{"work":{"id":"f9cc095b-e709-4c96-bccc-ffcef3254dbe","openalex_id":null,"doi":null,"arxiv_id":"1202.3665","raw_key":null,"title":"emcee: The MCMC Hammer","authors":null,"authors_text":"Daniel Foreman-Mackey, David W. Hogg, Dustin Lang, Jonathan Goodman","year":2012,"venue":"astro-ph.IM","abstract":"We introduce a stable, well tested Python implementation of the affine-invariant ensemble sampler for Markov chain Monte Carlo (MCMC) proposed by Goodman & Weare (2010). The code is open source and has already been used in several published projects in the astrophysics literature. The algorithm behind emcee has several advantages over traditional MCMC sampling methods and it has excellent performance as measured by the autocorrelation time (or function calls per independent sample). One major advantage of the algorithm is that it requires hand-tuning of only 1 or 2 parameters compared to $\\sim N^2$ for a traditional algorithm in an N-dimensional parameter space. In this document, we describe the algorithm and the details of our implementation and API. Exploiting the parallelism of the ensemble method, emcee permits any user to take advantage of multiple CPU cores without extra effort. The code is available online at http://dan.iel.fm/emcee under the MIT License.","external_url":"https://arxiv.org/abs/1202.3665","cited_by_count":null,"metadata_source":"pith","metadata_fetched_at":"2026-05-25T03:30:17.337937+00:00","pith_arxiv_id":"1202.3665","created_at":"2026-05-09T04:54:56.188334+00:00","updated_at":"2026-05-25T03:30:17.337937+00:00","title_quality_ok":false,"display_title":"emcee: The MCMC Hammer","render_title":"emcee: The MCMC Hammer"},"hub":{"state":{"work_id":"f9cc095b-e709-4c96-bccc-ffcef3254dbe","tier":"hub","tier_reason":"10+ Pith inbound or 1,000+ external citations","pith_inbound_count":81,"external_cited_by_count":null,"distinct_field_count":15,"first_pith_cited_at":"2019-07-10T14:23:27+00:00","last_pith_cited_at":"2026-05-22T16:50:05+00:00","author_build_status":"not_needed","summary_status":"needed","contexts_status":"needed","graph_status":"needed","ask_index_status":"not_needed","reader_status":"not_needed","recognition_status":"not_needed","updated_at":"2026-05-25T04:15:25.106529+00:00","tier_text":"hub"},"tier":"hub","role_counts":[{"context_role":"method","n":14},{"context_role":"background","n":11},{"context_role":"dataset","n":1},{"context_role":"other","n":1}],"polarity_counts":[{"context_polarity":"use_method","n":13},{"context_polarity":"background","n":10},{"context_polarity":"unclear","n":2},{"context_polarity":"support","n":1},{"context_polarity":"use_dataset","n":1}],"runs":{"context_extract":{"job_type":"context_extract","status":"succeeded","result":{"enqueued_papers":25},"error":null,"updated_at":"2026-05-21T03:22:38.490022+00:00"},"graph_features":{"job_type":"graph_features","status":"succeeded","result":{"co_cited":[{"title":"Planck 2018 results. VI. Cosmological parameters","work_id":"eeae0089-7b56-4c63-ace2-a31de468f6c5","shared_citers":21},{"title":"DESI DR2 Results II: Measurements of Baryon Acoustic Oscillations and Cosmological Constraints","work_id":"d54834f3-4944-4c0b-9fa4-3008a4c0122d","shared_citers":12},{"title":"DESI 2024 VI: Cosmological Constraints from the Measurements of Baryon Acoustic Oscillations","work_id":"abaeab8e-c412-45ac-813e-f53689657023","shared_citers":9},{"title":"GetDist: a Python package for analysing Monte Carlo samples","work_id":"f8674deb-70f7-4881-a67c-f177a925133d","shared_citers":9},{"title":"quiescent","work_id":"b11220ca-ed53-44f4-8db3-b381c41c3d0a","shared_citers":8},{"title":"A Comprehensive Measurement of the Local Value of the Hubble Constant with 1 km/s/Mpc Uncertainty from the Hubble Space Telescope and the SH0ES Team","work_id":"5dac78be-e29e-4ec2-9a41-4adf05fc6a6a","shared_citers":7},{"title":"Alam , M","work_id":"fd19ed7c-981b-4e7f-bdf4-3b82b6e29714","shared_citers":7},{"title":"Measurements of Omega and Lambda from 42 High-Redshift Supernovae","work_id":"2900b415-082f-4e4e-ad2c-7cba94498108","shared_citers":7},{"title":"The Pantheon+ Analysis: Cosmological Constraints","work_id":"fafba1aa-4e44-42c3-8b9a-f0e78a38ad17","shared_citers":7},{"title":"arXiv , Author =:1307.6212 , Journal =","work_id":"02354d9b-ad6f-4a4e-9720-06d21c2feba5","shared_citers":6},{"title":"In the Realm of the Hubble tension $-$ a Review of Solutions","work_id":"bfa51bc2-7198-4351-9514-5ef4574eed61","shared_citers":6},{"title":"M., & Rasio, F","work_id":"fbe25bd0-9403-4539-b873-fcf6743fba70","shared_citers":6},{"title":"M., & Spurzem, R","work_id":"cc97efa9-0c9d-45a7-ba7f-7bcd7663f68e","shared_citers":6},{"title":"M., Spurzem, R., & Lee, M","work_id":"fc06b3ca-a49a-40e5-a61e-b9d148b54d70","shared_citers":6},{"title":"Price-Whelan, B.M","work_id":"88a88437-2e0a-4987-83f9-ce7a5d386437","shared_citers":6},{"title":"The Completed SDSS-IV extended Baryon Oscillation Spectroscopic Survey: Cosmological Implications from two Decades of Spectroscopic Surveys at the Apache Point observatory","work_id":"5fcfe3bb-de1b-4cd2-868e-f137ebf5c19b","shared_citers":6},{"title":"2010, MNRAS, 401, 2318, doi: 10.1111/j.1365-2966.2009.15845.x","work_id":"23562d5d-936a-4b08-a6f0-15466b0fcd7c","shared_citers":5},{"title":"2012, MNRAS, 426, 1324, doi: 10.1111/j.1365-2966.2012.21757.x","work_id":"91cc99ad-bdb6-42b2-afbd-e26dad5ec964","shared_citers":5},{"title":"Ade et al","work_id":"a36b7348-19d6-4af7-9627-ad462b9f2c8a","shared_citers":5},{"title":"Beutler, C","work_id":"89f0655a-dc1a-456b-8618-650645b54428","shared_citers":5},{"title":"Chaussidon, C","work_id":"d43c2fc2-af93-4054-b7b9-c12335b6b184","shared_citers":5},{"title":"Data Release 1 of the Dark Energy Spectroscopic Instrument","work_id":"a536800a-8f7c-4eb1-b9ac-29718dc7c858","shared_citers":5},{"title":"F., Kramer M., 2008, @doi [ ] 10.1111/j.1365-2966.2008.14045.x , https://ui.adsabs.harvard.edu/abs/2008MNRAS.391.2009K 391","work_id":"a585747b-4fbc-4555-a22d-a4a769e4179d","shared_citers":5},{"title":"Raichoor, J","work_id":"e0b6cf9c-b50d-414a-a5ed-3ab4cd03d390","shared_citers":5}],"time_series":[{"n":1,"year":2019},{"n":1,"year":2020},{"n":1,"year":2021},{"n":1,"year":2024},{"n":13,"year":2025},{"n":42,"year":2026}],"dependency_candidates":[{"n":1,"role":"method","polarity":"use_method","paper_title":"No evidence for phantom crossing: local goodness-of-fit improvements do not persist under global Bayesian model comparison","primary_cat":"astro-ph.CO","context_text":"CMB+DESI DR2+DES-Dovekie−7.7−1.4 Information Criterion (DIC) and the Log Bayesian ev- idence lnZ[32, 33]. The DIC combines a measure of goodness-of-fit with a certain penalty for model complex- ity, where the effective number of parameters is estimated from the posterior distribution. In this work, DIC is com- puted from the MCMC posterior obtained withemcee [34], with averages taken over the chains and the refer- ence point set to the posterior mean. In contrast, the Bayesian evidence lnZis computed using nested sam- pling viaPyMultiNest[35], which naturally incorporates an Occam penalty by accounting for the full prior volume. We interpret|∆ lnZ|using the Jeffreys scale: values be- low 1 are inconclusive, 1-2.","citing_arxiv_id":"2605.13546"},{"n":1,"role":"method","polarity":"use_method","paper_title":"Constraints on Einstein-aether gravity from the precision timing of PSR J1738+0333","primary_cat":"gr-qc","context_text":"Vela[64], which implements the full non-linear pul- sar timing and noise model with efficient parallelization, provides aPythonbinding (pyvela) and handles data I/O, clock corrections, and ephemeris computation via pint[66, 67], and is designed to work with both narrow- band and wideband analysis paradigms. WithinVela, we employed the ensemble sampleremcee[68] to draw samples from the likelihood. C. From timing to theory parameters: the resampling strategy The Bayesian timing analysis described in the previ- ous section gives a posterior distribution in the form of Eq. (71), where ⃗λ={ ⃗θ, ⃗ζ}collects all timing parame- ters, with ⃗ζdenoting spin and astrometric parameters and ⃗θ={ ⃗θorb, ⃗θPK}the orbital and post-Keplerian pa-","citing_arxiv_id":"2605.01436"},{"n":1,"role":"method","polarity":"use_method","paper_title":"Confronting Color Glass Condensate at next-to-leading order with HERA data","primary_cat":"hep-ph","context_text":"1: Comparison between emulator prediction and model calculation for the total reduced cross section (left) and charm contribution (right). The relative difference averaged over all training samples is presented. section and the charm quark contribution). The cali- brated emulator becomes a faster model substitute for the MCMC algorithm to use. Theemcee[46] imple- mentation of the MCMC involves an ensemble of ran- dom walkers whose advancements depend on informa- tion from other walkers. A walker proposes their next step and is accepted with a probability similar to that of a Metropolis-Hastings algorithm:α∼P(θ 1)/P(θ 0), whereθ 0 is the walker's current position in the parame- ter space,θ 1 is the proposed step andPis the likelihood.","citing_arxiv_id":"2604.22332"},{"n":1,"role":"dataset","polarity":"use_dataset","paper_title":"AI-assisted modeling and Bayesian inference of unpolarized quark transverse momentum distributions from Drell-Yan data","primary_cat":"hep-ph","context_text":"contains 9 nonperturbative fit parameters: {c0, c1, BNP, λ1, λ2, λ3, ANP, x0, σx}.(2.34) - 8 - Dataset Process Observable √s[GeV] Fiducial cut Q[GeV] y Npts STAR [57] p-p dσ/dqT 510 pT >25 GeV |η|<1 73-114 |y|<1 7 CDF Run I [58] p-¯ p dσ/dqT 1800 Inclusive 66-116 Inclusive 25 CDF Run II [59] p-¯ p dσ/dqT 1960 Inclusive 66-116 Inclusive 26 D0 Run I [60] p-¯ p dσ/dqT 1800 Inclusive 75-105 Inclusive 12 D0 Run II [61] p-¯ p σ−1 dσ/dqT 1960 Inclusive 70-110 Inclusive 5 D0 Run II (µ +µ−) [62] p-¯ p σ−1 dσ/dqT 1960 pT >15 GeV |η|<1.7 65-115 Inclusive 3 ATLAS 7 TeV [63] p-p σ−1 dσ/dqT 7000 pT >20 GeV |η|<2.4 66-116 |y|<1 1<|y|<2 2<|y|<2.4 6 6 6 ATLAS 8 TeV [64] p-p dσ/dqT 8000 pT >20 GeV |η|<2.4 66-116","citing_arxiv_id":"2604.14133"},{"n":1,"role":"method","polarity":"use_method","paper_title":"Cosmologically viable non-polynomial quasi-topological gravity: explicit models, $\\Lambda$CDM limit and observational constraints","primary_cat":"gr-qc","context_text":"For the BAO sector, we utilize the dataset of [89], which incorporates fiducial cosmology corrections through the ratior d/rfid, allowing for a consistent com- parison between theoretical predictions and observations. The exploration of the parameter space is carried out using the Markov Chain Monte Carlo (MCMC) method, implemented through the publicly available Python pack- ageemcee[90]. In this context, the SNIa nuisance pa- rameterMis treated as a free parameter of the Pan- theon dataset (see [88] and references therein), while the radiation density parameter Ωr0 is neglected, as it is sub- dominant at late times. The above statistical framework is applied to the quar- tic and power-law models, as well as to the standard ΛCDM scenario, allowing for a direct and consistent com-","citing_arxiv_id":"2604.13002"},{"n":1,"role":"method","polarity":"use_method","paper_title":"$\\boldsymbol{B_c}$ Meson Spectroscopy from Bayesian MCMC: Probing Confinement and State Mixing","primary_cat":"hep-ph","context_text":"We apply this non-linear form unchanged to the mass spectra from both the Cornell and modified Cornell potentials, 2 enabling a direct comparison of the resulting Regge behavior between the two cases. The parameters of both the linear (Eq. (28)) and non-linear (Eq. (29)) trajectories are extracted by fitting the MCMC-derived mass spectra using theiminuit package [54], based on the MINUIT optimization algorithm [55]. The resulting trajectories are discussed in Sec. III. III. NUMERICAL RESULTS AND DISCUSSION We present the numerical results for theB c mass spectrum obtained from the Cornell potential (Eq. (1)) and its logarithmic extension (Eq. (2)), hereafter Potential I and Poten- tial II, respectively. The role of the logarithmic term is quantified through ther-dependent","citing_arxiv_id":"2604.04846"},{"n":1,"role":"method","polarity":"use_method","paper_title":"Probing soft signals of gravitational-wave memory with space-based interferometers","primary_cat":"gr-qc","context_text":"same SNR, possibly due to its stronger degeneracy dis- cussed in Sec. IV B. However, the FIM estimate cannot be reliably trusted at low SNR, which is likely a realistic regime. In the following, we present results from Bayesian parameter- estimation simulations, focusing on the case of displace- ment memory. For sampling the posterior, we use the MCMC sampler emcee [92] and plot the posterior dis- tributions using corner [93]. For concreteness, we con- sider the measurements using Taiji, LISA-Taiji network (described in Fig. 8) and BBO. For LISA-Taiji, we set ηTaiji = ηLISA + 2 π/9, Φ0 = 0 , and ηLISA = π/4; the log-likelihood is taken as the sum of those of LISA and Taiji. As mentioned in Sec. IV A, the signals at a sin-","citing_arxiv_id":"2603.28689"},{"n":1,"role":"method","polarity":"background","paper_title":"Model-independent test of the cosmic distance duality relation with recent observational data","primary_cat":"astro-ph.CO","context_text":"Nong, H. Wang, B. Zhang, Z. Li and N. Liang,Constraints on cosmological models with gamma-ray bursts in cosmology-independent way,Int. J. Mod. Phys. D34(2025) 2450073 [2307.16467]. [96] H. Wang and N. Liang,Constraints from Fermi observations of long gamma-ray bursts on cosmological parameters,Mon. Not. Roy. Astron. Soc.533(2024) 743 [2405.14357]. [97] B. Zhang, H. Wang, X. Nong, G. Wang, P. Wu and N. Liang,Model-independent gamma-ray bursts constraints on cosmological models using machine learning,Astrophys. Space Sci.370 (2025) 10 [2312.09440]. [98] Z. Huang, Z. Xiong, X. Luo, G. Wang, Y. Liu and N. Liang,Gamma-ray bursts calibrated from the observational H(z) data in artificial neural network framework,JHEAp47(2025)","citing_arxiv_id":"2603.27616"},{"n":1,"role":"method","polarity":"use_method","paper_title":"New Way to Date Globular Clusters: Brown Dwarf Cooling Sequences","primary_cat":"astro-ph.SR","context_text":"[139]andVasilievandBaumgardt[157]for47Tuc. Ihavetakenthemtoberepresentativeofallglobular clustersconsideredinthiswork. Thecalculationswererepeatedfortwotemporalbaselinesof2and5years. TheastrometricandphotometricerrorswereextractedfromthesimulatedobservationsbyfittingthePSFmodelusingtheGoodman-Weare [93]MarkovChainMonteCarlo(MCMC)algorithmimplementedinthe emceePythonpackage[104]. ThemodelPSFwasconstructedusingthe sameprocedureasthesimulatedobservationsbutwithoutapplyingnoisetotheimage. PSFfittingwascarriedoutonthesimulatedF150W2and F322W2imagesofthestarsimultaneously. Thefitterwasconstrainedtorequirethestartohavethesamepositioninbothbands. Thestandard errors in magnitudes and positions were taken as the half-difference between the84th and16th percentiles (±1 -sigma range) of the MCMC","citing_arxiv_id":"2603.07481"},{"n":1,"role":"method","polarity":"use_method","paper_title":"Cosmological analysis of the DESI DR1 Lyman alpha 1D power spectrum","primary_cat":"astro-ph.CO","context_text":"The resultingP 1D mocks are assigned the same covariance matrix as the one em- ployed in the DESI DR1 analysis. These mocks do not include contaminants or systematics; however, we analyze them with the model presented in the previous section, which accounts for both. We extract cosmological constraints using the publicly available affine-invariant Markov chain Monte Carlo (MCMC) ensemble sampleremcee 7 [118]. At each step,emceeproposes 7https://emcee.readthedocs.io/en/stable/ - 23 - 0.9 1.0 1.1 z = 2.2, 2 = 34.58, ndata=49 z = 2.4, 2 = 48.92, ndata=52 z = 2.6, 2 = 55.51, ndata=55 0.9 1.0 1.1 z = 2.8, 2 = 62.34, ndata=58 z = 3.0, 2 = 78.8, ndata=61 z = 3.2, 2 = 73.69, ndata=63 0.75 1.00 1.25 z = 3.4, 2 = 69.05, ndata=65 z = 3.6, 2 = 99.18, ndata=67","citing_arxiv_id":"2601.21432"},{"n":1,"role":"method","polarity":"use_method","paper_title":"Interacting $k$-essence field with non-pressureless Dark Matter: Cosmological Dynamics and Observational Constraints","primary_cat":"astro-ph.CO","context_text":"consistent solutions across cosmic epochs, and adopt uniform priors as listed in Tab. II. For sampling and likelihood eval- uation, we employ the nested samplerPolyChord, which is well suited for high-dimensional parameter spaces compared to affine-invariant MCMC samplers such asemcee[99-101]. The chains are analyzed usingGetDistto extract the best-fit values [102, 103]. Finally, we compare our models against flat ΛCDM using information criteria such as the Akaike Informa- tion Criterion (AIC) and Bayesian Information Criterion (BIC) [104-106], following the standard methodology described in [107]. V. RESULTS In this section, we present the results obtained by imple- menting the models described in Tab. I with the considered","citing_arxiv_id":"2509.09202"},{"n":1,"role":"method","polarity":"use_method","paper_title":"Interacting Scalar Fields as Dark Energy and Dark Matter in Einstein scalar Gauss Bonnet Gravity","primary_cat":"gr-qc","context_text":"PLANCK + PP, and (ii) CC + DESBAO + PLANCK + DES, as −2 lnLtot = 𝜒2 tot. (6.8) We estimate the likelihood by implementing the model in Python and use the publicly available affine-invariant Markov Chain Monte Carlo (MCMC) ensemble sampler emcee [145] to obtain the posterior distributions of the model parameters. The resulting samples are then analyzed usingGetDist [146] to obtain marginalized 1D and 2D