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Ryosuke Shibukawa, [Shoichi Matsuda](https://orcid.org/0000-0002-0640-3404), Kazuha Nakamura, [Ryo Tamura](https://orcid.org/0000-0002-0349-358X), [Koji Tsuda](https://orcid.org/0000-0002-4288-1606)

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[Collecting diverse near-optimal samples via nested Thompson sampling](https://mdr.nims.go.jp/datasets/b762cd44-32f2-472e-b62a-a47d025be615)

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npj | computationalmaterials ArticlePublished in partnership with the Shanghai Institute of Ceramics of the Chinese Academy of Scienceshttps://doi.org/10.1038/s41524-026-02067-0Collecting diverse near-optimal samplesvia nested Thompson samplingCheck for updatesRyosuke Shibukawa1, Shoichi Matsuda2 , Kazuha Nakamura2, Ryo Tamura1,3,4 & Koji Tsuda1,3,4Self-driving laboratories (SDLs) that combine automated experiments with machine learning haveaccelerated data-driven discovery. Although Bayesian optimization (BO) is widely used in SDLs toautonomously propose experimental conditions, many real systems require sampling diverse near-optimal candidates rather than identifying a single optimum.We propose nested Thompson sampling(NTS), a batchBOmethod that enhances diversity by incorporating the concept of nested sampling. InNTS, regions where the posterior exceeds a likelihood threshold are uniformly sampled, enablingexploration of multiple promising regions while requiring only one hyperparameter, that is, thethreshold schedule. Benchmark studies using materials datasets demonstrated that NTS achieveshigher sample diversity than a conventional batch BOmethod. Furthermore, the application of NTS toautomated electrolyte exploration in an SDL successfully produced diverse experimental samples.The NTS algorithm is implemented in the NIMO package, providing a practical framework forautonomous and diverse materials exploration.Research innovation is being driven by self-driving laboratories (SDLs),which realize closed-loop experimentation by combining automatedexperiments with machine learning-based experimental conditionproposals1,2. As an algorithm for proposing next experimental conditionswithout human intervention, Bayesian optimization (BO) is most com-monly used and has been adopted in many SDLs3–15. Furthermore, becausemany automated experimental systems are capable of parallel experiments,there is an increasing demand for batch BO16–19, which simultaneouslyproposes multiple experimental conditions. For example, as a conventionalbatch BO method, Snoek et al.20 proposed a sequential process, where thescores near already selected points are reduced, theoretically encouragingthe selection of diverse points across the batch. This approach can be viewedas a sampling-based extension of the kriging believer strategy proposed byGinsbourger et al.21When performing self-driving optimization in a completely unex-plored experimental system, the true objective is not always to find only thesingle optimal solution22–25. For instance, inmaterials science,materials withslightly inferior performance but simpler fabrication processes, or thosewith lower cost, are oftenmore valuable candidates than the optimal one. Insuch cases, the goal is to sample many near-optimal solutions with diverseexperimental conditions. To address this, batch BO using a determinantalpoint process (DPP) is a promising candidate26–28. DPP ensures diversityamong simultaneously proposed experimental conditions throughMarkovchain Monte Carlo (MCMC)-based computations. Nevertheless, DPP-based algorithms require the predefinition ofmany hyperparameters beforestarting the closed-loop experiments, making their application to newproblems nontrivial. Therefore, a method capable of proposing diversesamples with as few hyperparameters as possible is required in SDLs.We propose nested Thompson sampling (NTS), a method thatachieves batch BOwhile maintaining sample diversity by incorporating theconcept of nested sampling. Nested sampling is a technique that gradually“shrinks” the sample space by raising the likelihood threshold step by step,and the only hyperparameter is the schedule for increasing thisthreshold29–33. In NTS, first, models are sampled from the Gaussian processused as the surrogate model in BO (Thompson sampling). Next, when thelikelihood exceeds the threshold, the posterior is treated as a constant value;otherwise, it is set to zero (see Fig. 1). This enables sampling from multiplepeaks in a multimodal landscape, resulting