves et al. 2020 Deistler et al. 2021 Glöckler et al., 2022
Multicompartment model of a single neuron Finally, we turn to a landmark problem in neuro- science for which the posterior has not yet been identified: A morphologically detailed model of thick-tufted layer pyramidal cell (LSPC) from the neocortex [Ramaswamy et al.) 2015 Markram al. 2015 Van Geit et al. 2016). The model deseribes the response of neuron current stimuli of different strengths. The model has approximately 7000 separate compartments which compose the anatomy of the cell (Fig. (8) Each compartment has dynamics based on the Hodgkin- Hodgkin-Huxley Huxley equations [Hodgkin and Huxley 1952] and contains multiple ion channels (details in Van Geit et al. (2016). The model has 20 free parameters which are the maximal channel conductances and time constants of the ion channels. The simulation consists of four separate simulations corresponding to experimental protocols which describe the voltage response to different stimuli. For the first three protocols (Step 1, Step 2, Step 3), each voltage response is characterized by 10 summary statistics. The fourth protocol models the back- bockkpropagation of the voltage response through the dendritic tree and is captured by 5 additional summary statistics (bAP soma, bAP dend. 1, bAP dend. 2). In total, the model produces 35 summary statistics, to which we add Gaussian noise with diagonal covariance matrix capturing the response variability of previously reported measurements [Hay et al. 2011]
Our goal is infer the posterior distribution over 20 parameters given 35 summary statistics that were simulated and thus have a known ground-truth parameter set- to resemble experimentally observed activity (Fig. (p). The prior is uniform distribution within previously established bounds (Van Geit et al.] 2016). A major difficulty in fitting this model is that large fraction of prior samples generate summary statistics that are very different from the observed data: particular, about 99.989...9999 90.9% of prior predictives contain at least one summary statistic that is undefined, c.g., time to first spike is

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undefined in the absence of spikes (Fig. (p). When a summary statistic is undefined, we assign it a value that is substantially outside the range of the observed data (Appendix Sec. 6.14).
We ran TSNPE over six rounds (hyperparameters in Appendix Sec. 6.15) In each round, we ran 30k simulations, leading to a total of 180k simulations. After every round, we evaluated the expected
coverage with SBCC. After the first round, the approximate posterior exhibited poor expected coverage (Fig. O top). Therefore, as suggested by [Hermans et al. [02TT we used an ensemble of 10 neural density estimators to ensure that the approximate posterior is sulficiently broad (Fig. - bottom). Although the approximate posterior remains underconfident, the empirical expected coverage closely matches the confidence level for high-confidence levels, which is crucial for TSNPE (Sec. B.3)
The TSNPE-posterie has several parameters with broad marginals, demonstrating that this model exhibits 'degenercy'', a widespread phenomenon in biological systems [Marder and Taylor 201 (Fig. Ge. Appendix Sec. 6. for parameter names). Other marginals are narrow, demonstrating that the model is sensitive to changes in these parameters. Posterior predictive samples closely match the observed data (Fig. gg more samples in Appendix Fig. 22 We emphasize that fitting such neuron models is a challenging and widespread problem in neuroscience, one for which commonly used methods [e.g.. genetic algorithms, Druckmann al. 2007 Van Geil et al. 2016] are often simulation-ineffica
morphologically-detille
or do not estimate the full posterior distribution. We show promising results, suggesting that TSNPE could be applied other complex single-neuron models.
In contrast, with 'out of the box' APT, none of the 10M upproximate peproximate-posstrioor samples was within the prior bounds after the second round. This rendered third round too computationally expensive. Overall, the results on the pyloric network model and on the multicompartment model demonstrate that TSNPE is efficient and robust method that scales to complex and tighdiimensionl models that were inaccessible to the state-of-the-an method APT.
5 Discussion
We presented a new method to perform Bayesian inference in implicit models, which we call Trun- cated Sequential Neural Posterior Estimation (TSNPE). Like previous methods, TSNPE adaptively selects parameters to improve with many parameters. The key ingredient is that TSNPE samples parameters from a 'truncated' region of the prior, and thus overcomes instabilities of previous methods while maintaining simulation efficiency. In order to diagnose potential errors of TSNPE, we developed coverage test that can be run quickly and at every round of TSNPE. TSNPE presents new variant of SNPE which is at least as powerful as previous variants on benchmark tasks, but provides a powerful alternative which is able to solve inference problems on which the state-of-the-arr method APT failed.