posterior distributions. Finally, we compare the statistical preference of the current model relative to the flat ΛCDM model using information criteria such as the Akaike Information Criterion (AIC) and the Bayesian Information Criterion (BIC) [147-149]: AIC = −2 lnLmax + 2𝑘, (6.9) BIC = −2 lnLmax + 𝑘 ln 𝑁, (6.10)","citing_arxiv_id":"2507.05207"},{"n":1,"role":"method","polarity":"use_method","paper_title":"DESI 2024 III: Baryon Acoustic Oscillations from Galaxies and Quasars","primary_cat":"astro-ph.CO","context_text":"package that provides a common framework for writing DESI likelihoods. The BAO the- ory and likelihood is implemented in JAX [126] 20. Even though gradient-based sampling methods were implemented, we found that with analytic marginalization over broadband parameters that leaves a few sampled parameters, and using Jax just-in-time compilation and parallelization capabilities, the ensemble sampler emcee [127]21 provided well-sampled posterior estimates in a just a few minutes. In addition to MCMC sampling, we also perform posterior profiling using desilike's wrapping of Minuit [128]. During the course of this work, we also used/developed a fully independent galaxy BAO fitting pipeline (Barry22 [129]), for some of the supporting papers and with which we tested","citing_arxiv_id":"2404.03000"},{"n":1,"role":"method","polarity":"use_method","paper_title":"Inferring the properties of a population of compact binaries in presence of selection effects","primary_cat":"astro-ph.IM","context_text":"RPG-2019-350, and Royal Society Grant No. RGS-R2-202004. LIGO was con- structed by the California Institute of Technology and Massachusetts Institute of Technology with funding from the National Science Foundation and operates un- der cooperative agreement PHY-0757058. We acknowledge use of iPython [86], Matplotlib [87, 88], NumPy [89], SciPy [90], emcee [91] and SeaBorn [92]. This is LIGO Document P2000231 Glossary and main symbols α(⃗λ ) The fraction of physical sources that are detectable by our experiment, ac- cording to some detection threshold. It depends on the shape population hyper parameters⃗λ. ∧ Logical \"and\" of two propositions. A ∧ B is true if and only if both A and B are true. Background An (usually uninteresting) background event that might be produced","citing_arxiv_id":"2007.05579"}]},"error":null,"updated_at":"2026-05-21T03:22:31.168512+00:00"},"identity_refresh":{"job_type":"identity_refresh","status":"succeeded","result":{"items":[{"title":"Qwen3 Technical Report","outcome":"unchanged","work_id":"25a4e30c-1232-48e7-9925-02fa12ba7c9e","resolver":"local_arxiv","confidence":0.98,"old_work_id":"25a4e30c-1232-48e7-9925-02fa12ba7c9e"}],"counts":{"fixed":0,"merged":0,"unchanged":1,"quarantined":0,"needs_external_resolution":0},"errors":[],"attempted":1},"error":null,"updated_at":"2026-05-21T03:22:35.671019+00:00"},"summary_claims":{"job_type":"summary_claims","status":"succeeded","result":{"title":"emcee: The MCMC Hammer","claims":[{"claim_text":"We introduce a stable, well tested Python implementation of the affine-invariant ensemble sampler for Markov chain Monte Carlo (MCMC) proposed by Goodman & Weare (2010). The code is open source and has already been used in several published projects in the astrophysics literature. The algorithm behind emcee has several advantages over traditional MCMC sampling methods and it has excellent performance as measured by the autocorrelation time (or function calls per independent sample). One major advantage of the algorithm is that it requires hand-tuning of only 1 or 2 parameters compared to $\\sim","claim_type":"abstract","evidence_strength":"source_metadata"},{"claim_text":"The resultingP 1D mocks are assigned the same covariance matrix as the one em- ployed in the DESI DR1 analysis. These mocks do not include contaminants or systematics; however, we analyze them with the model presented in the previous section, which accounts for both. We extract cosmological constraints using the publicly available affine-invariant Markov chain Monte Carlo (MCMC) ensemble sampleremcee 7 [118]. At each step,emceeproposes 7https://emcee.readthedocs.io/en/stable/ - 23 - 0.9 1.0 1.1 ","claim_type":"method","confidence":0.95,"evidence_strength":"citation_context"},{"claim_text":"For the BAO sector, we utilize the dataset of [89], which incorporates fiducial cosmology corrections through the ratior d/rfid, allowing for a consistent com- parison between theoretical predictions and observations. The exploration of the parameter space is carried out using the Markov Chain Monte Carlo (MCMC) method, implemented through the publicly available Python pack- ageemcee[90]. In this context, the SNIa nuisance pa- rameterMis treated as a free parameter of the Pan- theon dataset (see","claim_type":"method","confidence":0.95,"evidence_strength":"citation_context"},{"claim_text":"RPG-2019-350, and Royal Society Grant No. RGS-R2-202004. LIGO was con- structed by the California Institute of Technology and Massachusetts Institute of Technology with funding from the National Science Foundation and operates un- der cooperative agreement PHY-0757058. We acknowledge use of iPython [86], Matplotlib [87, 88], NumPy [89], SciPy [90], emcee [91] and SeaBorn [92]. This is LIGO Document P2000231 Glossary and main symbols α(⃗λ ) The fraction of physical sources that are detectable by ","claim_type":"method","confidence":0.95,"evidence_strength":"citation_context"},{"claim_text":"PLANCK + PP, and (ii) CC + DESBAO + PLANCK + DES, as −2 lnLtot = 𝜒2 tot. (6.8) We estimate the likelihood by implementing the model in Python and use the publicly available affine-invariant Markov Chain Monte Carlo (MCMC) ensemble sampler emcee [145] to obtain the posterior distributions of the model parameters. The resulting samples are then analyzed usingGetDist [146] to obtain marginalized 