研究重點主要著眼於不相容條件模型之探討與應用，特別是使用不相容條件模型之吉氏取樣（Gibbs sampling）所衍生的結果，不僅打破傳統吉氏取樣的觀念，同時也加快取樣的收斂速度。另一方面，應用不相容吉氏取樣也能在插補法與高維多樣本檢定等議題上有特別的助益。另一研究重點則是解決部分網絡分析上的題材，諸如網絡分群、ERGM之參數估計、單一網絡特徵值之信賴區間估計等。

1. K.-L. Kuo and Y.J. Wang (2024). The Gibbs sampler revisited from the perspective of conditional modeling. Under review.

   Abstract Funding

   The Gibbs sampler (GS) is a crucial algorithm for approximating complex calculations, and it is justified by Markov chain theory, the alternating projection theorem, and I-projection, separately. We explore the equivalence between these three operators. Partially collapsed Gibbs sampler (PCGS) and pseudo-Gibbs sampler (PGS) are two generalizations of GS. For PCGS, the associated Markov chain is heterogeneous with varying state spaces, and we propose the iterative conditional replacement algorithm (ICR) to prove its convergence. In addition, ICR can approximate the multiple stationary distributions modeled by a PGS. Our approach highlights the benefit of using one operator for one conditional distribution, rather than lumping all the conditionals into one operator. Because no Markov chain theory is required, this approach simplifies the understanding of convergence.
   National Science and Technology Council, Taiwan, NSTC 112-2118-M-390-002.
2. S.-H. Chang, K.-L. Kuo, C.-C. Song, and T.J. Jiang (2024). Exploring pseudo-Gibbs sampler convergence and generalized compatibility in bivariate discrete conditional distributions. To be submitted.

   Abstract Funding

   Ministry of Science and Technology, Taiwan, MOST 111-2118-M-390-002.

1. K.-L. Kuo and Y.J. Wang (2023). Analytical computation of pseudo-Gibbs distributions for dependency networks. Methodology and Computing in Applied Probability, 25, 29.

   Abstract Link Funding

   Dependency network (DN) aims at using a collection of conditional distributions to identify a joint pdf. When the DN is compatible (self-consistent), the Gibbs sampler (GS) has been the algorithm to approximate the joint pdf. Without compatibility, GS will have multiple stationary distributions, named pseudo-Gibbs distributions (PGD), associated with different updating orders. To increase the computational efficiency and stability, we propose computing the marginal distributions. Closed-form marginal transition matrix is unearthed from DN. Thus, it becomes possible to compute the marginal distribution of PGD, which will be paired with a conditional distribution to obtain a PGD. We also show that multiple PGDs can be derived from one PGD. When the support is a union of disjoint regions, GS could not converge because the stationary pdf is a mixture of several joint distributions. Examples here show that our approach can obtain correct PGDs even for partitioned support. A new way to verify compatibility, under such circumstances, will also be proposed.
   Ministry of Science and Technology, Taiwan, MOST 107-2118-M-390-003 and MOST 108-2118-M-390-004-MY2.
2. K.-L. Kuo and Y.J. Wang (2019). Pseudo-Gibbs sampler for discrete conditional distributions. Annals of the Institute of Statistical Mathematics, 71, 93-105.

   Abstract Link Cited by 4 Funding

   Conditionally specified models offers a higher level of flexibility than the joint approach. Regression switching in multiple imputation is a typical example. However, reasonable-seeming conditional models are generally not coherent with one another. Gibbs sampler based on incompatible conditionals is called pseudo-Gibbs sampler, whose properties are mostly unknown. This article investigates the richness and commonalities among their stationary distributions. We show that Gibbs sampler replaces the conditional distributions iteratively, but keep the marginal distributions invariant. In the process, it minimizes the Kullback–Leibler divergence. Next, we prove that systematic pseudo-Gibbs projections converge for every scan order, and the stationary distributions share marginal distributions in a circularly fashion. Therefore, regardless of compatibility, univariate consistency is guaranteed when the orders of imputation are circularly related. Moreover, a conditional model and its pseudo-Gibbs distributions have equal number of parameters. Study of pseudo-Gibbs sampler provides a fresh perspective for understanding the original Gibbs sampler.
   • K.-L. Kuo and Y.J. Wang (2023). Analytical computation of pseudo-Gibbs distributions for dependency networks. Methodology and Computing in Applied Probability, 25, 29.
   • L. Rendsburg, A. Kristiadi, P. Hennig, and U. von Luxburg (2022). Discovering inductive bias with Gibbs priors: a diagnostic tool for approximate Bayesian inference. Proceedings of The 25th International Conference on Artificial Intelligence and Statistics, PMLR 151, 1503-1526.
   • M. Bazzi, L.G.S. Jeub, A. Arenas, S.D. Howison, and M.A. Porter (2020). A framework for the construction of generative models for mesoscale structure in multilayer networks. Physical Review Research, 2, 023100.
   • J. Mure (2019). Optimal compromise between incompatible conditional probability distributions, with application to Objective Bayesian Kriging. ESAIM: Probability and Statistics, 23, 271-309.
   Ministry of Science and Technology, Taiwan, MOST 104-2118-M-390-001 and MOST 105-2118-M-390-002.
3. K.-L. Kuo and Y.J. Wang (2018). Simulating conditionally specified models. Journal of Multivariate Analysis, 167, 171-180.

