Autovalidating c

This solves the open problem of rigorously drawing independent and identically distributed samples from the posterior distribution over rooted and unrooted small tree spaces (3 or 4 taxa) based on any multiply-aligned sequence data.

Math [email protected]@[email protected]@ =feaagaart1ev2aaat Cv AUf Ktt Learu Wr P9MDH5MBPb Iq V92Aae Xat Lx BI9g Baebbnrfif Hh DYfgasaac PC6x Ni=x H8vi VGI8Gi=h Eeeu0x Xdbba9fr Fj0xb9qqp G0d Xdb9aspe I8k8fi I fs Y=rq Gq Vepae9pg0db9vqai Vg Fr0xfr=xfr=xc9adbaqaae Gaci Gaaiaabeqaaeqabi Waaa Gcba Wefv3y SLgznfg DOjdaryqr1ng BPrginfg DObcv39gaiqaacq [email protected]@Math [email protected]@[email protected]@ =feaagaart1ev2aaat Cv AUf Ktt Learu Wr P9MDH5MBPb Iq V92Aae Xat Lx BI9g Baebbnrfif Hh DYfgasaac PC6x Ni=x H8vi VGI8Gi=h Eeeu0x Xdbba9fr Fj0xb9qqp G0d Xdb9aspe I8k8fi I fs Y=rq Gq Vepae9pg0db9vqai Vg Fr0xfr=xfr=xc9adbaqaae Gaci Gaaiaabeqaaeqabi Waaa Gcba Wefv3y SLgznfg DOjdaryqr1ng BPrginfg DObcv39gaiqaacq [email protected]@.

We only derive the likelihood functions for the simplest model with just two characters as it is thought to well-represent the core problems in phylogenetic estimation (see for e.g.

[Math [email protected]@[email protected]@ =feaagaart1ev2aaat Cv AUf Ktt Learu Wr P9MDH5MBPb Iq V92Aae Xat Lx BI9g Baebbnrfif Hh DYfgasaac PC6x Ni=x H8vi VGI8Gi=h Eeeu0x Xdbba9fr Fj0xb9qqp G0d Xdb9aspe I8k8fi I fs Y=rq Gq Vepae9pg0db9vqai Vg Fr0xfr=xfr=xc9adbaqaae Gaci Gaaiaabeqaaeqabi Waaa Gcba Waa Wraa Sqabeaatuu DJXw AKz KCHTg D1jharyqr1ng BPrgigjxy Rrx DYbacea Gae8Nc WVeaamrr1ng BPrwt Hrh AYaqehuu DJXw AKbst Hrh AGq1DVbacfa Gccq [email protected]@Math [email protected]@[email protected]@ =feaagaart1ev2aaat Cv AUf Ktt Learu Wr P9MDH5MBPb Iq V92Aae Xat Lx BI9g Baebbnrfif Hh DYfgasaac PC6x Ni=x H8vi VGI8Gi=h Eeeu0x Xdbba9fr Fj0xb9qqp G0d Xdb9aspe I8k8fi I fs Y=rq Gq Vepae9pg0db9vqai Vg Fr0xfr=xfr=xc9adbaqaae Gaci Gaaiaabeqaaeqabi Waaa Gcba Waa Wraa Sqabeaatuu DJXw AKz KCHTg D1jharyqr1ng BPrgigjxy Rrx DYbacea Gae8Nc WVeaamrr1ng BPrwt Hrh AYaqehuu DJXw AKbst Hrh AGq1DVbacfa Gccq [email protected]@Math [email protected]@[email protected]@ =feaagaart1ev2aaat Cv AUf Ktt Learu Wr P9MDH5MBPb Iq V92Aae Xat Lx BI9g Baebbnrfif Hh DYfgasaac PC6x Ni=x H8vi VGI8Gi=h Eeeu0x Xdbba9fr Fj0xb9qqp G0d Xdb9aspe I8k8fi I fs Y=rq Gq Vepae9pg0db9vqai Vg Fr0xfr=xfr=xc9adbaqaae Gaci Gaaiaabeqaaeqabi Waaa Gcba Waa Wraa Sqabeaatuu DJXw AKz KCHTg D1jharyqr1ng BPrgigjxy Rrx DYbacea Gae8Nc WVeaamrr1ng BPrwt Hrh AYaqehuu DJXw AKbst Hrh AGq1DVbacfa Gccq [email protected]@Math [email protected]@[email protected]@ =feaagaart1ev2aaat Cv AUf Ktt Learu Wr P9MDH5MBPb Iq V92Aae Xat Lx BI9g Baebbnrfif Hh DYfgasaac PC6x Ni=x H8vi VGI8Gi=h Eeeu0x Xdbba9fr Fj0xb9qqp G0d Xdb9aspe I8k8fi I fs Y=rq Gq Vepae9pg0db9vqai Vg Fr0xfr=xfr=xc9adbaqaae Gaci Gaaiaabeqaaeqabi Waaa Gcba Waa Wraa Sqabeaatuu DJXw AKz KCHTg D1jharyqr1ng BPrgigjxy Rrx DYbacea Gae8Nc WVeaamrr1ng BPrwt Hrh AYaqehuu DJXw AKbst Hrh AGq1DVbacfa Gccq [email protected]@ is a union of distinct topologies then we let our prior be an equally weighted finite mixture of uniform densities over large boxes in each topology.