1D and 2D posterior distributions. Finally, we compare the statistical preference of the current model r","claim_type":"method","confidence":0.9,"evidence_strength":"citation_context"},{"claim_text":"consistent solutions across cosmic epochs, and adopt uniform priors as listed in Tab. II. For sampling and likelihood eval- uation, we employ the nested samplerPolyChord, which is well suited for high-dimensional parameter spaces compared to affine-invariant MCMC samplers such asemcee[99-101]. The chains are analyzed usingGetDistto extract the best-fit values [102, 103]. Finally, we compare our models against flat ΛCDM using information criteria such as the Akaike Informa- tion Criterion (AIC) a","claim_type":"method","confidence":0.9,"evidence_strength":"citation_context"},{"claim_text":"package that provides a common framework for writing DESI likelihoods. The BAO the- ory and likelihood is implemented in JAX [126] 20. Even though gradient-based sampling methods were implemented, we found that with analytic marginalization over broadband parameters that leaves a few sampled parameters, and using Jax just-in-time compilation and parallelization capabilities, the ensemble sampler emcee [127]21 provided well-sampled posterior estimates in a just a few minutes. In addition to MCMC ","claim_type":"method","confidence":0.9,"evidence_strength":"citation_context"}],"why_cited":"Pith tracks emcee: The MCMC Hammer because it crossed a citation-hub threshold. Current citing contexts most often use it as method evidence (13 contexts).","role_counts":[{"n":13,"context_role":"method"},{"n":11,"context_role":"background"},{"n":1,"context_role":"dataset"},{"n":1,"context_role":"other"}]},"error":null,"updated_at":"2026-05-21T03:22:31.066402+00:00"}},"summary":{"title":"emcee: The MCMC Hammer","claims":[{"claim_text":"We introduce a stable, well tested Python implementation of the affine-invariant ensemble sampler for Markov chain Monte Carlo (MCMC) proposed by Goodman & Weare (2010). The code is open source and has already been used in several published projects in the astrophysics literature. The algorithm behind emcee has several advantages over traditional MCMC sampling methods and it has excellent performance as measured by the autocorrelation time (or function calls per independent sample). One major advantage of the algorithm is that it requires hand-tuning of only 1 or 2 parameters compared to $\\sim","claim_type":"abstract","evidence_strength":"source_metadata"},{"claim_text":"The resultingP 1D mocks are assigned the same covariance matrix as the one em- ployed in the DESI DR1 analysis. These mocks do not include contaminants or systematics; however, we analyze them with the model presented in the previous section, which accounts for both. We extract cosmological constraints using the publicly available affine-invariant Markov chain Monte Carlo (MCMC) ensemble sampleremcee 7 [118]. At each step,emceeproposes 7https://emcee.readthedocs.io/en/stable/ - 23 - 0.9 1.0 1.1 ","claim_type":"method","confidence":0.95,"evidence_strength":"citation_context"},{"claim_text":"For the BAO sector, we utilize the dataset of [89], which incorporates fiducial cosmology corrections through the ratior d/rfid, allowing for a consistent com- parison between theoretical predictions and observations. The exploration of the parameter space is carried out using the Markov Chain Monte Carlo (MCMC) method, implemented through the publicly available Python pack- ageemcee[90]. In this context, the SNIa nuisance pa- rameterMis treated as a free parameter of the Pan- theon dataset (see","claim_type":"method","confidence":0.95,"evidence_strength":"citation_context"},{"claim_text":"RPG-2019-350, and Royal Society Grant No. RGS-R2-202004. LIGO was con- structed by the California Institute of Technology and Massachusetts Institute of Technology with funding from the National Science Foundation and operates un- der cooperative agreement PHY-0757058. We acknowledge use of iPython [86], Matplotlib [87, 88], NumPy [89], SciPy [90], emcee [91] and SeaBorn [92]. This is LIGO Document P2000231 Glossary and main symbols α(⃗λ ) The fraction of physical sources that are detectable by ","claim_type":"method","confidence":0.95,"evidence_strength":"citation_context"},{"claim_text":"PLANCK + PP, and (ii) CC + DESBAO + PLANCK + DES, as −2 lnLtot = 𝜒2 tot. (6.8) We estimate the likelihood by implementing the model in Python and use the publicly available affine-invariant Markov Chain Monte Carlo (MCMC) ensemble sampler emcee [145] to obtain the posterior distributions of the model parameters. The resulting samples are then analyzed usingGetDist [146] to obtain marginalized 1D and 2D posterior distributions. Finally, we compare the statistical preference of the current model r","claim_type":"method","confidence":0.9,"evidence_strength":"citation_context"},{"claim_text":"consistent solutions across cosmic epochs, and adopt uniform priors as listed in Tab. II. For sampling and likelihood eval- uation, we employ the nested samplerPolyChord, which is well suited for high-dimensional parameter spaces compared to affine-invariant MCMC samplers such asemcee[99-101]. The chains are analyzed usingGetDistto extract the best-fit values [102, 103]. Finally, we compare our models against flat ΛCDM using information criteria such as the Akaike Informa- tion Criterion (AIC) a","claim_type":"method","confidence":0.9,"evidence_strength":"citation_context"},{"claim_text":"package that provides a common framework for writing DESI likelihoods. The BAO the- ory and likelihood is implemented in JAX [126] 20. Even though gradient-based sampling methods were implemented, we found that with analytic marginalization over broadband parameters that leaves a few sampled parameters, and using Jax just-in-time compilation and parallelization capabilities, the