   Abstract Link Cited by 3 Funding

   Expert systems routinely use conditional reasoning. Conditionally specified statistical models offer several advantages over joint models; one is that Gibbs sampling can be used to generate realizations of the model. As a result, full conditional specification for multiple imputation is gaining popularity because it is flexible and computationally straightforward. However, it would be restrictive to require that every regression/classification must involve all of the variables. Feature selection often removes some variables from the set of predictors, thus making the regression local. A mixture of full and local conditionals is referred to as a partially collapsed Gibbs sampler, which often achieves faster convergence due to reduced conditioning. However, its implementation requires choosing a correct scan order. Using an invalid scan order will bring about an incorrect transition kernel, which leads to the wrong stationary distribution. We prove a necessary and sufficient condition for Gibbs sampling to correctly sample the joint distribution. We propose an algorithm that identifies all of the valid scan orders for a given conditional model. A forward search algorithm is discussed. Checking compatibility among conditionals of different localities is also discussed.
   • R.C. Nethery, N. Katz-Christy, M.-A. Kioumourtzoglou, R.M. Parks, A. Schumacher, and G.B. Anderson (2023). Integrated causal-predictive machine learning models for tropical cyclone epidemiology. Biostatistics, 24, 449-464.
   • T. Park and S. Lee (2022). Improving the Gibbs sampler. WIREs Computational Statistics, 14, e1546.
   • H.Y. Chen (2022). Semiparametric Odds Ratio Model and its Applications. Chapman and Hall/CRC.
   Ministry of Science and Technology, Taiwan, MOST 105-2118-M-390-002.
4. K.-L. Kuo and T.J. Jiang (2018). A revisit of the distribution of linear combinations of Dirichlet components. Communications in Statistics - Theory and Methods, 47, 509-520.

   Abstract Link Cited by 1 Funding

   Provost and Cheong (2000) show the importance of the distribution of linear combinations of components of a Dirichlet random vector to quadratic forms and their ratios in statistics, which can be applied in a variety of contexts. The c-characteristic function has been shown to be very useful and more practical in some distributions that are hard to manage with the traditional characteristic functions. The importance of the distribution of linear combinations of components of a Dirichlet random vector to quadratic forms and their ratios in statistics, which can be applied in a variety of contexts, is well known. We first provide its inversion formula which is practical in determining the distribution function of a random variable when its c-characteristic function is known. We then use this inversion formula to find an expression of probability density function of linear combinations of components of any Dirichlet vector. This would generalize the currently well known results.
   • P. Forrester and J. Zhang (2020). Corank-1 projections and the randomised Horn problem. Tunisian Journal of Mathematics, 3, 55-73.
   Ministry of Science and Technology, Taiwan.
5. K.-L. Kuo, C.-C. Song and T.J. Jiang (2017). Exactly and almost compatible joint distributions for high-dimensional discrete conditional distributions. Journal of Multivariate Analysis, 157, 115-123.

