An introduction to probabilistic graphical models

An introduction to probabilistic graphical models

Probabilistic graphical models: principles and applications

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A graphical model, also known as a probabilistic graphical model (PGM) or organized probabilistic model, is a probabilistic model in which the conditional dependency structure between random variables is represented by a graph. Probability theory, statistics, especially Bayesian statistics, and machine learning all use them.
In general, probabilistic graphical models encode a distribution over a multi-dimensional space using a graph-based representation, which is a compact or factorized representation of a collection of independences that hold in the particular distribution. Bayesian networks and Markov random fields are two types of graphical representations of distributions that are widely used. Both families share factorization and independence properties, but they vary in the number of independences they can encode and the distribution factorization they cause. 1st

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For several tasks, representing each message by its form or class (i.e., if the message is spam/ham) and the presence of specific keywords in the message (e.g., “cash” or “thanks”) is sufficient.
The large number of “parameters” required by probabilistic models was deemed a major impediment to the use of probability theory in the early days of research in automated decision support systems.
Making additional assumptions is the secret to getting around such a stumbling block. In truth, we’ve already seen such a simplifying assumption: that objects are defined by their characteristics. Another possibility is to make the following assumption:
If all we want to do is compute class probabilities given all features, the model will suffice. We may use more expressive classes of functions for the class conditional probability, such as generalized linear models, decision trees, or neural networks, if desired, at the cost of an increase in the number of parameters.
Let’s assume we’re interested in the likelihood that a certain keyword, such as “thanks,” appears in ham messages. The model is then incomplete and useless. The argument is that the joint probability distribution has not been completely defined.

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and Thomas Wiecki has a few excellent blog posts on using pymc3 to build Bayesian neural networks.

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Introduction to graphical models

[two] If you’re interested in learning how these two principles can be combined, I highly recommend reading these three blogs. [Zero] http://blog.twiecki.github.io/2016/06/01/bayesian-deep-learn… 1st http://blog.twiecki.github.io/2016/07/05/bayesian-deep-learn… [two] http://blog.twiecki.github.io/2017/03/14/random-walk-deep-ne…
Back-propagation does not have full Bayesian inference (although there are some tricks [0]). Instead, they employ variational inference[1, which enables rapid inference of continuous PGMs. [Zero] http://mlg.eng.cam.ac.uk/yarin/blog 2248.html http://mlg.eng.cam.ac.uk/yarin/blog 2248.html 1st 1603.00788 https://arxiv.org/abs/1603.00788
PGMs are fantastic, but my experience from Koller’s course has taught me that identifying situations where they can be used is extremely difficult.
Part of the explanation is that you need to know the causal relationships (coarse grained, i.e. direction) between your variables before you start.
If you’re doing machine learning, you probably don’t know such causal relationships to begin with.
Physics, for example, is a good match since the rules are well-known.

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Graphical representations of probability distributions are known as probabilistic graphical models. Complex probability distributions are encountered in many science and engineering applications, and such models are versatile in representing them.
The next step will be focused on probabilistic reasoning, which will account for uncertainty as well as answer current deep learning applications, such as providing decision explanations, ethical AI, and so on.
(1) “Probabilistic Graphical Models” by Daphne Koller and Nir Friedman (MIT Press 2009), (ii) Chris Bishop’s “Pattern Recognition and Machine Learning” (Springer 2006), which has a chapter on PGMs that serves as a basic introduction, and (iii) “Deep Learning” by Goodfellow, et.al. (MIT Press, 2016), which has several chapters on graphical models.