ensemble sampler emcee [127]21 provided well-sampled posterior estimates in a just a few minutes. In addition to MCMC ","claim_type":"method","confidence":0.9,"evidence_strength":"citation_context"}],"why_cited":"Pith tracks emcee: The MCMC Hammer because it crossed a citation-hub threshold. Current citing contexts most often use it as method evidence (13 contexts).","role_counts":[{"n":13,"context_role":"method"},{"n":11,"context_role":"background"},{"n":1,"context_role":"dataset"},{"n":1,"context_role":"other"}]},"graph":{"co_cited":[{"title":"Planck 2018 results. VI. Cosmological parameters","work_id":"eeae0089-7b56-4c63-ace2-a31de468f6c5","shared_citers":21},{"title":"DESI DR2 Results II: Measurements of Baryon Acoustic Oscillations and Cosmological Constraints","work_id":"d54834f3-4944-4c0b-9fa4-3008a4c0122d","shared_citers":12},{"title":"DESI 2024 VI: Cosmological Constraints from the Measurements of Baryon Acoustic Oscillations","work_id":"abaeab8e-c412-45ac-813e-f53689657023","shared_citers":9},{"title":"GetDist: a Python package for analysing Monte Carlo samples","work_id":"f8674deb-70f7-4881-a67c-f177a925133d","shared_citers":9},{"title":"quiescent","work_id":"b11220ca-ed53-44f4-8db3-b381c41c3d0a","shared_citers":8},{"title":"A Comprehensive Measurement of the Local Value of the Hubble Constant with 1 km/s/Mpc Uncertainty from the Hubble Space Telescope and the SH0ES Team","work_id":"5dac78be-e29e-4ec2-9a41-4adf05fc6a6a","shared_citers":7},{"title":"Alam , M","work_id":"fd19ed7c-981b-4e7f-bdf4-3b82b6e29714","shared_citers":7},{"title":"Measurements of Omega and Lambda from 42 High-Redshift Supernovae","work_id":"2900b415-082f-4e4e-ad2c-7cba94498108","shared_citers":7},{"title":"The Pantheon+ Analysis: Cosmological Constraints","work_id":"fafba1aa-4e44-42c3-8b9a-f0e78a38ad17","shared_citers":7},{"title":"arXiv , Author =:1307.6212 , Journal =","work_id":"02354d9b-ad6f-4a4e-9720-06d21c2feba5","shared_citers":6},{"title":"In the Realm of the Hubble tension $-$ a Review of Solutions","work_id":"bfa51bc2-7198-4351-9514-5ef4574eed61","shared_citers":6},{"title":"M., & Rasio, F","work_id":"fbe25bd0-9403-4539-b873-fcf6743fba70","shared_citers":6},{"title":"M., & Spurzem, R","work_id":"cc97efa9-0c9d-45a7-ba7f-7bcd7663f68e","shared_citers":6},{"title":"M., Spurzem, R., & Lee, M","work_id":"fc06b3ca-a49a-40e5-a61e-b9d148b54d70","shared_citers":6},{"title":"Price-Whelan, B.M","work_id":"88a88437-2e0a-4987-83f9-ce7a5d386437","shared_citers":6},{"title":"The Completed SDSS-IV extended Baryon Oscillation Spectroscopic Survey: Cosmological Implications from two Decades of Spectroscopic Surveys at the Apache Point observatory","work_id":"5fcfe3bb-de1b-4cd2-868e-f137ebf5c19b","shared_citers":6},{"title":"2010, MNRAS, 401, 2318, doi: 10.1111/j.1365-2966.2009.15845.x","work_id":"23562d5d-936a-4b08-a6f0-15466b0fcd7c","shared_citers":5},{"title":"2012, MNRAS, 426, 1324, doi: 10.1111/j.1365-2966.2012.21757.x","work_id":"91cc99ad-bdb6-42b2-afbd-e26dad5ec964","shared_citers":5},{"title":"Ade et al","work_id":"a36b7348-19d6-4af7-9627-ad462b9f2c8a","shared_citers":5},{"title":"Beutler, C","work_id":"89f0655a-dc1a-456b-8618-650645b54428","shared_citers":5},{"title":"Chaussidon, C","work_id":"d43c2fc2-af93-4054-b7b9-c12335b6b184","shared_citers":5},{"title":"Data Release 1 of the Dark Energy Spectroscopic Instrument","work_id":"a536800a-8f7c-4eb1-b9ac-29718dc7c858","shared_citers":5},{"title":"F., Kramer M., 2008, @doi [ ] 10.1111/j.1365-2966.2008.14045.x , https://ui.adsabs.harvard.edu/abs/2008MNRAS.391.2009K 391","work_id":"a585747b-4fbc-4555-a22d-a4a769e4179d","shared_citers":5},{"title":"Raichoor, J","work_id":"e0b6cf9c-b50d-414a-a5ed-3ab4cd03d390","shared_citers":5}],"time_series":[{"n":1,"year":2019},{"n":1,"year":2020},{"n":1,"year":2021},{"n":1,"year":2024},{"n":13,"year":2025},{"n":42,"year":2026}],"dependency_candidates":[{"n":1,"role":"method","polarity":"use_method","paper_title":"No evidence for phantom crossing: local goodness-of-fit improvements do not persist under global Bayesian model comparison","primary_cat":"astro-ph.CO","context_text":"CMB+DESI DR2+DES-Dovekie−7.7−1.4 Information Criterion (DIC) and the Log Bayesian ev- idence lnZ[32, 33]. The DIC combines a measure of goodness-of-fit with a certain penalty for model complex- ity, where the effective number of parameters is estimated from the posterior distribution. In this work, DIC is com- puted from the MCMC posterior obtained withemcee [34], with averages taken over the chains and the refer- ence point set to the posterior mean. In contrast, the Bayesian evidence lnZis computed using nested sam- pling viaPyMultiNest[35], which naturally incorporates an Occam penalty by accounting for the full prior volume. We interpret|∆ lnZ|using the Jeffreys scale: values be- low 1 are inconclusive, 1-2.","citing_arxiv_id":"2605.13546"},{"n":1,"role":"method","polarity":"use_method","paper_title":"Constraints on Einstein-aether gravity from the precision timing of PSR J1738+0333","primary_cat":"gr-qc","context_text":"Vela[64], which implements the full non-linear pul- sar timing and noise model with efficient parallelization, provides aPythonbinding (pyvela) and handles data I/O, clock corrections, and ephemeris computation via pint[66, 67], and is designed to work with both narrow- band and wideband analysis paradigms. WithinVela, we employed the ensemble sampleremcee[68] to draw samples from the likelihood. C. From timing to theory parameters: the resampling strategy The Bayesian timing analysis described in the previ- ous section gives a posterior distribution in the form of Eq. (71), where ⃗λ={ ⃗θ, ⃗ζ}collects all timing parame- ters, with ⃗ζdenoting spin and astrometric parameters and ⃗θ={ ⃗θorb, ⃗θPK}the orbital and post-Keplerian pa-","citing_arxiv_id":"2605.01436"},{"n":1,"role":"method","polarity":"use_method","paper_title":"Confronting