A Peter Orbanz Introduction to Statistics and Statistical Machine Learning


Peter Orbanz is an Assistant Professor in Columbia University’s Department of Statistics. His research interests lie within machine learning and artificial intelligence. Additionally, his interest lies in studying discrete objects and structures such as permutation graphs, partitions, and binary sequences for their statistical properties.

His research lies at the crossroads of information theory, statistical physics, and theoretical computer science. He aims to gain engineering insight into practical problems by formulating and solving mathematical models.

Bayesian nonparametric

Bayesian nonparametrics is an integral field of statistical machine learning and related research, providing non-parametric methods for creating posterior probability distributions without using parametric assumptions as their basis. Applications of Bayesian nonparametrics include image segmentation, data mining, and genetic variation studies.

One significant result is the uniform consistency theorem, which states that a consistent estimator will have an extensive neighborhood around every point in its sample space for any finite sample size. This robust result establishes the connection between asymptotic behavior and limited sample error.

Another key result is the generalization of the Bernstein-von Mises theorem to random partition models, providing a general framework for nonparametric inference. It relies on the notion of metric spaces, applying to models with both continuous and discrete variables; furthermore, this allows more general nonparametric models and more efficient Monte Carlo inference algorithms to be created.

Bayesian nonparametrics is further advanced by the Pitman-Yor process, an infinite mixture of the Gaussian model. This approach has many applications in machine learning – for instance, a hierarchical Bayesian language model uses this model to represent sentences and lexicons, while random partition models have also been utilized as genetic variation models in species studies and dynamic clustering images for video segmentation.

Graphical models

Graphical models provide a general framework for formulating and visualizing statistical models. All kinds of statistical models, from structured causal ones such as path analysis to more abstract expert system models, can be constructed within this framework. One application for graphical models is diagnosis and troubleshooting problems – seen through applications like the Microsoft Answer Wizard in Office 95 or later versions of Windows or its bouncy paperclip helper feature on Windows Vista; other examples may include genetic counseling, information retrieval, medical monitoring weather forecasting manufacturing digital communication or machine vision.

Probabilistic graphical models (PGMs) use graphs to represent complex multivariate probability distributions, which makes the relationships among variables easier to express concisely. Nodes in a PGM represent random variables, while edges between nodes represent conditional independence assumptions. PGMs are used extensively in fields like statistics, machine learning, and artificial intelligence, as inference and learning tasks can be accomplished more efficiently with knowledge of their structure.

There are two general classes of graphical models: undirected and directed. Undirected models are most widely used in physics and computer vision communities, while directed graphical models tend to be utilized more in AI and statistics fields. Both undirected and directed graphical models can be described as Markov random fields; however, their properties vary. Undirected models define independence, while directed ones require an arc containing direction information to link each pair of nodes.

Neural networks

Neural networks are an example of machine learning algorithms that emulate the structure of human neural networks, with thousands or millions of superficial processing nodes densely interconnected and trained using one of several algorithms – backpropagation being one such algorithm – which are then introduced. Neural networks have proven invaluable for use in computer vision, speech recognition, natural language processing, and other tasks requiring complex models of relationships or patterns to identify complex nonlinear data sets quickly while learning from volatile or inconsistent data streams at lightning speed. Neural networks also possess classification and cluster data classification capabilities that are ten times faster than human experts can.

One of the best-known examples of neural network technology is Google’s search algorithm, which employs deep learning techniques based on neural networks. While not perfect, this technology has substantially increased search quality.

A neural net consists of weights that link input with output. Each node in the network receives its unique weight, with larger ones contributing more. When a result from one node multiplies by its assigned weight and is added together, that sum will be fed back into another node, producing new work until all final output has been reached.

Adaptive learning

Adaptive learning is an innovative method of instruction that allows students to progress at a pace best suited for them, using various technological programs for personalized learning paths and to identify those struggling. Teachers and administrators can then make continuous improvements by comparing student data across semesters; it may even reduce cheating as students are more likely to persevere when their instructor and class peers appreciate their efforts.

Adaptive learning in online courses can increase student engagement, which has many advantages for businesses. According to Gallup research, companies with highly engaged employees were 17% more productive and had 41% fewer absenteeism cases. Furthermore, adaptive learning can improve employee retention, saving money and time over time.

Adaptive learning works by tailoring content based on each student’s current knowledge and abilities, considering their preferences, and extracting insights from accurate data that resonates with them.

Adaptive learning creates a customized educational journey for each student, teaching them exactly what they need to know to increase motivation for mastering skills and knowledge while helping them feel more capable and confident about themselves and their abilities. According to The eLearning Industry, adaptive learning enables an individualized journey for all participants.