   Abstract Link Cited by 8 Funding

   A conditional model is a set of conditional distributions, which may be compatible or incompatible, depending on whether or not there exists a joint distribution whose conditionals match the given conditionals. In this paper, we propose a new mathematical tool called a “structural ratio matrix” (SRM) to develop a unified compatibility approach for discrete conditional models. With this approach, we can find all joint pdfs after confirming that the given model is compatible. In practice, it is most likely that the conditional models we encounter are incompatible. Therefore, it is important to investigate approximated joint distributions for them. We use the concept of SRM again to construct an almost compatible joint distribution, with consistency property, to represent the given incompatible conditional model.
   • K.-L. Kuo and Y.J. Wang (2023). Analytical computation of pseudo-Gibbs distributions for dependency networks. Methodology and Computing in Applied Probability, 25, 29.
   • K. Dong, S. Li, and D. Li (2022). Some properties of fractional cumulative residual entropy and fractional conditional cumulative residual entropy. Fractal and Fractional, 6, 400.
   • D. Petturiti and B. Vantaggi (2022). Probability envelopes and their Dempster-Shafer approximations in statistical matching. International Journal of Approximate Reasoning, 150, 199-222.
   • L. Rendsburg, A. Kristiadi, P. Hennig, and U. von Luxburg (2022). Discovering inductive bias with Gibbs priors: a diagnostic tool for approximate Bayesian inference. Proceedings of The 25th International Conference on Artificial Intelligence and Statistics, PMLR 151, 1503-1526.
   • E. Miranda and M. Zaffalon (2020). Compatibility, desirability, and the running intersection property. Artificial Intelligence, 283, 103274.
   • L. Burigana and M. Vicovaro (2020). Compatibility of distributions in probabilistic models: an algebraic frame and some characterizations. Algebraic Statistics, 11, 213-246.
   • J. Mure (2019). Optimal compromise between incompatible conditional probability distributions, with application to Objective Bayesian Kriging. ESAIM: Probability and Statistics, 23, 271-309.
   • S. van Buuren (2018). Flexible Imputation of Missing Data. Second Edition, CRC Press.
   Ministry of Science and Technology, Taiwan, MOST 104-2118-M-390-001.
6. J.M. Dickey, T.J. Jiang and K.-L. Kuo (2013). Distribution of functionals of a Ferguson-Dirichlet process over an n-dimensional ball. Journal of Multivariate Analysis, 120, 216-225.

   Abstract Link Funding

   The c-characteristic function has been shown to have properties similar to those of the Fourier transformation. We now give a new property of the c-characteristic function of the spherically symmetric distribution. With this property, we can easily determine whether a distribution is spherically symmetric. The exact probability density function of the random mean of a spherically symmetric Ferguson–Dirichlet process with parameter measure over an n-dimensional spherical surface and that over an n-dimensional ball are given. We further give the exact probability density function of the random mean of a Ferguson–Dirichlet process with parameter measure over an n-dimensional ellipsoidal surface and that over an n-dimensional ellipsoidal solid.
   National Science Council, Taiwan.
7. K.-L. Kuo and Y.J. Wang (2013). A fresh look at the running time analysis for the Gibbs sampler. Communications in Statistics - Simulation and Computation, 42, 1815-1823.

   Abstract Link Funding

   As the Gibbs sampler has become one of the standard tools in computing, the practice of burn-in is almost the default option. Because it takes a certain number of iterations for the initial distribution to reach stationarity, supporters of burn-in will throw away an initial segment of the samples and argue that such a practice ensures unbiasedness. Running time analysis studies the question of how many samples to be thrown away. Basically, it equates the number of iterations to stationarity with the number of initial samples to be discarded. However, many practitioners have found that burn-in wastes potentially useful samples and the practice is inefficient, and thus unnecessary. For the example considered, a single chain without burn-in offers both efficiency and accuracy superior to multiple chains with burn-in. We show that the Gibbs sampler uses odds to generate samples. Because the correct odds are used from the onset of the iterative process, the observations generated by the Gibbs sampler are identically distributed as the target distribution; thus throwing away those valid samples is wasteful. When the chain of distributions and the trajectory (sample path) of the chain are considered based on their separate merits, the disagreement can be settled. We advocate carefully choosing the initial state, but without burn-in to quicken the formation of the stationary distribution.
   National Science Council, Taiwan, NSC 100-2118-M-390-003.
8. K.-L. Kuo and Y.J. Wang (2011). A simple algorithm for checking compatibility among discrete conditional distributions. Computational Statistics and Data Analysis, 55, 2457-2462.