snqullation-fffcienne and allow posterior inference in complex models
Related work TSNPE differs from automatic posterior transformation (APT. SNPE-C) in its proposal and its loss function: TSNPE uses a truncated prior as proposal, while APT can flexibly use any proposal, which, c.g., allows for more sophisticated active learning rules [Lueckmann et al. 2019 Järvenpää et al. 2019]. However, APT's flexibility requires a modification of NPE loss function, which can be an impediment to its usage in practice: First, the modification can lead to 'leakage', which can make it prohibitive to draw samples within prior bounds. Second, APT loss requires an explicit prior and thus, cannot be applied to models in which the prior can only be sampled [Ramesh] et al. 2022]. Third, current formulations of APT cannot discard parameters leading to invalid simulations as the posterior mass would 'leak' into parameter regions which only produce invalid simulations (Appendix Sec. 6.5). It might be possible that these issues are resolved using modified formulation of APT, c.g., by combining its atomic loss with additional loss terms, or preventing leakage by penalizing 'bad' parameters Greenberg et al. 2019 In cases in which leakage prevents application of APT (in particular, in high-dimensional problems), TSNPE provides an alternative.
Our method is inspired by previous work that introduced mechanism to post-hoc correct samples obtained by an Approximate Bayesian Computation (ABC) algorithm [Blum and François 2010). i.e., 'regression adjustment ABC'. Their method draws samples from a truncated region of the prior to avoid correction terms, but estimates the posterior density with ABC samples-rathe than using a flexible neural density estimator-and and estimates the support by training a dedicated support-vector machine. In addition, the method runs a single round of truncation and retraining, whereas we demonstrate that TSNPE can be robustly applied across 10 rounds.

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Truncated proposals have also been proposed for neural ratio estimation [Trunccaeed Marginal Neural Ratio Estimation (TMNRE) Miller et al. 202T]: TMNRE uses truncated proposals to efficiently infer selected posterior marginals while being amortised around the observation, allowing to test the coverage properties of the selected marginals, e.g., with SBC (Cook et all 2006 Talts et al. 2018 In addition, truncating based on the marginals allows TMRNE to sample from the truncated proposal without rejection or SIR sampling. In contrast, TSNPE aims at efficiently inferring the full posterior distribution by proposals that avoid the correction of SNPE loss function. Truncating the proposal based on the full posterior rather than on the marginals can lead to drastically narrower proposals: E.g., on the pyloric network problem, truncation based on posterior marginals rejects 20% of prior samples versus 99.94% rejection based on the posterior joint. In addition. while TMNRE uses the expected coverage to test the consistency of the posterior marginals, TSNPE can test the expected coverage of the full posterior distribution.
Possible failure modes The main failure mode of TSNPE will occur if the truncated proposal excludes significant portions of density mass of the true posterior (e.g.. if the estimate misses posterior modes). In these cases, the learned approximate posterior will put systematically too little mass the excluded regions. We recommend the use of diagnostic tools such as SBCC to identify such failures
[Cook et al. 2006 Millr et al., 2021 Hermans et al. 2021 Rozet et al. 2021
In addition, if the true posterior has unbounded support, any finite values of E > 0 will lead to biased approximate posterior which puts too little weight in the posterior tails. In that case, and when running TSNPE across many rounds, the errors from each individual round could accumulate. Although we did not observe this bias to significantly affect the algorithm performance on several benchmark tasks, we cannot exclude the possibility of a substantial performance degradation when running TSNPE for a larger number of rounds (> 10).
Finally, unlike SNPE methods that use the previous estimate of the posterior as the proposal distribu- tion, our method requires scheme to sample from a truncated proposal. If the sampling scheme is inaccurate (i.e.. if it does not produce proposal distribution that is proportional to the prior within the truncated region), the results of TSNPE will be biased. To avoid this, we recommend using rejection sampling by default and using SIR or sequential Monte-Carlo methods only if rejection sampling is too computationally expensive. For SIR, it is important to use large oversampling factor K (e.g..
