The Annals of Statistics

Algebraic algorithms for sampling from conditional distributions

Persi Diaconis and Bernd Sturmfels

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We construct Markov chain algorithms for sampling from discrete exponential families conditional on a sufficient statistic. Examples include contingency tables, logistic regression, and spectral analysis of permutation data. The algorithms involve computations in polynomial rings using Gröbner bases.

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Ann. Statist., Volume 26, Number 1 (1998), 363-397.

First available in Project Euclid: 28 August 2002

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Zentralblatt MATH identifier

Primary: 6E17 13P10: Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)

Conditional inference Monte Carlo Markov chain exponential families Gröbner bases


Diaconis, Persi; Sturmfels, Bernd. Algebraic algorithms for sampling from conditional distributions. Ann. Statist. 26 (1998), no. 1, 363--397. doi:10.1214/aos/1030563990.

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  • AGRESTI, A. 1990. Categorical Data Analy sis. Wiley, New York. Z.
  • AGRESTI, A. 1992. A survey of exact inference for contingency tables. Statist. Sci. 7 131 177. Z.
  • ALDOUS, D. 1987. On the Markov chain stimulation method for uniform combinatorial distributions and simulated annealing. Probab. Engrg. Infom. Sci. 1 33 46. Z.
  • ANDREWS, D. and HERZBERG, A. 1985. Data. Springer, New York. Z.
  • BAGLIVIO, J., OLIVIER, D. and PAGANO, M. 1988. Methods for the analysis of contingency tables with large and small cell counts. J. Amer. Statist. Assoc. 83 1006 1013. Z.
  • BAGLIVIO, J., OLIVIER, D. and PAGANO, M. 1992. Methods for exact goodness-of-fit tests. J. Amer. Statist. Assoc. 87 464 469.Z.
  • BAGLIVIO, J., OLIVIER, D. and PAGANO, M. 1993. Analy sis of discrete data: rerandomization methods and complexity. Technical report, Dept. Mathematics, Boston College. Z.
  • BAy ER, D. and STILLMAN, M. 1989. MACAULAY: a computer algebra sy stem for algebraic geometry. Available via anony mous ftp from Z.
  • BELISLE, C., ROMEIJN, H. and SMITH, R. 1993. Hit and run algorithms for generating multivariate distributions. Math. Oper. Res. 18 255 266. Z.
  • BESAG, J. and CLIFFORD, P. 1989. Generalized Monte Carlo significance tests. Biometrika 76 633 642. Z.
  • BIRCH, B. W. 1963. Maximum likelihood in three-way contingency tables. J. Roy. Statist. Soc. Ser. B 25 220 233. Z.
  • BISHOP, Y., FINEBERG, S. and HOLLAND, P. 1975. Discrete Multivariate Analy sis. MIT Press. Z.
  • BJORNER, A., LAS VERGNAS, M., STURMFELS, B., WHITE, N. and ZIEGLER, G. 1993. Oriented ¨ Matroids. Cambridge Univ. Press. Z.
  • BOKOWSKI, J. and RICHTER-GEBERT, J. 1990. On the finding of final poly nomials. European J. Combin. 11 21 34. Z.
  • BOKOWSKI, J. and RICHTER-GEBERT, J. 1991. On the classification of non-realizable oriented matroids. II. Preprint, T. H. Darmstadt. Z.
  • BOKENHOLT, U. 1993. Applications of Thurstonian models to ranking data. Probability Models ¨ and Statistical Analy sis for Ranking Data. Lecture Notes in Statist. 80 157 172. Springer, New York. Z.
  • BROWN, L. D. 1990. An ancillarity paradox which appears in multiple linear regression. Ann. Statist. 18 471 538. Z.
  • CHRISTENSEN, R. 1990. Log-Linear Models. Springer, New York. Z.