Here we introduce an auto-validating version of the rejection sampler, via interval analysis, to rigorously draw samples from posterior distributions over small phylogenetic tree spaces.

The posterior samples from the auto-validating sampler are used to rigorously (i) estimate posterior probabilities for different rooted topologies based on mitochondrial DNA from human, chimpanzee and gorilla, (ii) conduct a non-parametric test of rate variation between protein-coding and t RNA-coding sites from three primates and (iii) obtain a posterior estimate of the human-neanderthal divergence time.

Using one of the latest data sets we obtain a rigorous posterior estimate of the human-neanderthal divergence time.

More formally, the method employs an interval extension of the target posterior shape Math [email protected]@[email protected]@ =feaagaart1ev2aaat Cv AUf Ktt Learu Wr P9MDH5MBPb Iq V92Aae Xat Lx BI9g Baebbnrfif Hh DYfgasaac PC6x Ni=x H8vi VGI8Gi=h Eeeu0x Xdbba9fr Fj0xb9qqp G0d Xdb9aspe I8k8fi I fs Y=rq Gq Vepae9pg0db9vqai Vg Fr0xfr=xfr=xc9adbaqaae Gaci Gaaiaabeqaaeqabi Waaa Gcba Wefv3y SLgznfg DOjdaryqr1ng BPrginfg DObcv39gaiqaacq [email protected]@Math [email protected]@[email protected]@ =feaagaart1ev2aaat Cv AUf Ktt Learu Wr P9MDH5MBPb Iq V92Aae Xat Lx BI9g Baebbnrfif Hh DYfgasaac PC6x Ni=x H8vi VGI8Gi=h Eeeu0x Xdbba9fr Fj0xb9qqp G0d Xdb9aspe I8k8fi I fs Y=rq Gq Vepae9pg0db9vqai Vg Fr0xfr=xfr=xc9adbaqaae Gaci Gaaiaabeqaaeqabi Waaa Gcba Wefv3y SLgzgjxy Rrx DYbqeguu DJXw AKb Ir Yf2A0v NCa Gabaiab=nb8ujabc Qda6iabg2da9iabc Uha7Hqadiab rha0naa Caaaleqaba Gaeiika GIae Gymae Jaeiyka Kcaa OGaeiila WIae4h Daq3aa Wbaa Sqabeaacq GGOaakcqa IYa Gmcq GGPaqkaa Gccq GGSaalcq GGUa Glcq GGUa Glcq GGUa Glcq GGSaalcq GF0ba Ddaahaa Wcbeqaaiabc Ica Oiabc Yha8jab=nb8ujabc Yha8jabc Mca [email protected]@Math [email protected]@[email protected]@ =feaagaart1ev2aaat Cv AUf Ktt Learu Wr P9MDH5MBPb Iq V92Aae Xat Lx BI9g Baebbnrfif Hh DYfgasaac PC6x Ni=x H8vi VGI8Gi=h Eeeu0x Xdbba9fr Fj0xb9qqp G0d Xdb9aspe I8k8fi I fs Y=rq Gq Vepae9pg0db9vqai Vg Fr0xfr=xfr=xc9adbaqaae Gaci Gaaiaabeqaaeqabi Waaa Gcba Wefv3y SLgznfg DOjdaryqr1ng BPrginfg DObcv39gaiqaacq [email protected]@Math [email protected]@[email protected]@ =feaagaart1ev2aaat Cv AUf Ktt Learu Wr P9MDH5MBPb Iq V92Aae Xat Lx BI9g Baebbnrfif Hh DYfgasaac PC6x Ni=x H8vi VGI8Gi=h Eeeu0x Xdbba9fr Fj0xb9qqp G0d Xdb9aspe I8k8fi I fs Y=rq Gq Vepae9pg0db9vqai Vg Fr0xfr=xfr=xc9adbaqaae Gaci Gaaiaabeqaaeqabi Waaa Gcba Wefv3y SLgznfg DOjdaryqr1ng BPrginfg DObcv39gaiqaacq [email protected]@Math [email protected]@[email protected]@ =feaagaart1ev2aaat Cv AUf Ktt Learu Wr P9MDH5MBPb Iq V92Aae Xat Lx BI9g Baebbnrfif Hh DYfgasaac PC6x Ni=x H8vi VGI8Gi=h Eeeu0x Xdbba9fr Fj0xb9qqp G0d Xdb9aspe I8k8fi I fs Y=rq Gq Vepae9pg0db9vqai Vg Fr0xfr=xfr=xc9adbaqaae Gaci Gaaiaabeqaaeqabi Waaa Gcba Wefv3y SLgznfg DOjdaryqr1ng BPrginfg DObcv39gaiqaacq [email protected]@Math [email protected]@[email protected]@ =feaagaart1ev2aaat Cv AUf Ktt Learu Wr P9MDH5MBPb Iq V92Aae Xat Lx BI9g Baebbnrfif Hh DYfgasaac PC6x Ni=x H8vi VGI8Gi=h Eeeu0x Xdbba9fr Fj0xb9qqp G0d Xdb9aspe I8k8fi I fs Y=rq Gq Vepae9pg0db9vqai Vg Fr0xfr=xfr=xc9adbaqaae Gaci Gaaiaabeqaaeqabi Waaa Gcba Wefv3y SLgzgjxy Rrx DYbqeguu DJXw AKb Ir Yf2A0v NCa [email protected]@] we efficiently propose samples from this normalized step-function envelope for von Neumann rejection sampling.The length of an edge (branch length) connecting two nodes (species) in the tree represents the amount of evolutionary time (divergence) between the two species. During the likelihood computation, one needs to integrate over all possible states at the unobserved ancestral nodes. Our rooted triplets (Figure Math [email protected]@[email protected]@ =feaagaart1ev2aaat Cv AUf Ktt Learu Wr P9MDH5MBPb Iq V92Aae Xat Lx BI9g Baebbnrfif Hh DYfgasaac PC6x Ni=x H8vi VGI8Gi=h Eeeu0x Xdbba9fr Fj0xb9qqp G0d Xdb9aspe I8k8fi I fs Y=rq Gq Vepae9pg0db9vqai Vg Fr0xfr=xfr=xc9adbaqaae Gaci Gaaiaabeqaaeqabi Waaa Gcba Wefv3y SLgzgjxy Rrx DYbqeguu DJXw AKb Ir Yf2A0v NCa Gabaiab=rb8vjabg2da9iabc Uha7jabdgga Hnaa Baaaleaacqa IXaqmaeqaa OGaeiila WIaemyyae2aa Sbaa Sqaaiabikda Yaqaba Gccq GGSaalcq GGUa Glcq GGUa Glcq GGUa Glcq GGSaalcq WGHbqyda Wgaa Wcba Gaeii Fa WNae8hf Wx Laeii Fa [email protected]@Math [email protected]@[email protected]@ =feaagaart1ev2aaat Cv AUf Ktt Learu Wr P9MDH5MBPb Iq V92Aae Xat Lx BI9g Baebbnrfif Hh DYfgasaac PC6x Ni=x H8vi VGI8Gi=h Eeeu0x