K = 1024) and use diagnostic tools such as effective sample size (Appendix Sec. 6.12
Fig. 13
Simulation- coverage calibration In order to diagnose whether the approximate posterior is broader than the true posterior, applied SBCC, coverage test for TSNPE (Cook et al., (006) Rozet et al. 202T]. SBCC evaluates the expected coverage of the approximate posterior without evaluating it on a grid Daalmasoo et al. 2020, Hermans al. 202T and, unlike diagnostic tools for methods based on learning the likelihood( ( does not require MCMC runs for multiple observations x [Miller al. 202T] This allows SBCC to be run quickly and for models with many parameters. In addition, in contrast to diagnostic tools for likelihood-free inference with Approximate Bayesian Computation, SBCC does not require an additional step to estimate the density of approximate posterior samples Prangle et al. [014). We note that, since SBCC is a variation of SBC (Cook et al., 2006 Talts et al., 2018], only ensures that the HPR, correct on average across observations, not for a particular observation. In principle, SBCC could be applied to other SNPE variants, although empirically the impact of arbitrary proposals on SBCC performance is currently unclear.
Conclusion Overall, TSNPE combines the simulation-efficiency of sequential neural posterior estimation with the robustness and coverage-tests of non-sequential methods. We demonstrated that it allows scale neural posterior estimation complex and high-dimensional scientific problems.
Acknowledgments and Disclosure of Funding
We thank Poornima Ramesh, Cornelius Schröder, Marcel Nonnenmacher, David Greenberg, and Jan-Matthis Lueckmann for discussions and feedback. We also thank the International Max Planck Research School for Intelligent Systems (IMPRS-IS) for supporting MD. This work was funded by the German Research Foundation (DFG; Germany's Excellence Strategy MICoE - EXC number 2064/1 PN 390727645) and the German Federal Ministry of Education and Research (BMBF; Tübingen Al Center, FKZ: (UIS18039A)
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Checklist
1. For all authors...
(a) Do the main claims made in the abstract and introduction accurately reflect the paper's
contributions and scope? [Yes]
(b) Did you describe the limitations of your work? [Yes]
(c) Did you discuss any potential negative societal impacts of your work? [N/A]
(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
2. If you are including theoretical results...
(a) Did you state the full set of assumptions of all theoretical results? [Yes]
(b) Did you include complete proofs of all theoretical results? [Yes]
3. If you experiments..
(a) Did you include the code, data, and instructions needed to reproduce the main experi- mental results (either the supplemental material or as URL)? [Yes]
(b) Did you specify all the training details (e.g., data splits. hyperparameters, how they were chosen)? [Yes]
(c) Did you report error bars (e.g.. with respect the random seed after running experi- ments multiple times)? [Yes]
(d) Did you include the total amount of compute and the type of resources used (c.g., type of GPUs, internal cluster, or cloud provider)? [Yes]
4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
(a) If your work uses existing assets, did you cite the creators? [Yes]
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5. If you used crowdsourcing or conducted research with human subjects...
(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]
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(c) Did you include the estimated hourly wage paid to participants and the total amount spent participant compensation? [N/A]
15


Reviews:
[{"review_id": "WWwvxu7P2I", "reviewer": "Reviewer_Afw3", "summary": "The manuscript presents a new sequential neural posterior estimation (SNPE) algorithm for simulation-based inference. Simulation-based inference refers to the task of inferring parameters of interest from a real observation, based on a simulator. Neural posterior estimation (NPE) refers to methods aiming to build a surrogate for the posterior (typically a normalizing flow) based on samples from the simulator. Sequential NPE refers to their sequential counterparts that alternate between training and sampling phases with the objective of sampling simulated observations close to the real observation of interest and hence typically sample parameters to use for simulation from the current posterior surrogate. However, current SNPE methods suffer from two issues limiting their applicability:\n* As training samples are sampled from the current posterior surrogate, they are non-amortized, meaning that the algorithms must be fully rerun for each observation. In opposition, amortized methods reuse the same model to perform inference on multiple observations. Non-amortized methods hence have the drawback of being inefficient to diagnose. \n* As parameters used for simulation are not sampled from the prior, parameters with zero prior probability could be sampled. This phenomenon is called leakage.