Within the Kolmogorov theory of probability, Bayes' rule allows one to perform statistical inference by relating conditional probabilities to unconditional probabilities. As we show here, however, there is a continuous set of alternative inference rules that yield the same results, and that may have computational or practical advantages for certain problems. We formulate generalized axioms for probability theory, according to which the reverse conditional probability distribution P(B|A) is not specified by the forward conditional probability distribution P(A|B) and the marginals P(A) and P(B). Thus, in order to perform statistical inference, one must specify an additional "inference axiom," which relates P(B|A) to P(A|B), P(A), and P(B). We show that when Bayes' rule is chosen as the inference axiom, the axioms are equivalent to the classical Kolmogorov axioms. We then derive consistency conditions on the inference axiom, and thereby characterize the set of all possible rules for inference. The set of "first-order" inference axioms, defined as the set of axioms in which P(B|A) depends on the first power of P(A|B), is found to be a 1-simplex, with Bayes' rule at one of the extreme points. The other extreme point, the "inversion rule," is studied in depth.
The twenty-first century has seen a breathtaking expansion of statistical methodology, both in scope and in influence. "Big data," "data science," and "machine learning" have become familiar terms in the news, as statistical methods are brought to bear upon the enormous data sets of modern science and commerce. How did we get here? And where are we going? This book takes us on a journey through the revolution in data analysis following the introduction of electronic computation in the 1950s.
Keith I don't disagree with this, but it is the tip of an iceberg. Where there are standardized credentialing exams, it becomes simple – nurses, physicians, pharmacists, CPAs, etc. either have the credential or not (and sometimes a particular degree is required, sometimes it is just based on an exam and/or experience). In some fields, the academic degree stands as a proxy for some body of knowledge – true for many STEM degrees. However, there are plenty of academic disciplines where career preparedness has little relationship with the degree: e.g., it is hard to know what a person with a sociology (or political science, or economics, or history, or……) degree has the capability to accomplish by virtue of their degree. I am not downgrading such degrees, merely pointing out that the definition of the field does not map very well into a set of particular skills.
In the last few months, I have had several people contact me about their enthusiasm for venturing into the world of data science and using Machine Learning (ML) techniques to probe statistical regularities and build impeccable data-driven products. However, I've observed that some actually lack the necessary mathematical intuition and framework to get useful results. This is the main reason I decided to write this blog post. Recently, there has been an upsurge in the availability of many easy-to-use machine and deep learning packages such as scikit-learn, Weka, Tensorflow etc. Machine Learning theory is a field that intersects statistical, probabilistic, computer science and algorithmic aspects arising from learning iteratively from data and finding hidden insights which can be used to build intelligent applications. Despite the immense possibilities of Machine and Deep Learning, a thorough mathematical understanding of many of these techniques is necessary for a good grasp of the inner workings of the algorithms and getting good results.
This (by Jason Torchinsky, from Stay Free magazine, around 1998?) is just hilarious. We used to have both those shake-out-the-powder cans, Comet and that parmesan cheese, in our house when I was growing up. The post What's powdery and comes out of a metallic-green cardboard can? appeared first on Statistical Modeling, Causal Inference, and Social Science. The post What's powdery and comes out of a metallic-green cardboard can? appeared first on All About Statistics.