Xdbba9fr Fj0xb9qqp G0d Xdb9aspe I8k8fi I fs Y=rq Gq Vepae9pg0db9vqai Vg Fr0xfr=xfr=xc9adbaqaae Gaci Gaaiaabeqaaeqabi Waaa Gcba Wefv3y SLgzgjxy Rrx DYbqeguu DJXw AKb Ir Yf2A0v NCa [email protected]@Math [email protected]@[email protected]@ =feaagaart1ev2aaat Cv AUf Ktt Learu Wr P9MDH5MBPb Iq V92Aae Xat Lx BI9g Baebbnrfif Hh DYfgasaac PC6x Ni=x H8vi VGI8Gi=h Eeeu0x Xdbba9fr Fj0xb9qqp G0d Xdb9aspe I8k8fi I fs Y=rq Gq Vepae9pg0db9vqai Vg Fr0xfr=xfr=xc9adbaqaae Gaci Gaaiaabeqaaeqabi Waaa Gcba Waaubeaeqaleaacq WGIb Gyda Wgaaqaaiabd Uga Rbadbeaaa Sqab0qaamaa Ceaaoeqaleaacq WGRb WAaaqcda Kaemi [email protected]@Math [email protected]@[email protected]@ =feaagaart1ev2aaat Cv AUf Ktt Learu Wr P9MDH5MBPb Iq V92Aae Xat Lx BI9g Baebbnrfif Hh DYfgasaac PC6x Ni=x H8vi VGI8Gi=h Eeeu0x Xdbba9fr Fj0xb9qqp G0d Xdb9aspe I8k8fi I fs Y=rq Gq Vepae9pg0db9vqai Vg Fr0xfr=xfr=xc9adbaqaae Gaci Gaaiaabeqaaeqabi Waaa Gcba Waa Wraa Sqabeaacq WGRb WAaa Wefv3y SLgznfg DOjdaryqr1ng BPrginfg [email protected]@Math [email protected]@[email protected]@ =feaagaart1ev2aaat Cv AUf Ktt Learu Wr P9MDH5MBPb Iq V92Aae Xat Lx BI9g Baebbnrfif Hh DYfgasaac PC6x Ni=x H8vi VGI8Gi=h Eeeu0x Xdbba9fr Fj0xb9qqp G0d Xdb9aspe I8k8fi I fs Y=rq Gq Vepae9pg0db9vqai Vg Fr0xfr=xfr=xc9adbaqaae Gaci Gaaiaabeqaaeqabi Waaa Gcba Gaemiuaa1aa Sbaa Sqaaiabdgga Hnaa Baaameaacq WGPbq Aaeqaa SGaeiila WIaemyyae2aa Sbaa Wqaaiabd Qga Qbqabaaaleqaa OGaeiika GIaemi Daq Naeiyka [email protected]@Math [email protected]@[email protected]@ =feaagaart1ev2aaat Cv AUf Ktt Learu Wr P9MDH5MBPb Iq V92Aae Xat Lx BI9g Baebbnrfif Hh DYfgasaac PC6x Ni=x H8vi VGI8Gi=h Eeeu0x Xdbba9fr Fj0xb9qqp G0d Xdb9aspe I8k8fi I fs Y=rq Gq Vepae9pg0db9vqai Vg Fr0xfr=xfr=xc9adbaqaae Gaci Gaaiaabeqaaeqabi Waaa Gcba Wefv3y SLgzgjxy Rrx DYbqeguu DJXw AKb Ir Yf2A0v NCa [email protected]@Math [email protected]@[email protected]@ =feaagaart1ev2aaat Cv AUf Ktt Learu Wr P9MDH5MBPb Iq V92Aae Xat Lx BI9g Baebbnrfif Hh DYfgasaac PC6x Ni=x H8vi VGI8Gi=h Eeeu0x Xdbba9fr Fj0xb9qqp G0d Xdb9aspe I8k8fi I fs Y=rq Gq Vepae9pg0db9vqai Vg Fr0xfr=xfr=xc9adbaqaae Gaci Gaaiaabeqaaeqabi Waaa Gcba Wefv3y SLgzgjxy Rrx DYbqeguu DJXw AKb Ir Yf2A0v NCa [email protected]@Math [email protected]@[email protected]@ =feaagaart1ev2aaat