\n\nTruncated SNPE (TSNPE) addresses both of those issues by using,  as proposal, a truncated version of the prior in place of the current surrogate posterior. The prior is truncated by removing parameters with surrogate posterior density below a given threshold, hence focusing on parameters of high posterior density. Consequently:\n* Parameters sampled are always within the prior support, solving the leakage issue. \n* It makes the method locally amortized as the built surrogate is valid for observations resulting from parameters sampled from a region of the prior instead of a single observation and hence enables efficient diagnostics. The paper introduces the simulation-based coverage calibration (SBCC) diagnostic that leverages this property.", "questions": "None", "limitations": "I do not see any potential negative societal impact of this work.", "rating": 7, "confidence": 3, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "### Major\n* (+) The paper is very well written and easy to follow.\n* (+) The method is sound\n* (+) It addresses two important issues in SNPE (leakage and diagnostics)\n* (+) Experiments are convincing.\n\n### Minor\n\n* (-) The SBCC diagnostic is quite known in the context of NPE and NRE (for example, see \"Arbitrary Marginal Neural Ratio Estimation\nfor Simulation-based Inference\", Rozet et al. 2021) and is a straightforward extension of SBC. Therefore, in my opinion, it should not be framed as a core contribution of the paper. The contribution rather lies in the development of an sbi algorithm that allows the use of this diagnostic (which is great).", "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": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "SevXSBJbY9L", "reviewer": "Reviewer_ue8u", "summary": "The authors present a new simulation-based inference (SBI) technique that sequentially learns a neural surrogate for the posterior. The key idea is to use a truncated version of the prior, rather than the current posterior, as a proposal distribution to generate new samples. This removes the need for an impractical importance weight during training, leading to two practical advantages: it solves a posterior leakage problem that other state-of-the-art sequential neural posterior methods (APT) have, and it simplifies the use of calibration diagnostics. There are some downsides, too: the truncation biases the posterior and sampling from the proposal distribution requires some rejection sampling, which can be inefficient.\n\nThis new method, TSNPE, is demonstrated on a suite of standard benchmarks and finally on two real-life problems from neuroscience. On the benchmarks it performs on par with other neural posterior estimation methods, while on the neuroscience problems it enables plausible solutions for the posterior where other methods failed.", "questions": "The authors present the posterior leakage of APT as a major motivation behind their work. As they point out, this posterior leakage can be fixed in APT through rejection sampling, but that can be inefficient. However, the proposed algorithm (TSNPE) also involves rejection sampling (plain or in the form of SIR). I would appreciate if the authors discuss in more detail of why the rejection sampling in TSNPE is more efficient (has a lower rejection rate) than the rejection sampling in the APT posterior. Could SIR be applied equally well to the APT posterior?\n\nIs it fair to say that in some sense, TSNPE is a compromise between vanilla NPE (no annoying importance factors in training) and SNPE (optimal use of past information in the posterior)? I'm trying to better understand how much information from past rounds is lost by sampling proportional to the prior rather than proportional to the current posterior (within some truncation region). Can the authors discuss this some more?\n\nPhrasing the same question in a different way: Compared to standard SNPE, the proposed algorithm presents two changes. 1) The proposal is truncated and 2) within the truncated support, we sample proportional to the prior rather than the posterior. It would be interesting to disentangle these two factors and understand how they individually contribute to the sample efficiency. If I'm not mistaken, only doing 2 (and not 1) corresponds to vanilla, non-sequential NPE, is that correct? How would an algorithm do that does 1 but not 2, i.e. truncates the posterior and then samples proportional to the posterior within this truncation region? Would this for all practical purposes be identical to APT?\n\nSome minor comments:\n- The authors do a good job of listing the most closely related references. Perhaps SBI has grown to be too large a field by now, but there are still a number of SBI works not cited here.\n- In the first sentence of the abstract, the authors describe SBI as \"learning posterior distributions\". As the authors themselves point out later, in many SBI methods other statistical objects like the likelihood or the likelihood ratio are learned, and other methods again do not have any explicit learning at all (like old-fashioned ABC). Maybe it is a good idea to rephrase this sentence. \n- The word \"likelihood\" is a bit overloaded and used both for the likelihood $p(x|\\theta)$, but often also to the density of some trained surrogate of the posterior $q(\\theta|x)$. This is to some extent unavoidable, as \"maximum likelihood\" is such a known term, but in other instance I found it slightly confusing (for instance in line 128). Perhaps the authors could refer to the \"probability density\" of the posterior rather than the \"likelihood\" of the posterior?\n- In the experiments, how big is the error of SIR due to finite K?\n- In the benchmark experiments, does the problem of APT posterior leakage occur? Do you fix it with rejection sampling (or in some other way)?