Probabilistic Graphical Model 1.2節

以下內容是我閱讀Probabilistic Graphical Model, Koller 2009一書的讀書筆記，未來將不定期新增內容，此技術屬AI人工智慧範疇。

1.2 Structured Probabilistic Models

既然要融入Uncertainty和Probability Theory，自然就會需要使用Random Variable來描述Value的不確定性，此外To reason probabilistically about the values of one or more of the variables, possibly given observations about some others，就會需要Joint Probability Distribution以及Posterior Distribution.

1.2.1 Probabilistic Graphical Models

當一個專業Domain需要用很多Random Variable描述時就會很棘手，因此This book describes the framework of probabilistic graphical models, which provides a mechanism for exploiting structure in complex distributions to describe them compactly, and in a way that allows them to be constructed and utilized effectively. Probabilistic graphical models use a graph-based representation as the basis for compactly encoding a complex distribution over a high-dimensional space. The nodes correspond to the variables in our domain, and the edges correspond to direct probabilistic interactions between them.

這本書會教你從Graph Representation出發來判斷機率分佈的Independencies以及從Graph的Skeleton來判斷Factorzation，因此：It turns out that these two perspectives — the graph as a representation of a set of independencies, and the graph as a skeleton for factorizing a distribution — are, in a deep sense, equivalent. The independence properties of the distribution are precisely what allow it to be represented compactly in a factorized form. Conversely, a particular factorization of the distribution guarantees that certain independencies hold.

此外這本書會介紹兩種Graph，分別是Bayesian Networks (Directed Graph)與Markov Networks (Undirected Graph)，Both representations provide the duality of independencies and factorization, but they differ in the set of independencies they can encode and in the factorization of the distribution that they induce.

1.2.2 Representation, Inference, Learning

Graph Representation的好處是：Transparent, in that a human expert can understand and evaluate its semantics and properties. This property is important for constructing models that provide an accurate reflection of our understanding of a domain. Models that are opaque can easily give rise to unexplained, and even undesirable answers.

另一個好處是做Inference的時候比較快：This book provides algorithms for computing the posterior probability of some variables given evidence on others. These inference algorithms work directly on the graph structure and are generally orders of magnitude faster than manipulating the joint distribution explicitly.

第三個好處是：Probabilistic graphical models support a data-driven approach to model construction that is very effective in practice. The models produced by this process are usually much better reflections of the domain than models that are purely hand-constructed. Moreover, they can sometimes reveal surprising connections between variables and provide novel insights about a domain.

總結來說Probabilistic Graphical Model提供的好處乃是：These three components — representation, inference, and learning — are critical components in constructing an intelligent system. We need a declarative representation that is a reasonable encoding of our world model. We need to be able to use this representation effectively to answer a broad range of questions that are of interest. And we need to be able to acquire this distribution, combining expert knowledge and accumulated data. Probabilistic graphical models are one of a small handful of frameworks that support all three capabilities for a broad range of problems.