Markov Chain Monte Carlo and Exact Logistic Regression

描述（由申请人提供）：逻辑回归是一个非常流行的 二进制数据分析模型，广泛适用于 物理、行为和生物医学科学。参数推断， 模型通常基于最大化无条件似然函数。 然而，无条件最大似然推理可以产生不一致的 点估计值、不准确的p值和不准确的置信区间 小的或不平衡的数据集和具有大量 参数相对于观测值的数量。有时这种方法会失败 完全是因为没有估计可以找到最大化的无条件 似然函数一种方法上合理的替代方法， 上述缺点都不是精确条件方法 一个生成足够的统计量的置换分布， 感兴趣的参数的条件是固定的充分统计， 剩余的干扰参数处于其观测值。主要的绊脚石 这种方法的障碍是它所带来的沉重的计算负担。Monte Carlo方法试图通过从参考值中取样来克服这个问题 一组可能的排列，而不是枚举它们。两个相互竞争 蒙特卡罗方法是基于网络的抽样和马尔可夫链蒙特卡罗方法 （MCMC）采样。网络采样受到内存限制，而MCMC 如果马尔可夫链不是遍历的，或者如果 该过程不处于稳定状态。我们提出了一种新的方法， 结合了网络和MCMC抽样，利用了每个 并克服其自身的局限性。我们建议实施这项措施， 混合网络MCMC方法在我们LogXact软件和外部程序 在SAS系统中。 拟定商业应用： 有大量的逻辑回归软件，可以处理小，稀疏或 不平衡数据集的精确方法。 我们的LogXact软件包是唯一一款 可以为不是“玩具问题”的数据集提供精确的推理。 还甚至 LogXact在中等规模的问题上可以快速分解。 新一代混合动力车 网络MCMC算法将处理更大的问题，但需要 精确推理 由于这类数据集很常见， 在科学研究中。

DESCRIPTION (provided by applicant): Logistic regression is a very popular model for the analysis of binary data with widespread applicability in the physical, behavioral and biomedical sciences. Parameter inference for this model is usually based on maximizing the unconditional likelihood function. However unconditional maximum likelihood inference can produce inconsistent point estimates, inaccurate p-values and inaccurate confidence intervals for small or unbalanced data sets and for data sets with a large number of parameters relative to the number of observations. Sometimes the method fails entirely as no estimates can be found that maximize the unconditional likelihood function. A methodologically sound alternative approach that has none of the aforementioned drawbacks is the exact conditional approach in which one generates the permutation distributions of the sufficient statistics for the parameters of interest conditional on fixing the sufficient statistics of the remaining nuisance parameters at their observed values. The major stumbling block to this approach is the heavy computational burden it imposes. Monte Carlo methods attempt to overcome this problem by sampling from the reference set of possible permutations instead of enumerating them all. Two competing Monte Carlo methods are network based sampling and Markov Chain Monte Carlo (MCMC) sampling. Network sampling suffers from memory limitations while MCMC sampling can produce incorrect results if the Markov chain is not ergodic or if the process is not in the steady state. We propose a novel approach which combines the network and MCMC sampling, draws upon the strengths of each of them and overcomes their individual limitations. We propose to implement this hybrid network-MCMC method in our LogXact software and as an external procedure in the SAS system. PROPOSED COMMERCIAL APPLICATION: There is great demand for logistic regression software that can handle small, sparse or unbalanced data sets by exact methods. Our LogXact package is the only software that can provide exact inference for data sets which are not "toy problems". Yet even LogXact quickly breaks down on moderate sized problems. The new generation of hybrid network-MCMC algorithms will handle substantially larger problems that nevertheless need exact inference. The commercial potential is considerable since such data sets are common in scientific studies.