\n- In the experiments, do the methods that rely on rejection sampling or SIR have a substantial overhead in runtime from that, or is that negligible?", "limitations": "The authors have addressed the limitations of their methods and discussed them well. Some aspects could use a more in-depth discussion, see the Questions section above.\n\nThe potential societal impact is not discussed, but I do not see any obvious issues.", "rating": 7, "confidence": 4, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "## Strengths\n\n- Scaling SBI methods to more complex problem is an important (and to some extent unsolved) problem, with potential impact in many areas of science.\n- The proposed method is, overall, sensible (with some caveats, see below) and can help make SBI work in more complex problems.\n- The downsides of the presented methods (in particular the bias from $\\epsilon > 0$ and the inefficiency of rejection sampling) are acknowledged and discussed (though this could be further improved, see below).\n- The empirical evaluation is strong. I appreciate that the authors both present benchmark problems (for which true posterior samples are available) and more challenging real-world neuroscience problem. In the latter, the authors do a good job of discussing the plausibility of the learned posterior. While the presented method does not lead to a huge improvement on the benchmark tasks, it does appear effective in the neuroscience problems, and that is ultimately more important.\n- The paper is very well written and clearly structured, quite a pleasure to read.\n- The figures (both schematics and experimental results) are well designed and very clear.\n- Related work is discussed in detail and the relation of the presented method to other approaches is made quite clear.\n- The authors have released the code to run the experiments.\n\n\n## Weaknesses\n\n- While TSNPE is a new algorithm, it is a modification of existing algorithms like SNPE and APT. Similarly, the proposed calibration tool builds strongly on existing work. This paper is thus somewhat iterative rather than revolutionary.\n- There are a couple of aspects that could use a more in-depth discussion, see the Questions section below. In particular, I would like to understand better why rejection sampling in TSNPE can be more efficient than rejection sampling based on the APT posterior. Also, it would be great if the authors could explain a bit more why sampling from a truncated prior isn't quite sample-inefficient compared to sampling from the posterior.\n\n## Bottom line\n\nI would like to thank the authors for a solid contribution to the toolbox of simulation-based inference and to a high-quality paper. Still, I think it can be made stronger by extending the discussion and addressing some questions (see below). ", "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": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "COnwU0dwLQ", "reviewer": "Reviewer_nSm9", "summary": "The paper introduces a simulation-based inference method for increasing the simulation-efficiency of Neural Posterior Estimation using an alternative proposal distribution. The proposal is constructed by truncating the prior to the HDR of the previously-estimated posterior for a particular observation of interest. They call their method Truncated Sequential Neural Posterior Estimation (TSNPE).\n\nTSNPE:\n- Overcomes issues with unstable training and leakage of posterior mass outside of the prior associated with other sequential version of NPE, namely so-called APT and SNPE-C.\n- Enables a specialized coverage test, which is dubbed Simulation-based Coverage Calibration (SBCC), which exploits the tractability of the approximate posterior and general empirical expected coverage testing on the truncated region due to \"local amortization.\"\n- 's empirical performance is shown on six benchmark tasks from the simulation-based inference benchmark as well as two complex neuroscience problems. The performance on benchmarks is not significantly distinguished from other methods. The previous versions of Sequential Neural Posterior Estimation cannot robustly identify the posterior on the neuroscience problems.", "questions": "- Please address the bullet points raised in the \"weaknesses\" section above.\n- Would you please clarify Algorithm 2 such that I can check whether or not it is novel?\n- Would you please quantify the effects of using SIR and opposed to rejection sampling for drawing from the proposal? An analysis of the variance of importance weights would help us to understand whether SIR produces useful samples for training.\n- Why do you consider computing coverage at every round to be optional? Truncating with an overconfident posterior, as is empirically produced by even the ensemble of networks in Figure 6, means that the proposal truncates too aggressively for the hyperparameter setting.\n- I am not convinced that training TSNPE based on data aggregated from all rounds during truncation converges to the HDR of the ground truth posterior. Could you please clarify where in the paper this is proven or prove it here (and include it in the paper)? Specifically, I am concerned by how a collection of data from many rounds implies a prior with mass modulated by the details of truncation regions of previous rounds. It also implies that the estimated posterior outside of the HDR is untrustworthy.