in higher sample diversity.Unlike conventional Thompson sampling, which selects the single pointmaximizing the posterior function as the next experimental condition, NTSuniformly samples the next experimental conditions from a broader regionwhere the posterior exceeds the threshold. Thus, batch BO with highdiversity can be performed by NTS.In this study, we investigated three strategies for increasing thethresholdLt, which is the only hyperparameter inNTS: the top 10%, 5%, and1% of previously obtained values were used as thresholds, named “con-servative”, “moderate”, and “aggressive”, respectively. These thresholds arecommonly used in statistical hypothesis testing (e.g., 1%, 5%, and 10%1Graduate School of Frontier Sciences, The University of Tokyo, Kashiwa, Chiba, Japan. 2Center for Green Research on Energy and Environmental Materials,National Institute forMaterials Science, Tsukuba, Ibaraki, Japan. 3Center for BasicResearch onMaterials, National Institute forMaterials Science, Tsukuba, Ibaraki,Japan. 4RIKEN Center for Advanced Intelligence Project, Tokyo, Japan. e-mail: matsuda.shoichi@nims.go.jp; tamura.ryo@nims.go.jp; tsuda@k.u-tokyo.ac.jpnpj Computational Materials |          (2026) 12:197 11234567890():,;1234567890():,;http://crossmark.crossref.org/dialog/?doi=10.1038/s41524-026-02067-0&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41524-026-02067-0&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41524-026-02067-0&domain=pdfmailto:matsuda.shoichi@nims.go.jpmailto:tamura.ryo@nims.go.jpmailto:tsuda@k.u-tokyo.ac.jpwww.nature.com/npjcompumatssignificance levels) and provide a practical range for balancing explorationand exploitation. To quantify the diversity of sampled conditions, we usedthe metric #Circles34,35. The definition of #Circles is as follows:#CirclesðS; d; rÞ ¼ maxC�SjCjs:t: dðxi; x0iÞ > r; 8xi; x0i 2 C; i≠i0; ð1Þwhere S = {x1,...,x∣S∣} is the dataset of experimental conditions for near-optimal samples,C is a subset of S, and r is radius.Here,dðxi; x0iÞ is a distancemetric between xi and x0i. As r increases,multiple points become included inthe same subset, and thus the #Circles metric decreasesmonotonically withrespect to r (see Fig. 2). As shown in Fig. 2, in the case of dense data points(left case), when the radius r is increased, a large number of pointsimmediately fall into the same subset, causing the #Circles to decreaserapidly. In contrast, for diverse data points (right case), increasing rdoes notimmediately cause data points to fall into the same subset, and #Circlesmetric bulges upward and lies in the upper-right part compared to the densecase. Therefore, a function that bulges upward in the r-dependency of the#Circles indicates that the dataset contains more diverse samples. Toquantify #Circles metric, we defined the area under the #Circles curve(Circles-AUC) as follows:I ¼Z 10#Circles ðS; rÞdr: ð2ÞA larger Circles-AUC indicates that more diverse near-optimal samples areobtained.We conducted a benchmark study using existing materials datasets:melting temperatures of an alloy system36 and grain-boundary energiescalculated by first-principles calculations37. As a result, NTS was found toproduce experimental conditions with higher diversity compared to aconventional batch BOmethod (Snoek et al.) implemented in the PHYSBOpackage38. Furthermore, NTS demonstrated better optimization perfor-mance than random sampling, confirming that it fulfills the motivation ofefficiently sampling many near-optimal and diverse samples. In addition,we applied NTS to electrolyte discovery experiments using an SDL andconfirmed that diverse samples were successfully obtained in real experi-ments as well. The NTS algorithm has been implemented in the NIMOpackage, which is a software for SDLs and is available on GitHub (https://github.com/NIMS-DA/nimo)39,40.ResultsBenchmark using materials datasetsWe used the melting-temperature data of Al-Mg-Zn alloys36 and thegrain boundary energy dataset37 as benchmark materials datasets. Thenumber of data points was 1326 and 17,232 for the former and lattercases, respectively. The dimensionality of the search space was two(compositions) for the melting-temperature dataset and three (grainboundary structural descriptors) for the grain boundary energy dataset.In the batch setting, the batch size was fixed at B = 3 and 24, respectively.We evaluated the average simple regret and the #Circles metrics over fiveindependent trials with the same initial samples but different randomseeds, as shown in Fig. 3.Fig. 