  • CHUNG, F., GRAHAM, R. and YAU, S. T. 1996. On sampling with Markov chains. Random Structures Algorithms 9 55 77. Z.
  • COHEN, A., KEMPERMAN, J. and SACKROWITZ, H. 1994. Unbiased testing in exponential family regression. Ann. Statist. 22 1931 1946. Z.
  • CONTI, P. and TRAVERSO, C. 1991. Buchberger algorithm and integer programming. Proceedings AAECC-9. Lecture Notes in Comp. Sci. 539 130 139. Springer, New York. Z.
  • COX, D. 1958. Some problems connected with statistical inference. Ann. Math. Statist. 29 357 372. Z.
  • COX, D. 1988. Some aspects of conditional and asy mptotic inference. Sankhy a Ser. A 50 314 337.
  • COX, D., LITTLE, J. and O'SHEA, D. 1992. Ideals, Varieties, and Algorithms. Springer, New York. Z.
  • CROON, M. 1989. Latent class models for the analysis of rankings. In New Developments in Z. Psy chological Choice Modeling G. De Solte, H. Feger and K. C. Klauer, eds. 99 121. North-Holland, Amsterdam. Z.
  • DARROCH, J., LAURITZEN, S. and SPEED T. 1980. Markov fields a log-linear interaction models for contingency tables. Ann. Statist. 8 522 539. Z.
  • DIACONIS, P. 1988. Group Representations in Probability and Statistics. IMS, Hay ward, CA. Z.
  • DIACONIS, P. 1989. A generalization of spectral analysis with application to ranked data. Ann. Statist. 17 949 979. Z.
  • DIACONIS, P. and EFRON, B. 1985. Testing for independence in a two-way table: new interpretations for the chi-square statistic. Ann. Statist. 13 845 905. Z.
  • DIACONIS, P. and EFRON, B. 1987. Probabilistic-geometric theorems arising from the analysis of contingency tables. In Contributions to the Theory and Application of Statistics: A Z. Volume in Honor of Herbert Solomon A. Gelfand, ed.. Academic Press, New York. Z.
  • DIACONIS, P., EISENBUD, D. and HOLMES, S. 1997. Speeding up algebraic random walks. Dept. Mathematics, Brandeis Univ. Preprint. Z.
  • DIACONIS, P., EISENBUD, D. and STURMFELS, B. 1996. Lattice walks and primary decompositions. Z. In Proceedings of the Rota Fest B. Sagan, ed.. To appear. Z.
  • DIACONIS, P. and FREEDMAN, D. 1987. A dozen deFinetti-sty le results in search of a theory. Ann. Inst. H. Poincare 23 397 423. ´Z.
  • DIACONIS, P. and GANGOLLI, A. 1995. Rectangular array s with fixed margins. In Discrete Z. Probability and Algorithms D. Aldous, et al., eds.. 15 41. Springer, New York. Z.
  • DIACONIS, P., GRAHAM, R. and STURMFELS, B. 1996. Primitive partition identities. Combinatorics. Paul Erdos Is Eighty 2 173 192. Z.
  • DIACONIS, P., HOLMES, S. and NEALE, R. 1997. A nonreversible Markov chain sampling method. Technical report, Biometry, Cornell Univ. Z.
  • DIACONIS, P. and RABINOWITZ, A. 1997. Conditional inference for logistic regression. Technical report, Stanford Univ. Z.
  • DIACONIS, P. and SALOFF-COSTE, L. 1995a. Random walk on contingency tables with fixed row and column sums. Dept. Mathematics, Harvard Univ., Preprint. Z.
  • DIACONIS, P. and SALOFF-COSTE, L. 1995b. What do we know about the Metropolis algorithm. Technical report, Dept. Mathematics, Harvard Univ. Z.
  • DIACONIS, P. and SALOFF-COSTE, L. 1996a. Nash inequalities for finite Markov chains. J. Theoret. Probab. 9 459 510. Z.
  • DIACONIS, P. and SALOFF-COSTE, L. 1996b. Logarithmic Sobolev inequalities for finite Markov chains. Ann. Appl. Probab. 6 695 750. Z.