\n- In Section 3.3 the authors claim that having (expected) coverage implies that the HDR of q_phi(t | x_0) is a super set of p(t | x_0). I don't think this is true. Would the authors please clarify or prove it?", "limitations": "- The simulation efficiency gained by truncation is marginal if the posterior is nearly as wide as the prior. This is mentioned in the result section, but should be put into the limitations section.\n- It is possible to miss modes with this method and inaccurately remove them from the analysis. This should be emphasized. Coverage alone does not stop this from happening, to my knowledge. (see questions / quality weaknesses.)\n- The limitations of computing coverage using the SBCC algorithms in high dimensions should be mentioned. Yes, drawing parameters is cheap from this method but it may be that a significant amount of proposed parameters are below the contour line of interest. I expect this to suffer from the curse of dimensionality.", "rating": 5, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "## Originality\n### Strengths\n- The combination of existing truncated proposals with existing Neural Posterior Estimation (NPE) is novel.\n- Addressing the issue of posterior \"leakage\" into the region without prior support in APT / SNPE-C is an important advance in the development of NPE-related techniques.\n- Although SIR is not novel, using it to sample from the truncated proposal in this context is new to me.\n\n### Weaknesses\n#### Relation to TMNRE\nA core weakness of originality in the paper is that the proposed method is merely the combination of Neural Posterior Estimation with a simplified version (only rejection / SIR sampling from the prior; no iterative bounding of prior for faster proposal sampling) of the proposal algorithm of [Truncated Marginal Neural Ratio Estimation (TMNRE)](https://arxiv.org/abs/2107.01214), another deep-learning based and highly-effective simulation-based inference algorithm. This holds when the marginals are taken to be the entire set of parameters; a mode which is suggested in the TMNRE paper on page 6. Unfortunately, this current paper relegates the exceptional similarity with TMNRE to a paragraph which emphasizes that both techniques are locally amortized (agreed) and writes that their aims are orthogonal (a point on which I disagree). Furthermore, just as in the TMNRE paper, the current paper emphasizes empirical expected coverage testing in order to avoid truncating the prior too aggressively; however, the TMNRE paper is never cited in this context. To the reviewer's knowledge, the TMNRE paper is first to use tests of coverage in this way and even uses [a very similar algorithm](https://github.com/bkmi/tmnre/blob/main/tmnre/coverage/oned.py) for computing coverage as the current paper's proposed SBCC that doesn't require integrating on a grid as the current paper claims (consult their code, which is available on github). Finally, relevant to TMNRE, several examples of applications of simulation-based inference to scientific problems are cited; however, relevant work applying TMNRE (which uses the truncated proposals suggested by the current paper) is not cited. For example: [Fast and Credible Likelihood-Free Cosmology with Truncated Marginal Neural Ratio Estimation](https://arxiv.org/abs/2111.08030) and [Estimating the warm dark matter mass from strong lensing images with truncated marginal neural ratio estimation](https://arxiv.org/abs/2205.09126). Also the workshop paper which predated TMNRE, [Simulation-efficient marginal posterior estimation with swyft: stop wasting your precious time](https://arxiv.org/abs/2011.13951), is not cited either.\n\nTo address this issue, I suggest the authors:\n- Clearly highlight the similarities between TMNRE and relationship to NPE. In particular, by emphasizing the similar methodology (in the case of parameters of interest = all parameters) applied to a different simulation-based inference technique. In my opinion, the connection is so close that it could be mentioned in the abstract along with the relationship to Blum and François [2010]. Also highlight the new key innovations contained in this paper.\n- Clarify that the SBCC algorithm (as best as I can understand it given the state of its clarity in the current draft, see Clarity) is quite similar to techniques used in the computation of coverage by TMNRE and include TMNRE in citations in the discussions of using coverage as a diagnostic tool. (A very similar algorithm is also used in the averting a crisis in sbi paper, see github code.)\n- Introduce the application / workshop paper citations suggested above.\n\n#### SBCC Algorithms\nThe current paper suggests that coverage testing has only been done by integrating on a grid in the cited papers; however, this is not the case. An example is shown above in the previous section for TMNRE. Here is an example from the [Averting a crisis in SBI](https://github.com/montefiore-ai/averting-a-crisis-in-simulation-based-inference/blob/a53f34ccfa2b520c58f46158e52e235766da681f/workflows/SNL/mg1/snl.py#L116-L129) paper.. Both TMNRE and the crisis in SBI papers use a similar algorithm to SBCC and the crisis in SBI paper computes coverage for the entire set of parameters. In addition to the stated similarity to simulation-based calibration, this shows that the SBCC algorithm is not novel; although its explicit enumeration is useful for the community. \n\nTo address the issue:\n- The paper should be updated to reflect that coverage is not generally computed by these other papers based on integrating on a grid.