Cv AUf Ktt Learu Wr P9MDH5MBPb Iq V92Aae Xat Lx BI9g Baebbnrfif Hh DYfgasaac PC6x Ni=x H8vi VGI8Gi=h Eeeu0x Xdbba9fr Fj0xb9qqp G0d Xdb9aspe I8k8fi I fs Y=rq Gq Vepae9pg0db9vqai Vg Fr0xfr=xfr=xc9adbaqaae Gaci Gaaiaabeqaaeqabi Waaa Gcba Waaubeaeqaleaacq WGIb Gyda Wgaaqaaiabd Uga Rbadbeaaa Sqab0qaamaa Ceaaoeqaleaacq WGRb WAaaqcda Kaemi [email protected]@Math [email protected]@[email protected]@ =feaagaart1ev2aaat Cv AUf Ktt Learu Wr P9MDH5MBPb Iq V92Aae Xat Lx BI9g Baebbnrfif Hh DYfgasaac PC6x Ni=x H8vi VGI8Gi=h Eeeu0x Xdbba9fr Fj0xb9qqp G0d Xdb9aspe I8k8fi I fs Y=rq Gq Vepae9pg0db9vqai Vg Fr0xfr=xfr=xc9adbaqaae Gaci Gaaiaabeqaaeqabi Waaa Gcba Gaemi Ba W2aa Sbaa Sqaaiabd Iga Obqaba Gccq GG6a Gocq GH9aqpcq GGOaakcq WGSba Bdaqhaa Wcba Gaemi Aa Ggaba Gaeiika GIaemyyae2aa Sbaa Wqaaiabigda Xaqaba Wccq GGPaqkaa Gccq GGSaalcq WGSba Bdaqhaa Wcba Gaemi Aa Ggaba Gaeiika GIaemyyae2aa Sbaa Wqaaiabikda Yaqaba Wccq GGPaqkaa Gccq GGSaalcq GGUa Glcq GGUa Glcq GGUa Glcq GGSaalcq WGSba Bdaqhaa Wcba Gaemi Aa Ggaba Gaeiika GIaemyyae2aa Sbaa Wqaaiabc Yha8nrr1ng BPr Mr Yf2A0v NCae Hbfv3y SLgz Gy KCHTg D1jhaiqaacq WFua Fvcq GG8ba Faeqaa SGaeiyka Kcaa OGaeiyka KIaeyic I48efv3y SLgznfg DOjdar Cqr1ng BPrginfg DObcv39gaiuaacq GFDe Iudaahaa Wcbeqaaiabc Yha8jab=rb8vjabc [email protected]@Math [email protected]@[email protected]@ =feaagaart1ev2aaat Cv AUf Ktt Learu Wr P9MDH5MBPb Iq V92Aae Xat Lx BI9g Baebbnrfif Hh DYfgasaac PC6x Ni=x H8vi VGI8Gi=h Eeeu0x Xdbba9fr Fj0xb9qqp G0d Xdb9aspe I8k8fi I fs Y=rq Gq Vepae9pg0db9vqai Vg Fr0xfr=xfr=xc9adbaqaae Gaci Gaaiaabeqaaeqabi Waaa Gcba Waaubeaeqaleaacq WGIb Gyda Wgaaqaaiabd Uga Rbadbeaaa Sqab0qaamaa Ceaaoeqaleaacq WGRb WAaaqcda Kaemi [email protected]@Math [email protected]@[email protected]@ =feaagaart1ev2aaat Cv AUf Ktt Learu Wr P9MDH5MBPb Iq V92Aae Xat Lx BI9g Baebbnrfif Hh DYfgasaac PC6x Ni=x H8vi VGI8Gi=h Eeeu0x Xdbba9fr Fj0xb9qqp G0d Xdb9aspe I8k8fi I fs Y=rq Gq Vepae9pg0db9vqai Vg Fr0xfr=xfr=xc9adbaqaae Gaci Gaaiaabeqaaeqabi Waaa Gcba Gaemiuaa1aa Sbaa Sqaaiabdgga Hnaa Baaameaacq WGPbq Aaeqaa SGaeiila WIaemyyae2aa Sbaa Wqaaiabd Qga Qbqabaaaleqaa OGaeiika GIaemi Daq Naeiyka [email protected]@Math [email protected]@[email protected]@ =feaagaart1ev2aaat Cv AUf Ktt Learu