  • DIACONIS, P. and STROOCK, D. 1991. Geometric bounds for eigenvalues of Markov chains. Ann. Appl. Probab. 1 36 61. Z.
  • Dy ER, R., KANNAN, R. and MOUNT, J. 1995. Sampling contingency tables. Random Structures Algorithms. To appear. Z.
  • EFRON, B. and HINKLEY, D. 1978. Assessing the accuracy of the MLE: observed versus expected Z. Fisher information with discussion. Biometrika 65 457 487. Z.
  • FARRELL, R. 1971. The necessity that a conditional procedure be almost every where admissible. Z. Wahrsch. Verw. Gebiete 19 57 66. Z. Z.
  • FISHER, R. 1925. Statistical Methods for Research Workers, 1st ed. 14th ed. 1970. Oliver and Boy d, Edinburgh. Z.
  • FISHER, R. 1950. The significance of deviations from expectation in a Poisson series. Biometrics 6 17 24. Z.
  • FISHER, R., THORNTON, H. and MACKENZIE, N. 1922. The accuracy of the plating method of estimating the density of bacterial populations. Ann. Appl. Biology 9 325 359. Z.
  • FULTON, W. 1993. Introduction to Toric Varieties. Princeton Univ. Press. Z.
  • GANGOLLI, A. 1991. Convergence bounds for Markov chains and applications to sampling. Ph.D. thesis, Dept. Computer Science, Stanford Univ.
  • GLONEK, G. 1987. Some aspects of log linear models. Ph.D. thesis, School of Math. Sciences, Flinders Univ. South Australia. Z.
  • GOODMAN, L. 1970. The multivariate analysis of qualitative data: interactions among multiple classifications. J. Amer. Statist. Assoc. 65 226 256. Z.
  • GUO, S. and THOMPSON, E. 1992. Performing the exact test for Hardy Weinberg proportion for multiple alleles. Biometrics 48 361 372. Z.
  • HABERMAN, S. 1978. Analy sis of Qualitative Data 1, 2. Academic Press, Orlando, FL. Z.
  • HAMMERSLY, J. and HANDSCOMB, D. 1964. Monte Carlo Methods. Wiley, New York. Z.
  • HARRIS, J. 1992. Algebraic Geometry: A First Course. Springer, New York. Z.
  • HERNEK, D. 1997. Random generation and counting of rectangular array s with fixed margins. Dept. Mathematics, Preprint, UCLA. Z.
  • HOLMES, R. and JONES, L. 1996. On uniform generation of two-way tables with fixed margins and the conditional volume test of Diaconis and Efron. Ann. Statist. 24 64 68. Z.
  • HOLMES, S. 1995. Examples for Stein's method. Preprint, Dept. Statistics, Stanford Univ. Z.
  • JENSEN, J. 1991. Uniform saddlepoint approximations and log-convex densities. J. Roy. Statist. Soc. Ser. B 157 172. Z. Z.
  • KIEFER, J. 1977. Conditional confidence statements and confidence estimators with discussion. J. Amer. Statist. Assoc. 72 789 827. Z.
  • KOLASSA, J. and TANNER, M. 1994. Approximate conditional inference in exponential families via the Gibbs sample. J. Amer. Statist. Assoc. 89 697 702. Z.
  • KOLASSA, J. and TANNER, M. 1996. Approximate Monte Carlo conditional inference. Dept. Statistics, Northwestern Univ. Preprint. Z.
  • KONG, F. 1993. Edgeworth expansions for conditional distributions in logistic regression models. Technical report, Dept. Statistics, Columbia Univ. Z.
  • KONG, F. and LEVIN, B. 1993. Edgeworth expansions for the sum of discrete random vectors and their applications in generalized linear models. Technical report, Dept. Statistics, Columbia Univ. Z.
  • LANGE, K. and LAZZERONI, L. 1997. Markov chains for Monte Carlo tests of genetic equilibrium in multidimensional contingency tables. Ann. Statist. To appear. Z.
  • LARNTZ, K. 1978. Small-sample comparison of exact levels for chi-squared goodness-of-fit statistics. J. Amer. Statist. Assoc. 73 253 263. Z.