Color Glass Condensate at next-to-leading order with HERA data","primary_cat":"hep-ph","context_text":"1: Comparison between emulator prediction and model calculation for the total reduced cross section (left) and charm contribution (right). The relative difference averaged over all training samples is presented. section and the charm quark contribution). The cali- brated emulator becomes a faster model substitute for the MCMC algorithm to use. Theemcee[46] imple- mentation of the MCMC involves an ensemble of ran- dom walkers whose advancements depend on informa- tion from other walkers. A walker proposes their next step and is accepted with a probability similar to that of a Metropolis-Hastings algorithm:α∼P(θ 1)/P(θ 0), whereθ 0 is the walker's current position in the parame- ter space,θ 1 is the proposed step andPis the likelihood.","citing_arxiv_id":"2604.22332"},{"n":1,"role":"dataset","polarity":"use_dataset","paper_title":"AI-assisted modeling and Bayesian inference of unpolarized quark transverse momentum distributions from Drell-Yan data","primary_cat":"hep-ph","context_text":"contains 9 nonperturbative fit parameters: {c0, c1, BNP, λ1, λ2, λ3, ANP, x0, σx}.(2.34) - 8 - Dataset Process Observable √s[GeV] Fiducial cut Q[GeV] y Npts STAR [57] p-p dσ/dqT 510 pT >25 GeV |η|<1 73-114 |y|<1 7 CDF Run I [58] p-¯ p dσ/dqT 1800 Inclusive 66-116 Inclusive 25 CDF Run II [59] p-¯ p dσ/dqT 1960 Inclusive 66-116 Inclusive 26 D0 Run I [60] p-¯ p dσ/dqT 1800 Inclusive 75-105 Inclusive 12 D0 Run II [61] p-¯ p σ−1 dσ/dqT 1960 Inclusive 70-110 Inclusive 5 D0 Run II (µ +µ−) [62] p-¯ p σ−1 dσ/dqT 1960 pT >15 GeV |η|<1.7 65-115 Inclusive 3 ATLAS 7 TeV [63] p-p σ−1 dσ/dqT 7000 pT >20 GeV |η|<2.4 66-116 |y|<1 1<|y|<2 2<|y|<2.4 6 6 6 ATLAS 8 TeV [64] p-p dσ/dqT 8000 pT >20 GeV |η|<2.4 66-116","citing_arxiv_id":"2604.14133"},{"n":1,"role":"method","polarity":"use_method","paper_title":"Cosmologically viable non-polynomial quasi-topological gravity: explicit models, $\\Lambda$CDM limit and observational constraints","primary_cat":"gr-qc","context_text":"For the BAO sector, we utilize the dataset of [89], which incorporates fiducial cosmology corrections through the ratior d/rfid, allowing for a consistent com- parison between theoretical predictions and observations. The exploration of the parameter space is carried out using the Markov Chain Monte Carlo (MCMC) method, implemented through the publicly available Python pack- ageemcee[90]. In this context, the SNIa nuisance pa- rameterMis treated as a free parameter of the Pan- theon dataset (see [88] and references therein), while the radiation density parameter Ωr0 is neglected, as it is sub- dominant at late times. The above statistical framework is applied to the quar- tic and power-law models, as well as to the standard ΛCDM scenario, allowing for a direct and consistent com-","citing_arxiv_id":"2604.13002"},{"n":1,"role":"method","polarity":"use_method","paper_title":"$\\boldsymbol{B_c}$ Meson Spectroscopy from Bayesian MCMC: Probing Confinement and State Mixing","primary_cat":"hep-ph","context_text":"We apply this non-linear form unchanged to the mass spectra from both the Cornell and modified Cornell potentials, 2 enabling a direct comparison of the resulting Regge behavior between the two cases. The parameters of both the linear (Eq. (28)) and non-linear (Eq. (29)) trajectories are extracted by fitting the MCMC-derived mass spectra using theiminuit package [54], based on the MINUIT optimization algorithm [55]. The resulting trajectories are discussed in Sec. III. III. NUMERICAL RESULTS AND DISCUSSION We present the numerical results for theB c mass spectrum obtained from the Cornell potential (Eq. (1)) and its logarithmic extension (Eq. (2)), hereafter Potential I and Poten- tial II, respectively. The role of the logarithmic term is quantified through ther-dependent","citing_arxiv_id":"2604.04846"},{"n":1,"role":"method","polarity":"use_method","paper_title":"Probing soft signals of gravitational-wave memory with space-based interferometers","primary_cat":"gr-qc","context_text":"same SNR, possibly due to its stronger degeneracy dis- cussed in Sec. IV B. However, the FIM estimate cannot be reliably trusted at low SNR, which is likely a realistic regime. In the following, we present results from Bayesian parameter- estimation simulations, focusing on the case of displace- ment memory. For sampling the posterior, we use the MCMC sampler emcee [92] and plot the posterior dis- tributions using corner [93]. For concreteness, we con- sider the measurements using Taiji, LISA-Taiji network (described in Fig. 8) and BBO. For LISA-Taiji, we set ηTaiji = ηLISA + 2 π/9, Φ0 = 0 , and ηLISA = π/4; the log-likelihood is taken as the sum of those of LISA and Taiji. As mentioned in Sec. IV A, the signals at a sin-","citing_arxiv_id":"2603.28689"},{"n":1,"role":"method","polarity":"background","paper_title":"Model-independent test of the cosmic distance duality relation with recent observational data","primary_cat":"astro-ph.CO","context_text":"Nong, H. Wang, B. Zhang, Z. Li and N. Liang,Constraints on cosmological models with gamma-ray bursts in cosmology-independent way,Int. J. Mod. Phys. D34(2025) 2450073 [2307.16467]. [96] H. Wang and N. Liang,Constraints from Fermi observations of long gamma-ray bursts on cosmological parameters,Mon. Not. Roy. Astron. Soc.533(2024) 743 [2405.14357]. [97] B. Zhang, H. Wang, X. Nong, G. Wang, P. Wu and N. Liang,Model-independent gamma-ray bursts constraints on cosmological models using machine learning,Astrophys. Space Sci.370 (2025) 10 [2312.09440]. [98] Z. Huang, Z. Xiong, X. Luo, G. Wang, Y. Liu and N. Liang,Gamma-ray bursts calibrated from the observational H(z) data in artificial neural network framework,JHEAp47(2025)","citing_arxiv_id":"2603.27616"},{"n":1,"role":"method","polarity":"use_method","paper_title":"New