811726高雄市楠梓區高雄大學路700號理學院314室

07-5919523(分機7523)

klkuo@nuk.edu.tw

國立高雄大學統計學研究所副教授(2018.02-)

國立高雄大學校務研究辦公室執行暨諮詢顧問(2015.08-2019.01)

國立高雄大學教務處招生組組長(2014.08-2019.01)

國立高雄大學統計學研究所助理教授(2011.08-2018.01)

中央研究院院級一般博士後研究學者(2008.08-2011.07)

國立政治大學應用數學系博士(姜志銘教授指導，2002.09-2008.07)

國立政治大學應用數學系碩士(姜志銘教授指導，1999.09-2002.06)

東海大學數學系學士(1995.09-1999.06)

- K.-L. Kuo and Y.J. Wang (2022).
Analytical computation of Pseudo-Gibbs distributions for dependency networks. Under review.
Both Bayesian network (BN) and dependency network (DN) use a collection of conditional distributions to identify a multivariate joint distribution (pdf). Statisticians call such approaches conditionally specified distribution (CSD), see Arnold and Sarabia (2022). In a BN, the propagations of dependence only go forward, while DN allows both forward and backward propagations, hence, has feedback loops. In terms of modeling, DN has some flexibility over BN, except its computation is more involved. When the DN is compatible (self-consistent), the Gibbs sampler (GS) is the algorithm to approximate the joint pdf. Thus, the first issue of DN is to verify compatibility. Without compatibility, the GS will have multiple stationary distributions associated with different updating orders. Heckerman et al. (2000) named the incompatible GS the ordered pseudo Gibbs sampler (OPGS), and they claimed that the differences among multiple pseudo-Gibbs distributions (PGD) can be ignored when all of the conditional models are estimated from a large data set. Moreover, the computational complexity of GS is known to be NP. Here, we show how to unearth closed-form marginal transition matrices from a DN. Our approach computes the marginal pdf which can be paired with DN to obtain PGDs. Compared with GS simulations, this route achieves dimension reduction, computation efficiency, and better accuracy. When the support is a union of disjoint regions, OPGS would not converge because the stationary distribution is a mixture of several joint distributions. Examples here show that the marginal approach can obtain correct PGDs even for partitioned support. The verification of compatibility under such circumstances will also be addressed.
- K.-L. Kuo and Y.J. Wang (2022+).
Iterative conditional replacement algorithm for conditionally specified models. In progress.
Conditionally specified model embeds a set of conditional distributions into a joint distribution. It popularity is partly due to the Gibbs sampler, which requires compatible full conditional distributions. Reasonable-seeming conditional distributions are generally neither full nor compatible. For an incompatible Gibbs sampler, its stationary distribution depends on the order that the variables are updated. Such dependence can be exploited to check the compatibility. This paper proposes the iterative conditional replacement (ICR) algorithm to compute multiple circularly-related stationary distributions. A conditional model is deemed compatible if this batch of stationary distributions are equal. Kuo and Wang (2019) showed that stationarity is characterized by the marginal commonalities among circularly-related distributions. These observations motivate ICR, and provide the metric to detect its convergence. ICR replaces both complex mathematical objects and fancy linear equations with straightforward computations. We highlight ICR's ability to deal with two situations: (i) not totally-positive conditional models and (ii) mixtures of incompatible full and non-full conditionals; both are fairly common but rarely addressed. The separate marginalization and multiplication steps make ICR flexible enough to embed some conditional models that neither the original Gibbs sampler nor systems of linear equations can attain. The connections with other algorithms will be discussed.
- K.-L. Kuo and Y.J. Wang (2022+). The Gibbs sampler: an interaction-based approach. In progress.
- K.-L. Kuo, C.-C. Song, T.J. Jiang, and S.-H. Chang (2022+). Stochastic properties for conditionally specified models. In progress.

- 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.
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.
- 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.

- K.-L. Kuo and Y.J. Wang (2018).
Simulating conditionally specified models.
Journal of Multivariate Analysis, 167, 171-180.
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.
- 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.
- R.C. Nethery, N. Katz-Christy, M.-A. Kioumourtzoglou, R.M. Parks, A. Schumacher, and G.B. Anderson (2021+). Integrated causal-predictive machine learning models for tropical cyclone epidemiology. Biostatistics. To appear.

- 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.
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.

- 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.
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.
- 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.

- 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.
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.
- 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.
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.
- 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.
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.

- 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.
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.

- Y.J. Wang and K.-L. Kuo (2010).
Compatibility of discrete conditional distributions with structural zeros.
Journal of Multivariate Analysis, 101, 191-199.
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.
- 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.

- 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.
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.

- 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.
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 (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.

- K.-L. Kuo (2009). Compatible non-normalized conditional densities. Journal of the Chinese Statistical Association, 47, 104-112.
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.
- 郭錕霖（2008）縮圖與Pick公式。數學傳播，32(4)，56-65。
- 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。
- 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。
- 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。

2019.08-2022.07 指數隨機圖模型的性質 MOST 108-2118-M-390-004-MY2

2018.08-2019.10 條件機率模型之探討 MOST 107-2118-M-390-003

2016.08-2017.09 動態取樣策略 MOST 105-2118-M-390-002

2015.08-2016.08 吉氏取樣：推廣與應用 MOST 104-2118-M-390-001

2014.08-2015.07 基於偽吉氏分佈之多變量K樣本檢定 MOST 103-2118-M-390-002

2013.08-2014.08 非全條件分配之吉氏取樣 NSC 102-2118-M-390-002

2012.08-2013.07 吉氏取樣法的收斂診斷樣 NSC 101-2118-M-390-001

2011.10-2012.07 不相容的吉氏取樣 NSC 100-2118-M-390-003

- 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.

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