1 | Brief description of nested Thompson sampling (NTS). From theGaussian process (left panel), one model is sampled (Thompson sampling). Inconventional BO, the proposed point is determined by the maximum posterior(middle panel). In NTS (right panel), the posterior takes a constant value when thesampled function value is higher than Lt, and becomes zero when it is lower than thethreshold value Lt. Multiple conditions can then be uniformly selected from theregions with constant and finite posterior values.Fig. 2 | Illustration of the Circles metric. A highercurve indicates amore diverse sample set. The areaunder the curve is defined as Circles-AUC, I, anddatasets with higher diversity exhibit larger areas,that is, I2 > I1.https://doi.org/10.1038/s41524-026-02067-0 Articlenpj Computational Materials |          (2026) 12:197 2https://github.com/NIMS-DA/nimohttps://github.com/NIMS-DA/nimowww.nature.com/npjcompumatsNTS was computed under three operating modes. The simple regretresults (see Fig. 3a and d) showed that the conventional batch BO(PHYSBO) outperformed random sampling and NTS for the melting-temperature dataset, whereas for the grain boundary dataset, the optimi-zation results were not significantly different from random sampling. Tocompute the #Circlesmetric, it is necessary to define the criterion for “near-optimal” points. Here, we assumed data points with target values greaterthan the 90th percentile of the entire dataset as near-optimal. The radiusdependence of #Circles is shown in Fig. 3b and e, where NTS expandstoward the upper right, indicating that the obtained samples exhibit higherdiversity. Here, to evaluate #Circles, we used the Euclidean distance as adistance metric. On the other hand, the results of the three NTS modes fallwithin the error bars for both regret and #Circles over almost the entirerange of radii, suggesting that the overall differences among the modes aresmall for these datasets.We compared the Circles-AUC values of the “conservative” mode ofNTS, PHYSBO, and random sampling, as shown in Fig. 3c and f. NTSyielded the highest values, confirming that the conditions sampled by NTSwere more diverse. Note that the results discussed in the SupplementaryNote A indicate that diverse sampling is maintained even when the batchsize is varied in the NTS method. Furthermore, from Fig. 3b, it can beinferred that the aggressive mode is preferable when faster optimization isdesired, although the appropriatemode shouldbe selecteddependingon theobjective of the exploration.Electrolyte exploration in self-driving laboratory using NTSIn this section, we present results for electrolyte explorations using NIMSautomated robotic electrochemical experiments (NAREE)7. NAREE is asystem that enables high-throughput screening of electrolytes by utilizing amicroplate-type electrochemical cell. Electrochemical cells were fabricatedusing LiFePO4 as the positive electrode andCu foil as the negative electrode.Here, the objective function value was set to discharge time (sec). Tomaximize the objective function, a combination of electrolyte additives wasoptimized. Five different additiveswere selected froma list of 32 compounds(Table 1) and injected into an electrochemical cell containing 1 M Lithiumbis(trifluoromethanesulfonyl)imide (LiTFSI) in propylene carbonate (PC).The total number of experimental conditions is 32C5 = 201, 376. To expressthe experimental condition as a vector, 32 binary variables are used: A valueof 1 is set if the additive is used, and 0 if it is not. Consequently, for eachvector, the number of variables with a value of 1 is always 5. SinceNAREE isconnected to some exploration algorithms through NIMO, the self-drivingelectrolyte explorations without human intervention can be performed.Thus, using NIMO, the closed-loop experiments of NAREE and NTS wereperformed (see Fig. 4a). Note that for binary variables, the squared Eucli-dean distance in the RBF kernel directly corresponds to the Hammingdistance, allowing the model to effectively capture similarities in discretespaces.Batch sizewas set toB=48, and 10 independent cycles were conductedusing conventional batch BO ("PHYSBO” inNIMO) andNTS. As an initialdataset, we selected 48 experimental conditions randomly. The NTSexploration was performed in the “conservative”mode. The dependence ofthe maximum value of discharge time on cycles is shown in Fig. 4b. Fromthese results, electrolytes whose discharge timewas larger than 0.14 secwereregarded as near-optimal. To examine the diversity of the