\n- The paper should be updated to reflect that the SBCC algorithm is not novel.\n\n**Caveat: Algorithm 2, which corresponds to SBCC has several typos. If this means that I didn't understand how it was novel, I would change my opinion here. Either way these typos must be fixed (see Clarity).**\n\n#### Nested Sampling\nThe paper does not mention nested sampling, although truncation is an important part of the process. Please consider citing the following review https://arxiv.org/abs/2101.09675, and these papers:\n- https://projecteuclid.org/journals/bayesian-analysis/volume-1/issue-4/Nested-sampling-for-general-Bayesian-computation/10.1214/06-BA127.full\n- https://arxiv.org/abs/0809.3437\n- https://arxiv.org/abs/1904.02180\n- https://arxiv.org/abs/1506.00171\n\n\n## Quality\n### Strengths\n- Addressing the leakage issue with APT / SNPE-C is demonstrated.\n- The neuroscience tasks are made possible using the algorithm.\n- The use of coverage testing to determine whether or not it is safe to truncate is a good idea and in line with previous work on the subject (TMNRE / crisis in sbi).\n\n### Weaknesses\n- The improvement on the benchmark is marginal, but the benchmark does not usually produce a clear \"winner\" among methods. Perhaps this should be stated within the paper.\n- In Figure 6 d (and corresponding text) the use of ensembles does not produce a conservative estimator of the posterior, although it improves matters compared to single neural networks. Truncating in this round would lead to an overconfident estimate of the HDR. This is not mentioned in the paper, but it should be highlighted as exactly the failure mode to avoid!\n- It should be made clear that computing coverage only in the final round does NOT tell the user if the algorithm truncated too aggressively since parameter samples are drawn from the already truncated prior. In this setting, it is like computing the coverage of different problem where the prior has been replaced by a truncated version.\n- The term coverage is generally misused throughout the paper and not well explained in the proposed SBCC. The paper computes the expected coverage. Coverage is a frequentist concept which characterizes the confidence sets, not the credible regions discussed in this paper. Expected coverage is discussed the averting a crisis in sbi paper and implies a coverage test for averaged over simulations from the proposal. (See questions)\n- Truncating is somehow \"dangerous\" when the first rounds of NPE miss a mode of the posterior, since it will not be represented in the subsequent rounds due to truncation. This is not sufficiently addressed and should have its own section. (See the limitations response prompt for more discussion.)\n- It looks like the parameter rejection rate for many benchmark problems goes to nearly zero at higher rounds. The amount of help SIR provides in this is not well quantified; does using K=1024 distort the proposal distribution due to high variance in the importance weights? Does it affect the final posterior? This is an importance nuance which is not sufficiently explored. (See questions.)\n\n## Clarity\n### Strengths\n- The main text is well written and easy to follow.\n- I am extremely familiar with this field of research. I feel that it has a satisfactory preamble and main paper description. I will give a positive score but look to other reviewers to see if it is clear for others who are less well acquainted. Namely, I believe the paper references concepts from the development of deep-learning based simulation-based inference which are clear to me but perhaps not to others.\n- I have not looked at the code presented in the paper, but hydra has been used for documenting the details of every run for reproducibility. That's very good!\n\n### Weaknesses\n- The proof of convergence in Section 6.2 is a difficult to follow. It could benefit from fewer words and more terse mathematical description.\n- Algorithm 2 uses symbols which are starred then removes the stars later. It also uses symbols which are not introduced. This makes the algorithm unclear. Perhaps a verbal description would be beneficial as a comment? (See questions.)\n- Algorithm 3 uses symbols which are not introduced. This makes the algorithm unclear. Perhaps a verbal description would be beneficial as a comment? (See questions.)\n- The statement about the region which is well calibrated in 6.12 is unclear to me. Is the implication that the estimated posterior is truncated based upon the most recent round, thereby the support of q handles any issues regarding sampling parameters from the prior? If this is the case, it is a nuance which I suspect to pass most readers by and should be clarified. (see questions)\n- It is not made clear that training on data drawn from all rounds should produce an accurate posterior. (Section 6.11) Drawing data from all rounds means that the \"implicit prior,\" due to the use of a mixed proposal, can be modulated by a sum of R step functions with arbitrary support. Consider that in the limit of large steps that implies the modulating function can be extremely expressive. (see questions)\n\n## Significance\n### Strengths\n- Although I am not an expert in neuroscience, the author's claims imply that the expanded relevance of NPE to these problems is significant.\n- Addressing leakage in APT / SNPE-C is quite practical.