   Abstract Link Cited by 12

   A distribution is said to be conditionally specified when only its conditional distributions are known or available. The very first issue is always compatibility: does there exist a joint distribution capable of reproducing all of the conditional distributions? We review five methods–mostly for two or three variables–published since 2002, and we conclude that these methods are either mathematically too involved and/or are too difficult (and in many cases impossible) to generalize to a high dimension. The purpose of this paper is to propose a general algorithm that can efficiently verify compatibility in a straightforward fashion. Our method is intuitively simple and general enough to deal with any full-conditional specifications. Furthermore, we illustrate the phenomenon that two theoretically equivalent conditional models can be different in terms of compatibilities, or can result in different joint distributions. The implications of this phenomenon are also discussed.
   • L. Rendsburg, A. Kristiadi, P. Hennig, and U. von Luxburg (2022). Discovering inductive bias with Gibbs priors: a diagnostic tool for approximate Bayesian inference. Proceedings of The 25th International Conference on Artificial Intelligence and Statistics, PMLR 151, 1503-1526.
   • L. Burigana and M. Vicovaro (2020). Compatibility of distributions in probabilistic models: an algebraic frame and some characterizations. Algebraic Statistics, 11, 213-246.
   • M. Bazzi, L.G.S. Jeub, A. Arenas, S.D. Howison, and M.A. Porter (2020). A framework for the construction of generative models for mesoscale structure in multilayer networks. Physical Review Research, 2, 023100.
   • K.-L. Kuo and Y.J. Wang (2019). Pseudo-Gibbs sampler for discrete conditional distributions. Annals of the Institute of Statistical Mathematics, 71, 93-105.
   • S. van Buuren (2018). Flexible Imputation of Missing Data. Second Edition, CRC Press.
   • I. Ghosh and S. Nadarajah (2017). On the construction of a joint distribution given two discrete conditionals. Studia Scientiarum Mathematicarum Hungarica, 54, 178-204.
   • K.-L. Kuo, C.-C. Song, and T.J. Jiang (2017). Exactly and almost compatible joint distributions for high-dimensional discrete conditional distributions. Journal of Multivariate Analysis, 157, 115-123.
   • I. Ghosh and S. Nadarajah (2016). An alternative approach for compatibility of two discrete conditional distributions. Communications in Statistics - Theory and Methods, 45, 4416-4432.
   • Y.-C. Yao, S.-C. Chen, and S.-H. Wang (2014). On compatibility of discrete full conditional distributions: A graphical representation approach. Journal of Multivariate Analysis, 124, 1-9.
   • G. Molenberghs, G. Fitzmaurice, M.G. Kenward, A. Tsiatis, and G. Verbeke (Eds.) (2014). Handbook of missing data methodology. CRC Press.
   • R.A. Hughes, I.R. White, S.R. Seaman, J.R. Carpenter, K. Tilling, and J.A. Sterne (2014). Joint modelling rationale for chained equations. BMC medical research methodology, 14, 1-10.
   • S.H. Chen, E.H. Ip, and Y.J. Wang (2013). Gibbs ensembles for incompatible dependency networks. WIREs Computational Statistics, 5, 478-485.
9. C.-C. Song, L.-A. Li, C.-H. Chen, T.J. Jiang and K.-L. Kuo (2010). Compatibility of finite discrete conditional distributions. Statistica Sinica, 20, 423-440.

   Abstract Link Cited by 7 Funding

   This paper provides new versions of necessary and sufficient conditions for compatibility of finite discrete conditional distributions, and of the uniqueness for those compatible conditional distributions. We note that the ratio matrix (the matrix C in Arnold and Press (1989)), after interchanging its rows and/or columns, can be rearranged to be an irreducible block diagonal matrix. We find that checking compatibility is equivalent to inspecting whether every block on the diagonal has a rank one positive extension, and that the necessary and sufficient conditions of the uniqueness, if the given conditional densities are compatible, is that the ratio matrix itself is irreducible. We show that each joint density, if it exists, corresponds to a rank one positive extension of the ratio matrix, and we characterize the set of all possible joint densities. Finally, we provide algorithms for checking compatibility, for checking uniqueness, and for constructing densities.
   • I. Ghosh (2018). A complete characterization of bivariate densities using the conditional percentile function. Metrika, 81, 485-492.
   • E. Dreassi and P. Rigo (2017). A note on compatibility of conditional autoregressive models. Statistics & Probability Letters, 125, 9-16.
   • I. Ghosh and S. Nadarajah (2017). On the construction of a joint distribution given two discrete conditionals. Studia Scientiarum Mathematicarum Hungarica, 54, 178-204.
   • K.-L. Kuo, C.-C. Song and T.J. Jiang (2017). Exactly and almost compatible joint distributions for high-dimensional discrete conditional distributions. Journal of Multivariate Analysis, 157, 115-123.
   • I. Ghosh and S. Nadarajah (2016). An alternative approach for compatibility of two discrete conditional distributions. Communications in Statistics - Theory and Methods, 45, 4416-4432.
   • Y.-C. Yao, S.-C. Chen, and S.-H. Wang (2014). On compatibility of discrete full conditional distributions: A graphical representation approach. Journal of Multivariate Analysis, 124, 1-9.
   • P. Berti, E. Dreassi, and P. Rigo (2014). Compatibility results for conditional distributions. Journal of Multivariate Analysis, 125, 190-203.
   National Science Council, Taiwan.
10. Y.J. Wang and K.-L. Kuo (2010). Compatibility of discrete conditional distributions with structural zeros. Journal of Multivariate Analysis, 101, 191-199.