near-optimalsamples, the #Circles metrics for the samples whose discharge time waslarger than 0.14 sec were plotted in Fig. 4c. Here, to evaluate #Circles, weused theHamming distance as a distancemetric. The results of #Circles as afunction of radius obtained by NTS extend further toward the upper-rightregion compared with those by conventional batch BO, indicating that thesamples generated byNTS possess higher diversity than conventional batchBO. In Fig. 4c, the Circles-AUC values are also shown, indicating that thediversity of samples by NTS is higher than that by conventional batch BO.Figure 4d also shows a two-dimensional mapping of the near-optimalsamples using t-SNE, where the NTS samples exhibit a more widely spreaddistribution. These results demonstrate that, while conventional batch BOmethods often prove insufficient, near-optimal sampling can be achievedFig. 3 | Benchmark results on materials datasets. The melting temperature andgrain boundary datasets are used for a–c and d–f, respectively. Simple regret as afunction of cycle (a) and d), number of circles (#Circles) as a function of radius (b)and (e), and Circles-AUC for near-optimal samples (c) and (f). Near-optimalsamples are defined as data points whose target values are greater than the 90thpercentile of the entire dataset. The lines represent the average value, and the shadedarea and error bars indicate the deviation from five independent trials.https://doi.org/10.1038/s41524-026-02067-0 Articlenpj Computational Materials |          (2026) 12:197 3www.nature.com/npjcompumatswithNTS in self-driving laboratories, even though the only hyperparameterthat needs to be specified externally is the exploration mode.DiscussionsTo efficiently collect many near-optimal samples, the nested Thompsonsampling (NTS) method was proposed. This method supports a batchapproach, making it suited for high-throughput parallel experimentation.Using materials datasets, we confirmed that the NTS method can obtainsamples with higher diversity than conventional batch Bayesian opti-mization (BO). We then conducted automated electrolyte exploration bycombining NTS with robotic electrochemical experiments, which suc-cessfully produced diverse samples in real-world experiments. Ratherthan focusing solely on the rapid identification of a single global opti-mum via conventional BO, a strategy that explores a diverse set of near-optimal candidates using NTS offers significant advantages in materialsdiscovery. In this field, it is rarely the case that a single target propertydetermines the overall excellence of a material. In practice, multiplecompeting factors, such as synthesis cost, stability, and environmentalimpact, must be balanced. Therefore, identifying a diverse set of near-optimal candidates is a more robust strategy. This approach enables theselection of the most suitable candidates from a broad pool of high-performing options, accounting for secondary criteria that may not beexplicitly included in the automated optimization cycle.The NTS method has been implemented in the NIMO package,enabling seamless integration into robotic experimental systems. By com-bining self-driving laboratories (SDLs) with NTS, it becomes possible toacquire numerous near-optimal samples, from which users can select themost suitable conditions according to their objectives, such as prioritizingsimpler processes or lower costs. This new approach is expected to be highlycompatible with SDLs, which can perform extensive experiments withouthuman intervention, andwebelieve itwill drive significant innovation in theresearch process.MethodsNested Thompson samplingNested sampling was originally proposed for high-dimensional integrationin Bayesian inference29,30. It has demonstrated success in a wide range ofapplications, including cosmology and particle physics41,42. Starting fromrandomly initialized points, the nested sampling algorithm progressivelyrestricts the search space to more promising regions by replacing low-performingmembers of the populationwith higher-performing ones. In thepresence ofmultimodality, the algorithm can simultaneously trackmultiplepeaks, thereby promoting the generation of solutions with greater diversity.By incorporating the idea of nested sampling into black-box optimi-zation, it becomes possible to collect diverse near-optimal samples. In ouralgorithm, Thompson sampling is employed to define a constrained pos-terior distribution that determines the next proposals, which we refer to asnestedThompson sampling (NTS).AGaussianprocess (GP) is aprobabilitydistributionwhere a function f(x) corresponds to the realizationof a randomvariable43. The fundamental idea of GP is that a function gives similaroutputs for similar inputs: the similarity between f(x1) and f(x2) is close tothat of x1 and x2. To measure the similarity between x1 and x2, a kernelfunction is needed. In thiswork, a popular radial basis function (RBF) kernelwas adopted. In Gaussian process (GP) regression, GP(f∣Dt) models thefunction f(x) as a probability distribution, allowing one to infer the likelyshapes of the underlying function from the observed datasetDt ¼ fðx1; y1Þ; . . . ; ðxjDt j; yjDt jÞg. In BO using Thompson sampling, afunction is sampled from the posterior GP distribution GP(f∣Dt), and thenext proposal is determined by the maximum of the sampled function.Instead of maximum distribution, we consider a constrained posteriordistribution πt(x), which is defined as the probability that x yields a functionvalue exceeding a threshold Lt. Then, πt(x) is given asπtðxÞ / Eef�GPðf jDt Þ1ef ðxÞ > LtðxÞ� �; ð3Þwhere Lt is a threshold andef ðxÞ is a sampled function from GP regression.1AðxÞ is indicator function defined as1AðxÞ ¼1 x 2 A0 x=2A�: ð4ÞIn addition,Eef�GPðf jDt Þ½�� is approximated by the sample average from theposterior GP. Thus, πt(x) is given by the sample average of the indicatorfunction. In our implementation, the number of generated samples is fixedas 100. In NTS, multiple candidates are generated by sampling from theconstrained posterior distribution πt(x), enabling simultaneous explorationof the diverse solution space. Algorithm 1 describes the whole procedure.Algorithm 1. Nested Thompson sampling1: Procedure NTS2: D0  fðx1; y1Þ; . . . ; ðxjD0j; yjD0jÞg3: t ← 04: repeat5: compute Gaussian process posterior GP(f ∣Dt)Table 1 | 32 additives used in electrolyte exploration in a self-driving laboratoryNo. Solution1 1M Lithium bis(pentafluoroethanesulfonyl)imide in PC2 1M Lithium bis(fluorosulfonyl)imide in PC3 1M Lithium trifluoromethanesulfonate in PC4 1M Lithium tetrafluoroborate in PC5 1M Lithium hexafluorophosphate in PC6 1M Lithium difluoro(oxalato)borate in PC7 1M Lithium perchlorate in PC8 50vol% Dimethylsulfamoyl fluoride in PC9 0.1M Lithium bis(pentafluoroethanesulfonyl)imide in PC10 0.1M Lithium bis(fluorosulfonyl)imide in PC11 0.1M Lithium trifluoromethanesulfonate in PC12 0.1M Lithium tetrafluoroborate in PC13 0.1M Lithium hexafluorophosphate in PC14 0.1M Lithium difluoro(oxalato)borate in PC15 0.1M Lithium perchlorate in PC16 5vol% Dimethylsulfamoyl fluoride in PC17 50vol% Trimethyl phosphate in PC18 50vol% Triethyl phosphate in PC19 50vol% N-methyl-2-pyrrolidone in PC20 50vol% Tetraethylene glycol dimethyl ether in PC21 50vol% Dimethyl sulfoxide in PC22 50vol% 1,3-Propane sultone in PC23 50vol% 2-Propynyl methanesulfonate in PC24 50vol% Sulfolane in (1M LiTFSI in PC)25 5vol% Trimethyl phosphate in PC26 5vol% Triethyl phosphate in PC27 5vol% N-methyl-2-pyrrolidone in PC28 5vol% Tetraethylene glycol dimethyl ether in PC29 5vol% Dimethyl sulfoxide in PC30 5vol% 1,3-Propane sultone in PC31 5vol% 2-Propynyl methanesulfonate in PC32 5vol% Sulfolane in (1M LiTFSI in PC)https://doi.org/10.1038/s41524-026-02067-0 Articlenpj Computational Materials |          (2026) 12:197 4www.nature.com/npjcompumats6: select a threshold Lt7: perform Thompson sampling multiple times and con-struct πt(x)8: sample the next batch B = {x(1),…,x∣batch_size∣} by drawing i.i.d.samples from πt9: DB ← {(x, f (x))∣x ∈ B}10: Dt+1 ← Dt ∪ DB11: t ← t + 112: Until until the predetermined number of cycles is reachedTo determine the value of Lt, we proposed three strategies termed“conservative”, “moderate”, and “aggressive”, which represent the onlyhyperparameter in NTS. In these strategies, the top 10%, 5%, and 1% ofpreviously obtained values were used as Lt, respectively. Consequently, Lt ismonotonically increased with each cycle.Calculation of #Circles MetricsAs a measure of the diversity of near-optimal samples, we adoptedthe #Circles metric that is currently proposed for evaluating thediversity of chemical space34,35, which is defined in Eq. (1). The#Circles metric corresponds to the packing number in graph theory34.In practice, one can construct an undirected graph G = (V, E) byadding an edge between xi; x0i 2 S if dðxi; x0iÞ≤ r. The #Circles metriccan then be obtained by computing a maximal independent set of Gusing any graph software. In this work, we employed the max-imal_independent_set method of NetworkX 3.4.244 to com-pute the #Circles metric.Implementation in NIMOWe implemented the NTS algorithm in NIMO39,40. NTS can be appliedthrough the nimo.selection function as followsnimo.selection(method = "NTS", input_file = “can-didates.csv”, output_file = “proposals.csv”, num_ob-jectives = 1, num_proposals = 48, sample_mode =“conservative”)The candidates are stored beforehand incandidates.csv, and theproposals are saved to proposals.csv. Subsequently, experiments willbe conducted according to the experimental conditions listed in propo-sals.csv. The parameter sample_mode specifies the strategy to con-trol a threshold Lt, where “conservative”, “moderate”, and “aggressive” canbe selected. If the data contains significant noise, using the “aggressive”mode with a strict threshold from the early stages may lead to the omissionof promising regions. Conversely, since the threshold increases mono-tonically even in “conservative”mode, it is essential to prepare a sufficientamount of initial data to ensure that potential candidate regions are notoverlooked.Data availabilityThe datasets generated and analyzed during the current study are availablein the GitHub repository at https://github.com/shibukawar/nimo.Code availabilityThe code supporting the findings of this study is available from the GitHubhttps://github.com/shibukawar/nimo.Fig. 4 | Exploration of electrolytes using theNAREE system and theNTSmethod.aWorkflow of the exploration process without human intervention. b Objectivefunction values (discharge time (sec)) as a function of cycle obtained by PHYSBOand the NTS method. c Number of circles (#Circles) as a function of radius.dDimensionality reduction results for near-optimal samples obtained using t-SNE.Near-optimal samples are defined as those having target values greater than 0.14.https://doi.org/10.1038/s41524-026-02067-0 Articlenpj Computational Materials |          (2026) 12:197 5https://github.com/shibukawar/nimohttps://github.com/shibukawar/nimowww.nature.com/npjcompumatsReceived: 7 December 2025; Accepted: 24 March 2026;References1. 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T.; methodology: R.S. and K.T.; code developmentand execution: R.S. and R.T.; experiments: S.M. and K.N.; writing—initialdraft: R.S. and R.T.; writing—review and editing: all authors.Competing interestsThe authors declare no competing interests.Additional informationSupplementary information The online version containssupplementary material available athttps://doi.org/10.1038/s41524-026-02067-0.Correspondence and requests for materials should be addressed toShoichi Matsuda, Ryo Tamura or Koji Tsuda.Reprints and permissions information is available athttp://www.nature.com/reprintsPublisher’s note Springer Nature remains neutral with regard tojurisdictional claims in published maps and institutional affiliations.Open Access This article is licensed under a Creative CommonsAttribution-NonCommercial-NoDerivatives 4.0 International License,which permits any non-commercial use, sharing, distribution andreproduction in any medium or format, as long as you give appropriatecredit to the original author(s) and the source, provide a link to the CreativeCommons licence, and indicate if you modified the licensed material. Youdo not have permission under this licence to share adapted materialderived from this article or parts of it. The images or other third partymaterial in this article are included in the article’s Creative Commonslicence, unless indicated otherwise in a credit line to thematerial. If materialis not included in thearticle’sCreativeCommons licenceandyour intendeduse is not permitted by statutory regulation or exceeds the permitted use,you will need to obtain permission directly from the copyright holder. Toview a copy of this licence, visit http://creativecommons.org/licenses/by-nc-nd/4.0/.© The Author(s) 2026https://doi.org/10.1038/s41524-026-02067-0 Articlenpj Computational Materials |          (2026) 12:197 7https://doi.org/10.1038/s41524-026-02067-0http://www.nature.com/reprintshttp://creativecommons.org/licenses/by-nc-nd/4.0/http://creativecommons.org/licenses/by-nc-nd/4.0/www.nature.com/npjcompumats Collecting diverse near-optimal samples via nested Thompson sampling Results Benchmark using materials datasets Electrolyte exploration in self-driving laboratory using NTS Discussions Methods Nested Thompson sampling Calculation of #Circles Metrics Implementation in NIMO Data availability Code availability References Acknowledgements Author contributions Competing interests Additional information