\n\n### Weaknesses\n- As discussed in the Originality section above, the novelty of the TSNPE is marginal.\n- Unless it is shown that training on data from all rounds is acceptable, then the proposed algorithm has a fundamental flaw. (It is addressed by training only on data drawn from that lies within the previous round's proposal, but then the performance on various tasks is unknown.)", "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": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "5xmwBTeSYv", "reviewer": "Reviewer_VtwR", "summary": "Given a generative model $\\mathcal P(\\theta, x)$ of the model parameters $\\theta$ and data $x$, the goal is to solve the inverse problem $\\mathcal P(x | \\theta)$. Neural Posterior Estimation (NPE) uses simulations from $\\mathcal P(\\theta, x)$ to train an approximate posterior distribution $q_\\phi (\\theta | x)$ that is tractable and has learnable parameters $\\phi$.\nIf one is interested in solving the inverse problem for a particular data realization $x_0$, Sequential Neural Posterior Estimation (SPNE) proposes a iterative sequence, starting at the prior, where the sampler for the model parameters is drawn from the previously learned $q_\\phi$.\n\nRather than plugging in the previous proposal. this paper proposes a sequence of truncated prior distributions.\nWith Truncated Sequential Neural Posterior Estimation (TSPNE), the sampler for $\\theta$ at each step is drawn from the prior distribution truncated at the highest-probability region (HPR) of the most recent approximate posterior.\nThis overcomes the drawback of SPNE that the maximizer of the loss function no longer converges to the true posterior.\nThe authors show that as long as the chosen HPR covers the support of the true posterior distribution, TSNPE provably converges to the true posterior.\nThe authors also introduce a metric called simulation-based coverage calculation (SBCC) that evaluates the coverage of their learned posterior approximation. This coverage is based on whether the quantiles of the learned log-density cover the log-density evaluated at the true parameter across repeated draws from the generative model.\n\nThe authors evaluate their procedure on six synthetic benchmarks and compare to the original NPE and APT, which is a specific variant of SPNE.\nThey show that their method mostly matches or outperforms APT and NPE, although APT seems to do better in two cases.\nThey then compare APT and their TSNPE on a simulation of pyloric networks in crabs.\nTheir TSNPE method does well in this example and is similar to previously known posteriors.\nThey also show that APT does poorly in this case.\nTheir final example is on a multicompartment model of a single neuron. Since only one proposal exhibited poor coverage based on their SBCC metric, they use an ensemble of 10 proposals and find this leads to much better coverage.\nThe competing APT method also does poorly in this case.\n", "questions": "+ Instead of an ensemble mixture to solve the poor coverage in the multicompartment experiment, would it work to form a defensive mixture by sampling from the untruncated prior with some probability? Even if this probability were small, it may help get around the issue of the HPR not covering the true posterior support.\n+ It has been suggested that the forward KL objective (used here) leads to an overdispersed posterior compared to the reverse KL typical of variational inference. Is this applicable to TSNPE?\n", "limitations": "The authors address the main limitation of their method that the estimated HPR may leave out the true posterior support, and they describe their proposed SBCC metric for checking poor coverage.\n", "rating": 7, "confidence": 4, "soundness": 4, "presentation": 4, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "## Strengths\nThis paper builds on the established Sequential Neural Posterior Estimation (SNPE) framework, but using a truncated prior distribution as a prior. The truncation is achieved by rejecting samples outside of the highest-probability region (HPR) of the previously-estimated posterior. This method is simple, yet it makes intuitive sense because it prevents sampling parameters that are not possible under the prior and parameters unlikely under the estimated posterior.\nMoreover, it corrects the main problem with SPNE by ensuring that the minimizer of the TSNPE objective at eatch stage is the true posterior.\nThis property hinges on the HPR covering the support of the true posterior, which may not be guarenteed in practice, but the authors develop the SBCC metric to verify the validity of the learned posteriors.\nThis lends confidence to the proposed method.\n\nThe pyloric network and multicompartment experiments demonstrate that TSNPE can work in challenging scenarios that the competing method (ANP) fails in. In the multicompartment experiment, it was nice to see that the authors' SBCC metric identified that a single neural network led to poor coverage and was corrected by an ensemble method.\n\n## Weaknesses\nThe main weakness to this work are the simple simulation experiments in section 4.1. While TSNPE appears to match the performance of other methods, it would be reassuring to re-create the failure of APT observed in the more complex neurological applications in a simple setting.\n", "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": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}]

Author rebuttal:

Decision: Accept