  • LAURITZEN, S. 1996. Graphical Models. Oxford Univ. Press. Z.
  • LEHMANN, E. 1986. Testing Statistical Hy potheses, 2nd ed. Wiley, New York. Z.
  • LEVIN, B. 1992. On calculations involving the maximum cell frequency. Comm. Statist. Z.
  • LEVIN, B. 1992. Tests of odds ratio homogeneity with improved power in sparse fourfold tables. Comm. Statist. Theory Methods 21 1469 1500. Z.
  • MARDEN, J. 1995. Analy zing and Modeling Rank Data. Chapman and Hall, London. Z.
  • MAy R, E. and MEy ER, A. 1982. The complexity of the word problem for commutative semigroups and poly nomial ideals. Adv. in Math. 46 305 329. Z.
  • MCCULLOGH, P. 1985. On the asy mptotic distribution of Pearson's statistic in linear exponential family models. International Statistical Review 53 61 67. Z.
  • MCCULLOUGH, P. 1986. The conditional distribution of goodness-to-fit statistics for discrete data. J. Amer. Statist. Assoc. 81 104 107. Z.
  • MEHTA, C. and PATEL, N. 1983. A network algorithm for performing Fisher's exact test in r c contingency tables. J. Amer. Statist. Assoc. 78 427 434. Z.
  • NEy MAN, J. 1937. Outline of a theory of statistical estimation based on the classical theory of probability. Philos. Trans. 236 333 380. Z.
  • ODOROFF, C. 1970. A comparison of minimum logit chi-square estimation and maximum likelihood estimation in 2 2 2 and 3 2 2 contingency tables: tests for interaction. J. Amer. Statist. Assoc. 65 1617 1631. Z.
  • PROPP, J. and WILSON, D. 1986. Exact sampling with coupled Markov chains and applications to statistical mechanics. Random Structures Algorithms 9 232 252. Z.
  • REID, N. 1995. The roles of conditioning in inference. Statist. Sci. 10 138 199. Z. Z.
  • SAVAGE, L. 1976. On rereading R. A. Fisher with discussion. Ann. Statist. 4 441 450. Z.
  • SCHRIJVER, A. 1986. Theory of Linear and Integer Programming. Wiley, New York.
  • SINCLAIR, A. 1993. Algorithms for Random Generation and Counting: A Markov Chain Approach. Birkhauser, Boston. ¨ Z.
  • SKOVGAARD, I. 1987. Saddlepoint expansions for conditional distributions. J. Appl. Probab. 24 875 887. Z. SNEE 1974. Graphical display of two-way contingency tables. Amer. Statist. 38 9 12. Z.
  • STANLEY, R. 1980. Decompositoin of rational convex poly topes. Ann. Discrete Math. 6 333 342. Z.
  • STEIN, C. 1986. Approximate Computation of Expectations. IMS, Hay ward, CA. Z.
  • STURMFELS, B. 1991. Grobner bases of toric varieties. Tohoko Math. J. 43 249 261. ¨ Z.
  • STURMFELS, B. 1992. Asy mptotic analysis of toric ideals. Mem. Fac. Sci. Ky ushu Univ. Ser. A 46 217 228. Z.
  • STURMFELS, B. 1996. Grobner Bases and Convex Poly topes. Amer. Math. Soc., Providence, RI. ¨ Z.
  • THOMAS, R. 1995. A geometric Buchberger algorithm for integer programming. Math. Oper. Res. 20 864 884. Z.
  • VIRAG, B. 1997. Random walks on finite convex sets of lattice points. Technical report, Dept. Statistics, Univ. California, Berkeley. Z.
  • WEISPFENNING, V. 1987. Admissible orders and linear forms. ACM SIGSAM Bulletin 21 16 18. Z. 2
  • YARNOLD, J. 1970. The minimum expectation in X goodness-of-fit tests and the accuracy of approximations for the null distribution. J. Amer. Statist. Assoc. 65 864 886. Z.
  • YATES, F. 1984. Tests of significance for 2 2 contingency tables. J. Roy. Statist. Soc. Ser. A 147 426 463.