本文將會討論一貝氏影像分析的生成模型。模型部分，我們考慮兩點：根據「合成性(compositioanlity)」的想法，建構一個關於影像解釋並具有脈絡訊息的先驗分布以及建構給定特定解釋下，影像像素的機率分布。我們也會介紹一個馬可夫鏈蒙地卡羅的推論算法---吉式抽樣。最後，我們將應用此模型與吉式方法進行五官樣態估計的實驗。

In this thesis, we discuss a generative model for Bayesian image analysis. In this model, we focus on building a prior of pares of an image with context information based on compositionality and a conditional model of image pixels given a particular interpretation.
Also, a MCMC inference algorithm, Gibbs sampler, is introduced. Finally,
Gibbs sampler and our model will be applied to a facial pose estimation experiment.

1 Introduction 1
1.1 Bayesian Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Bayesian Inference as image parsing . . . . . . . . . . . . . . . . . . 1
2 Inference Algorithms 2
2.1 Markov Chain Monte Carlo Sampling: The Gibbs Sampler . . . . . 2
3 Building The Prior: Hierarchical Modelling 5
3.1 Interpretation and Basic Notation . . . . . . . . . . . . . . . . . . . 6
3.2 Context-Free Grammar: Markove Backbone . . . . . . . . . . . . . 8
3.3 Context-Sensitive Grammar: The Perturbation Method . . . . . . . 8
3.4 Discussion: Existence of KL Distance Minimizer . . . . . . . . . . . 10
4 An Experiment of Pose Inference 12
4.1 The Model: Prior of the Poses . . . . . . . . . . . . . . . . . . . . . 12
4.2 The Model: Likelihood Term . . . . . . . . . . . . . . . . . . . . . . 14
4.3 Parameters Estimation . . . . . . . . . . . . . . . . . . . . . . . . . 16
4.4 Model Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4.5 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 22
5 Conclusion 23

[1] C. M. Bishop, Pattern recognition and machine learning, Springer, 2006.
[2] J. Pearl, Probabilistic reasoning in intelligent systems: Networks of plausible
inference, Morgan Kaufmann Publishers, San Francisco, CA, 1988.
[3] L.-B. Chang, Conditional modelling and conditional inference, Ph.D. thesis,
Brown University, Division of Applied Mathematics, 2010.
[4] A. G. Dabak and D. H. Johnson, Relations between Kullback-Leibler distance
and Fisher information, Manscript, 2002.
[5] I. A. Cosma and L. Evers, Markov Chains and Monte Carlo Methods, Lecture
Notes, 2010.
[6] S. Geman, D.F. Potter, and Z. Chi., Composition systems, Quarterly of Ap-
plied Mathematics LX (2002), 707736.
[7] N. Chater, J.B. Tenenbaum, and A. Yuille, Probabilistic models of cognition:
Conceptual foundations, Trends in cognitive sciences 10 (2006), 287-291.
[8] L.-B. Chang, E. Borenstein, W. Zhang, and Stuart Geman, Maximum Likeli-
hood Features for Generative Image Models, to appear.
[9] D. A. Levin, Y. Peres, and E. Wilmer, Markov chains and mixing times, Amer-
ican Mathematical Society, Providence, RI, 2009.
[10] P. Felzenszwalb, R. Girshick, and D. McAllester, Cascade Object Detec-
tion with Deformable Part Models, IEEE Conference on Computer Vision and
Pattern Recognition (CVPR), 2010.
[11] F. Sanchez-Vega, J. Eisner, L. Younes and D. Geman,Learning multivari-
ate distributions by competitive assembly of marginals, IEEE Transactions on
Pattern Analysis and Machine Intelligence 35 (2013), 398-410.
[12] I. Kokkinos and A.L. Yuille, Inference and Learning with Hierarchical Compo-
sitional Models, International Journal of Computer Vision 93 (2013), 201-225.
[13] J. Porway, Q.C. Wang, and S.C. Zhu, Hierarchical and Contextual Model
for Aerial Image Understanding, International Journal of Computer Vision 88
(2010), 254-283.