    Abstract Link Cited by 9

    A general algorithm is provided for determining the compatibility among full conditionals of discrete random variables with structural zeros. The algorithm is scalable and it can be implemented in a fairly straightforward manner. A MATLAB program is included in the Appendix and therefore, it is now feasible to check the compatibility of multi-dimensional conditional distributions with constrained supports. Rather than the linear equations in the restricted domain of Arnold et al. (2002) [11] Tian et al. (2009) [16], the approach is odds-oriented and it is a discrete adaptation of the compatibility check of Besag (1994) [17]. The method naturally leads to the calculation of a compatible joint distribution or, in the absence of compatibility, a nearly compatible joint distribution. Besag’s [5] factorization of a joint density in terms of conditional densities is used to justify the algorithm.
    • K.-L. Kuo and Y.J. Wang (2023). Analytical computation of pseudo-Gibbs distributions for dependency networks. Methodology and Computing in Applied Probability, 25, 29.
    • D. Petturiti and B. Vantaggi (2022). Probability envelopes and their Dempster-Shafer approximations in statistical matching. International Journal of Approximate Reasoning, 150, 199-222.
    • E. Miranda and M. Zaffalon (2020). Compatibility, desirability, and the running intersection property. Artificial Intelligence, 283, 103274.
    • L. Burigana and M. Vicovaro (2020). Compatibility of distributions in probabilistic models: an algebraic frame and some characterizations. Algebraic Statistics, 11, 213-246.
    • S. van Buuren (2018). Flexible Imputation of Missing Data. Second Edition, CRC Press.
    • I. Ghosh and S. Nadarajah (2017). On the construction of a joint distribution given two discrete conditionals. Studia Scientiarum Mathematicarum Hungarica, 54, 178-204.
    • K.-L. Kuo, C.-C. Song and T.J. Jiang (2017). Exactly and almost compatible joint distributions for high-dimensional discrete conditional distributions. Journal of Multivariate Analysis, 157, 115-123.
    • I. Ghosh and S. Nadarajah (2016). An alternative approach for compatibility of two discrete conditional distributions. Communications in Statistics - Theory and Methods, 45, 4416-4432.
    • Y.-C. Yao, S.-C. Chen, and S.-H. Wang (2014). On compatibility of discrete full conditional distributions: A graphical representation approach. Journal of Multivariate Analysis, 124, 1-9.
11. T.J. Jiang and K.-L. Kuo (2008). Distribution of a random functional of a Ferguson-Dirichlet process over the unit sphere. Electronic Communications in Probability, 13, 518-525.

    Abstract Link Cited by 2 Funding

    Jiang, Dickey, and Kuo (2004) gave the multivariate c-characteristic function and showed that it has properties similar to those of the multivariate Fourier transformation. We first give the multivariate c-characteristic function of a random functional of a Ferguson-Dirichlet process over the unit sphere. We then find out its probability density function using properties of the multivariate c-characteristic function. This new result would generalize that given by Jiang (1991).
    • L.D. Schiavo (2019). Characteristic functionals of Dirichlet measures. Electronic Journal of Probability, 24, 1-38.
    • J.M. Dickey, T.J. Jiang, and K.-L. Kuo (2013). Distribution of functionals of a Ferguson-Dirichlet process over an n-dimensional ball. Journal of Multivariate Analysis, 120, 216-225.
    National Science Council, Taiwan.
12. T.J. Jiang, J.M. Dickey and K.-L. Kuo (2004). A new multivariate transform and the distribution of a random functional of a Ferguson-Dirichlet process. Stochastic Processes and their Applications, 111, 77-95.

