Maximum Conditional Likelihood Method

• Maximum conditional likelihood method is a suite of techniques that estimates parameters by maximizing conditional likelihood with fixed sufficient statistics.
• It replaces the UMVUE with the MLE in sequential sampling, reducing computational complexity in log-linear, generalized linear, and graphical models.
• The method underpins efficient direct sampling in contingency tables and algebraic statistics, enabling scalable inference in large and complex datasets.

The maximum conditional likelihood method refers to a suite of statistical techniques for parameter estimation and sampling that maximize, or approximate maximization of, the conditional likelihood function in a parametric or semiparametric family, often under constraints imposed by observed sufficient statistics or events. The conditional likelihood is particularly relevant in models such as log-linear, generalized linear, or graphical models where inference is carried out in the presence of nuisance parameters or partially observed data. Recent formulations have extended maximum conditional likelihood to both exact and approximate direct sampling algorithms, inference in models with intractable joint likelihoods, and efficient subsampling schemes in large-scale data analysis.

1. Sequential Direct Sampling via Maximum Conditional Likelihood

The direct sampling algorithm for conditional distributions in log-affine models operates on a finite integer lattice (Markov lattice) representing tables or configurations with fixed sufficient statistics. The core mechanism is a sequential process: starting from the observed sufficient statistics $b$, the algorithm “peels off” counts sequentially, at each step selecting which cell to decrement based on transition probabilities proportional to estimators of expected counts.

The original transition probability for moving from the current state $\beta$ to the next $\beta - a_j$ is

$$P(\beta, \beta - a_j; x) = \frac{\tilde{\mu}_j(\beta; x)}{\deg(\beta)}$$

where

$$\tilde{\mu}_j(\beta; x) = \frac{Z_A(\beta - a_j; x)}{Z_A(\beta; x)} x_j$$

and $Z_A(\cdot; x)$ denotes the evaluation of the $A$--hypergeometric (toric) polynomial associated with the configuration matrix $A$, $x$ is a positive weight vector, and $\deg(\beta)$ is the sum of all $\tilde{\mu}_j$ at $\beta$.

A key contribution is the replacement of the uniformly minimum variance unbiased estimator (UMVUE) $\tilde{\mu}_j$ with the maximum likelihood estimator (MLE) $\hat{\mu}_j$, dramatically improving computational tractability while retaining desirable asymptotic properties in many settings (Mano, 2 Feb 2025).

2. Mathematical Structure and Estimation Equations

The log-affine model is governed by a configuration (design) matrix $A$ and parameter vector $\xi$. For sufficient statistic $b$, the MLE for expected counts $\mu$ is obtained by solving

$A \hat{\mu} = b$

subject to the model constraints, typically via an iterative proportional scaling (IPS) algorithm. The log-likelihood is

$l(\xi; b) = \sum_i b_i \xi_i - \psi(\xi)$

with

$$\psi(\xi) = |u| \cdot \log Z(e^\xi, x) + \text{const}$$

where $|u|$ is the total count and $Z$ is as above.

For decomposable graphical models and other “nice” log-linear models, the UMVUE and MLE coincide. In non-decomposable cases, the MLE serves as a consistent and computationally efficient proxy for the UMVUE, especially as the sample size grows.

Table: Comparison of UMVUE and MLE in Transition Probability Computation

Model Structure	Transition Probabilities Use	Computational Burden
Decomposable/Log-linear	UMVUE ≡ MLE, closed-form	Low
Non-decomposable	MLE approximates UMVUE, iterative	Moderate-High

3. Algorithmic Implementation and Complexity

The sequential direct sampling process is summarized as follows:

1. Initialize $\beta = b$, $\nu = n$ (total count).
2. For each decrement, compute transition probabilities for each $j$:

$$P(\beta, \beta - a_j; x) = \frac{\hat{\mu}_j(\beta; x)}{\deg(\beta)}$$

where $\hat{\mu}_j$ is obtained as the solution to $A \hat{\mu} = \beta$.

1. Select $j$ according to these probabilities, update $\beta \gets \beta - a_j$, $\nu \gets \nu - 1$.
2. Repeat until $\nu = 1$, return $u$ with $u_j$ equal to the number of times $j$ selected.

Significant computational effort is saved by avoiding repeated evaluation of A–hypergeometric polynomials and their shifts (which require computationally intensive computation of connection matrices and Gröbner bases). Instead, MLE computation via IPS proceeds with per-iteration complexity $O(dm)$, where $d$ is the dimension of $A$ and $m$ is the number of cells. Typically, only a moderate number of iterations is required for practical convergence.

4. Statistical Properties and Limitations

The approximate algorithm using the MLE delivers near-exact sampling in large sample regimes due to the asymptotically vanishing bias between MLE and UMVUE in non-decomposable models. However, for moderate sample sizes or in highly non-log-linear models, the residual bias in transition probabilities can introduce distortions. The following limitations are inherent:

• For models where UMVUE $\neq$ MLE, the sampling bias is non-zero but decays with increasing sample size.
• Convergence of IPS depends on tuning parameters and initial values, and may require more iterations in ill-conditioned cases.
• The method is not exactly unbiased for small samples unless the model is decomposable.

Potential improvements include faster MLE solvers (Newton’s method), bias-correction of the MLE for small samples, and development of error bounds for practical performance assessment.

5. Applications in Contingency Tables and Algebraic Statistics

The maximum conditional likelihood–based sequential sampling method is particularly suited to:

• Constructing exact or approximate independent samples from the conditional distribution of contingency tables with fixed margins (the fiber), a foundational step in goodness-of-fit testing and exact inference in categorical data analysis.
• Providing an alternative to Markov chain Monte Carlo methods (which suffer from slow mixing or lack of guarantees) in sampling from conditional distributions defined by fibers in discrete exponential families.
• Facilitating direct inference in algebraic statistics and toric models, especially where independence, graphical, or log-affine structures are present, by exploiting the efficient computation of the MLE even in complex scenarios.

This approach has enabled efficient large-scale inference in computational algebraic statistics for models with tens of thousands of cells, subject to the outlined computational tradeoffs.

6. Future Directions

Further research is suggested in:

• Accelerating the computation of MLEs in high-dimensional fibers by leveraging advanced optimization or distributed methods.
• Quantifying and, where feasible, correcting for the finite-sample bias when UMVUE and MLE diverge, particularly for models exhibiting severe non-log-linearity.
• Extending the algorithm to richer classes of exponential families where the fiber structure or additional constraints complicate direct sampling.
• Deriving sharp error bounds and diagnostics for the approximate algorithm to guide practitioners in choosing between exact and approximate methods in practice.

Advances in these areas will broaden the applicability and reliability of direct maximum conditional likelihood–based sampling in large and complex statistical models.