Wr P9MDH5MBPb Iq V92Aae Xat Lx BI9g Baebbnrfif Hh DYfgasaac PC6x Ni=x H8vi VGI8Gi=h Eeeu0x Xdbba9fr Fj0xb9qqp G0d Xdb9aspe I8k8fi I fs Y=rq Gq Vepae9pg0db9vqai Vg Fr0xfr=xfr=xc9adbaqaae Gaci Gaaiaabeqaaeqabi Waaa Gcba Wefv3y SLgzgjxy Rrx DYbqeguu DJXw AKb Ir Yf2A0v NCa [email protected]@Math [email protected]@[email protected]@ =feaagaart1ev2aaat Cv AUf Ktt Learu Wr P9MDH5MBPb Iq V92Aae Xat Lx BI9g Baebbnrfif Hh DYfgasaac PC6x Ni=x H8vi VGI8Gi=h Eeeu0x Xdbba9fr Fj0xb9qqp G0d Xdb9aspe I8k8fi I fs Y=rq Gq Vepae9pg0db9vqai Vg Fr0xfr=xfr=xc9adbaqaae Gaci Gaaiaabeqaaeqabi Waaa Gcba Wefv3y SLgzgjxy Rrx DYbqeguu DJXw AKb Ir Yf2A0v NCa [email protected]@Math [email protected]@[email protected]@ =feaagaart1ev2aaat Cv AUf Ktt Learu Wr P9MDH5MBPb Iq V92Aae Xat Lx BI9g Baebbnrfif Hh DYfgasaac PC6x Ni=x H8vi VGI8Gi=h Eeeu0x Xdbba9fr Fj0xb9qqp G0d Xdb9aspe I8k8fi I fs Y=rq Gq Vepae9pg0db9vqai Vg Fr0xfr=xfr=xc9adbaqaae Gaci Gaaiaabeqaaeqabi Waaa Gcba Gaemi Ba W2aa Sbaa Sqaaiabdsga Knaa Baaameaacq WIp M zcq GGSaalcq WGXbq Caeqaaa [email protected]@Math [email protected]@[email protected]@ =feaagaart1ev2aaat Cv AUf Ktt Learu Wr P9MDH5MBPb Iq V92Aae Xat Lx BI9g Baebbnrfif Hh DYfgasaac PC6x Ni=x H8vi VGI8Gi=h Eeeu0x Xdbba9fr Fj0xb9qqp G0d Xdb9aspe I8k8fi I fs Y=rq Gq Vepae9pg0db9vqai Vg Fr0xfr=xfr=xc9adbaqaae Gaci Gaaiaabeqaaeqabi Waaa Gcba Gaemi Ba W2aa0baa Sqaaiabd Iga Obqaaiabc Ica Oiabdgga Hnaa Baaameaacq WGPbq Aaeqaa SGaeiyka [email protected]@Math [email protected]@[email protected]@ =feaagaart1ev2aaat Cv AUf Ktt Learu Wr P9MDH5MBPb Iq V92Aae Xat Lx BI9g Baebbnrfif Hh DYfgasaac PC6x Ni=x H8vi VGI8Gi=h Eeeu0x Xdbba9fr Fj0xb9qqp G0d Xdb9aspe I8k8fi I fs Y=rq Gq Vepae9pg0db9vqai Vg Fr0xfr=xfr=xc9adbaqaae Gaci Gaaiaabeqaaeqabi Waaa Gcba Gaemi Ba W2aa0baa Sqaaiabd Iga Obqaaiabc Ica Oiabdgga Hnaa Baaameaacq WGQb GAaeqaa SGaeiyka [email protected]@Math [email protected]@[email protected]@ =feaagaart1ev2aaat Cv AUf Ktt Learu Wr P9MDH5MBPb Iq V92Aae Xat Lx BI9g Baebbnrfif Hh DYfgasaac PC6x Ni=x I8qi VKYPFj Yd Ha Vhbbf9v8qqaq Fr0xc9vq Fj0d Xdbba91qpepe I8k8fi I fs Y=rq Gq Vepae9pg0db9vqai Vg Fr0xfr=xfr=xc9adbaqaae Gaci Gaaiaabeqaaeqabi Waaa Gcba Gaemi Ba W2aa Sbaa Sqaaiabdsga Knaa Baaameaacq WIp M zcq GGSaalcq