    Abstract Link Cited by 8 Funding

    A new multivariate transformation is given, with various properties, e.g., uniqueness and convergence properties, that are similar to those of the Fourier transformation. The new transformation is particularly useful for distributions that are difficult to deal with by Fourier transformation, such as relatives of the Dirichlet distributions. The new multivariate transformation of the Dirichlet distribution can be expressed in closed form. With this result, we easily show that the marginal of a Dirichlet distribution is still a Dirichlet distribution. We also give a closed form for the filtered-variate Dirichlet distribution. A relation between the new characteristic function and the traditional characteristic function is given. Using this multivariate transformation, we give the distribution, on the region bounded by an ellipse, of a random functional of a Ferguson–Dirichlet process over the boundary.
    • L.D. Schiavo and E. Lytvynov (2023). A Mecke-type characterization of the Dirichlet–Ferguson measure. Electronic Communications in Probability, 28, 1-12.
    • L.D. Schiavo (2022). The Dirichlet–Ferguson diffusion on the space of probability measures over a closed Riemannian manifold. Annals of Probability, 50, 591-648.
    • H. Homei (2021). The stochastic linear combination of Dirichlet distributions. Communications in Statistics - Theory and Methods, 50, 2354-2359.
    • L.D. Schiavo (2019). Characteristic functionals of Dirichlet measures. Electronic Journal of Probability, 24, 1-38.
    • K.-L. Kuo and T.J. Jiang (2018). A revisit of the distribution of linear combinations of Dirichlet components. Communications in Statistics - Theory and Methods, 47, 509-520.
    • J.M. Dickey, T.J. Jiang, and K.-L. Kuo (2013). Distribution of functionals of a Ferguson-Dirichlet process over an n-dimensional ball. Journal of Multivariate Analysis, 120, 216-225.
    • L.F. James, A. Lijoi, and I. Prünster (2010). On the posterior distribution of classes of random means. Bernoulli, 16, 155-180.
    • T.J. Jiang and K.-L. Kuo (2008). Distribution of a random functional of a Ferguson-Dirichlet process over the unit sphere. Electronic Communications in Probability, 13, 518-525.
    National Science Council, Taiwan.

1. K.-L. Kuo (2009). Compatible non-normalized conditional densities. Journal of the Chinese Statistical Association, 47, 104-112.

   Abstract Link

   Checking compatibility of the specified conditional distributions is an important problem in statistics, especially in Bayesian computations. However, conditional density may be given without exact normalizing constant multiplier since this multiplier is hard to identify. In this article, we provide necessary and sufficient conditions for compatibility of non-normalized conditional densities. In addition, if they are compatible, we also discuss the uniqueness of the associated joint density which generates them.
2. 郭錕霖(2008)縮圖與Pick公式。數學傳播，32(4)，56-65。

   Link

3. 郭錕霖(2008)橢圓內部點與兩邊界點形成之面積。數學傳播，32(2)，13-18。

   Link

4. 姜志銘、宋傳欽、郭錕霖(2008)主成份分析在木球運動表現與教學上的運用。統計薪傳，8(1)，1-14。
5. 宋傳欽、姜志銘、郭錕霖(2003)影響青少年木球運動表現因素的研究。統計薪傳，3(2)，127-144。

• Efficient computation of pseudo-Gibbs distributions. European Meeting of Statisticians, Palermo, Italy, July 22-26, 2019.
• Simulating conditionally specified models. European Meeting of Statisticians, Helsinki, Finland, July 24-28, 2017.
• Pseudo-Gibbs distributions for incompatible conditional models. World Congress in Probability and Statistics, Toronto, Canada, July 11-15, 2016.
• Gibbs sampling on non-full conditional distributions. European Meeting of Statisticians, Amsterdam, Netherlands, July 06-10, 2015.
• Nonparametric inference for network statistics. International Congress of Mathematicians, Seoul, Korea, August 13-21, 2014.
• Pseudo-Gibbs distribution and its application on multivariate two sample test. European Meeting of Statisticians, Budapest, Hungary, July 20-25, 2013.
• A history-dependent algorithm for social structure and patterns of social interactions. Sunbelt Social Networks Conference, Trento, Italy, June 29 - July 04, 2010.
• Fundamental properties on Gibbs distributions. ISI-ISM-ISSAS Joint Conference, Kolkata, India. January 21-22, 2010.
• A new approach to the inverse Bayes formula of compatible conditional distributions. Joint Statistical Meetings, Salt Lake City, U.S.A., July 29 - August 02, 2007.

論文審查服務