Way to Date Globular Clusters: Brown Dwarf Cooling Sequences","primary_cat":"astro-ph.SR","context_text":"[139]andVasilievandBaumgardt[157]for47Tuc. Ihavetakenthemtoberepresentativeofallglobular clustersconsideredinthiswork. Thecalculationswererepeatedfortwotemporalbaselinesof2and5years. TheastrometricandphotometricerrorswereextractedfromthesimulatedobservationsbyfittingthePSFmodelusingtheGoodman-Weare [93]MarkovChainMonteCarlo(MCMC)algorithmimplementedinthe emceePythonpackage[104]. ThemodelPSFwasconstructedusingthe sameprocedureasthesimulatedobservationsbutwithoutapplyingnoisetotheimage. PSFfittingwascarriedoutonthesimulatedF150W2and F322W2imagesofthestarsimultaneously. Thefitterwasconstrainedtorequirethestartohavethesamepositioninbothbands. Thestandard errors in magnitudes and positions were taken as the half-difference between the84th and16th percentiles (±1 -sigma range) of the MCMC","citing_arxiv_id":"2603.07481"},{"n":1,"role":"method","polarity":"use_method","paper_title":"Cosmological analysis of the DESI DR1 Lyman alpha 1D power spectrum","primary_cat":"astro-ph.CO","context_text":"The resultingP 1D mocks are assigned the same covariance matrix as the one em- ployed in the DESI DR1 analysis. These mocks do not include contaminants or systematics; however, we analyze them with the model presented in the previous section, which accounts for both. We extract cosmological constraints using the publicly available affine-invariant Markov chain Monte Carlo (MCMC) ensemble sampleremcee 7 [118]. At each step,emceeproposes 7https://emcee.readthedocs.io/en/stable/ - 23 - 0.9 1.0 1.1 z = 2.2, 2 = 34.58, ndata=49 z = 2.4, 2 = 48.92, ndata=52 z = 2.6, 2 = 55.51, ndata=55 0.9 1.0 1.1 z = 2.8, 2 = 62.34, ndata=58 z = 3.0, 2 = 78.8, ndata=61 z = 3.2, 2 = 73.69, ndata=63 0.75 1.00 1.25 z = 3.4, 2 = 69.05, ndata=65 z = 3.6, 2 = 99.18, ndata=67","citing_arxiv_id":"2601.21432"},{"n":1,"role":"method","polarity":"use_method","paper_title":"Interacting $k$-essence field with non-pressureless Dark Matter: Cosmological Dynamics and Observational Constraints","primary_cat":"astro-ph.CO","context_text":"consistent solutions across cosmic epochs, and adopt uniform priors as listed in Tab. II. For sampling and likelihood eval- uation, we employ the nested samplerPolyChord, which is well suited for high-dimensional parameter spaces compared to affine-invariant MCMC samplers such asemcee[99-101]. The chains are analyzed usingGetDistto extract the best-fit values [102, 103]. Finally, we compare our models against flat ΛCDM using information criteria such as the Akaike Informa- tion Criterion (AIC) and Bayesian Information Criterion (BIC) [104-106], following the standard methodology described in [107]. V. RESULTS In this section, we present the results obtained by imple- menting the models described in Tab. I with the considered","citing_arxiv_id":"2509.09202"},{"n":1,"role":"method","polarity":"use_method","paper_title":"Interacting Scalar Fields as Dark Energy and Dark Matter in Einstein scalar Gauss Bonnet Gravity","primary_cat":"gr-qc","context_text":"PLANCK + PP, and (ii) CC + DESBAO + PLANCK + DES, as −2 lnLtot = 𝜒2 tot. (6.8) We estimate the likelihood by implementing the model in Python and use the publicly available affine-invariant Markov Chain Monte Carlo (MCMC) ensemble sampler emcee [145] to obtain the posterior distributions of the model parameters. The resulting samples are then analyzed usingGetDist [146] to obtain marginalized 1D and 2D posterior distributions. Finally, we compare the statistical preference of the current model relative to the flat ΛCDM model using information criteria such as the Akaike Information Criterion (AIC) and the Bayesian Information Criterion (BIC) [147-149]: AIC = −2 lnLmax + 2𝑘, (6.9) BIC = −2 lnLmax + 𝑘 ln 𝑁, (6.10)","citing_arxiv_id":"2507.05207"},{"n":1,"role":"method","polarity":"use_method","paper_title":"DESI 2024 III: Baryon Acoustic Oscillations from Galaxies and Quasars","primary_cat":"astro-ph.CO","context_text":"package that provides a common framework for writing DESI likelihoods. The BAO the- ory and likelihood is implemented in JAX [126] 20. Even though gradient-based sampling methods were implemented, we found that with analytic marginalization over broadband parameters that leaves a few sampled parameters, and using Jax just-in-time compilation and parallelization capabilities, the ensemble sampler emcee [127]21 provided well-sampled posterior estimates in a just a few minutes. In addition to MCMC sampling, we also perform posterior profiling using desilike's wrapping of Minuit [128]. During the course of this work, we also used/developed a fully independent galaxy BAO fitting pipeline (Barry22 [129]), for some of the supporting papers and with which we tested","citing_arxiv_id":"2404.03000"},{"n":1,"role":"method","polarity":"use_method","paper_title":"Inferring the properties of a population of compact binaries in presence of selection effects","primary_cat":"astro-ph.IM","context_text":"RPG-2019-350, and Royal Society Grant No. RGS-R2-202004. LIGO was con- structed by the California Institute of Technology and Massachusetts Institute of Technology with funding from the National Science Foundation and operates un- der cooperative agreement PHY-0757058. We acknowledge use of iPython [86], Matplotlib [87, 88], NumPy [89], SciPy [90], emcee [91] and SeaBorn [92]. This is LIGO Document P2000231 Glossary and main symbols α(⃗λ ) The fraction of physical sources that are detectable by our experiment, ac- cording to some detection threshold. It depends on the shape population hyper parameters⃗λ. ∧ Logical \"and\" of two propositions. A ∧ B is true if and only if both A and B are true. Background An (usually uninteresting) background event that might be produced","citing_arxiv_id":"2007.05579"}]},"authors":[]}}