WGXbq Caeqaaa Wcbeaakiabc Ica Omaa Ceaaleqaba Gaem4Aa Sgaa OGaemi Daq Naeiyka KIaeyypa0Jaemi Ba W2aa Sbaa Sqaaiabdkha Ybqaba Gccq GH9aqpdaaeqbqaaiabc Ica Oiabec8a Wjabc Ica Oiabdgga Hnaa Baaaleaacq WGPbq Aaeqaa OGaeiyka KIaeyy XICTaemi Ba W2aa0baa Sqaaiabdkha Ybqaaiabc Ica Oiabdgga Hnaa Baaameaacq WGPbq Aaeqaa SGaeiyka Kcaaaqaaiabdgga Hnaa Baaameaacq WGPbq Aaeqaa SGaeyic I48efv3y SLgzgjxy Rrx DYbqeguu DJXw AKb Ir Yf2A0v NCa Gabaiab=rb8vbqab0Gaeyye Iuoakiabc Mca Piabc6ca [email protected]@Math [email protected]@[email protected]@ =feaagaart1ev2aaat Cv AUf Ktt Learu Wr P9MDH5MBPb Iq V92Aae Xat Lx BI9g Baebbnrfif Hh DYfgasaac PC6x Ni=x I8qi VKYPFj Yd Ha Vhbbf9v8qqaq Fr0xc9vq Fj0d Xdbba91qpepe I8k8fi I fs Y=rq Gq Vepae9pg0db9vqai Vg Fr0xfr=xfr=xc9adbaqaae Gaci Gaaiaabeqaaeqabi Waaa Gcba Waa Cbeaeaacy GGHbqycq GGYb GCcq GGNb Wzcy GGTbq Bcq GGHbqycq GG4ba Ea Sqaamaa Ceaameqaba Gaem4Aa Sgaa SGaemi Daq Naeyic I48aa Wraa Wqabeaatuu DJXw AKz KCHTg D1jharyqr1ng BPrgigjxy Rrx DYbacea Gae8Nc WVeaamrr1ng BPrwt Hrh AYaqehuu DJXw AKbst Hrh AGq1DVbacfa Wccq GFtcpvaeqaa OGaemi Ba W2aa Sbaa Sqaaiabdsga Kbqaba Gccq GGOaakdaahba Wcbeqaaiabd Uga Rbaakiabdsha0jabc Mca Piabc6ca [email protected]@Math [email protected]@[email protected]@ =feaagaart1ev2aaat Cv AUf Ktt Learu Wr P9MDH5MBPb Iq V92Aae Xat Lx BI9g Baebbnrfif Hh DYfgasaac PC6x Ni=x H8vi VGI8Gi=h Eeeu0x Xdbba9fr Fj0xb9qqp G0d Xdb9aspe I8k8fi I fs Y=rq Gq Vepae9pg0db9vqai Vg Fr0xfr=xfr=xc9adbaqaae Gaci Gaaiaabeqaaeqabi Waaa Gcba Wefv3y SLgzgjxy Rrx DYbqeguu DJXw AKb Ir Yf2A0v NCa [email protected]@.We introduce three such models employed in this study next.Moreover, a practical and efficient envelope function has to be as close to the target shape as possible from above.When an envelope function is constructed using point-valued methods, except for simple classes of targets, one cannot guarantee that the envelope function dominates the target shape globally.Unlike many conventional samplers, each sample produced by our method is equivalent to a computer-assisted proof that it is drawn from the desired target, up to the pseudo-randomness of the underlying, deterministic, pseudo-random number generator.

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