In the winter of 1958, a 30-year-old psychologist named Frank Rosenblatt was en route from Cornell University to the Office of Naval Research in Washington DC when he stopped for coffee with a journalist. Rosenblatt had unveiled a remarkable invention that, in the nascent days of computing, created quite a stir. It was, he declared, "the first machine which is capable of having an original idea". Rosenblatt's brainchild was the Perceptron, a program inspired by human neurons that ran on a state-of-the-art computer: a five-tonne IBM mainframe the size of a wall. Feed the Perceptron a pile of punch cards and it could learn to distinguish those marked on the left from those marked on the right.

In statistical inference, uncertainty is unknown and all models are wrong. That is to say, a person who makes a statistical model and a prior distribution is simultaneously aware that both are fictional candidates. To study such cases, statistical measures have been constructed, such as cross validation, information criteria, and marginal likelihood, however, their mathematical properties have not yet been completely clarified when statistical models are under- and over- parametrized. We introduce a place of mathematical theory of Bayesian statistics for unknown uncertainty, which clarifies general properties of cross validation, information criteria, and marginal likelihood, even if an unknown data-generating process is unrealizable by a model or even if the posterior distribution cannot be approximated by any normal distribution. Hence it gives a helpful standpoint for a person who cannot believe in any specific model and prior. This paper consists of three parts. The first is a new result, whereas the second and third are well-known previous results with new experiments. We show there exists a more precise estimator of the generalization loss than leave-one-out cross validation, there exists a more accurate approximation of marginal likelihood than BIC, and the optimal hyperparameters for generalization loss and marginal likelihood are different.

This article is a review of theoretical advances in the research field of algebraic geometry and Bayesian statistics in the last two decades. Many statistical models and learning machines which contain hierarchical structures or latent variables are called nonidentifiable, because the map from a parameter to a statistical model is not one-to-one. In nonidentifiable models, both the likelihood function and the posterior distribution have singularities in general, hence it was difficult to analyze their statistical properties. However, from the end of the 20th century, new theory and methodology based on algebraic geometry have been established which enables us to investigate such models and machines in the real world. In this article, the following results in recent advances are reported. First, we explain the framework of Bayesian statistics and introduce a new perspective from the birational geometry. Second, two mathematical solutions are derived based on algebraic geometry. An appropriate parameter space can be found by a resolution map, which makes the posterior distribution be normal crossing and the log likelihood ratio function be well-defined. Third, three applications to statistics are introduced. The posterior distribution is represented by the renormalized form, the asymptotic free energy is derived, and the universal formula among the generalization loss, the cross validation, and the information criterion is established. Two mathematical solutions and three applications to statistics based on algebraic geometry reported in this article are now being used in many practical fields in data science and artificial intelligence.

This course is about the discussion of sampling and exploring data, as well as basic probability theory and Bayes' rule. A variety of exploratory data analysis techniques will be covered, including numeric summary statistics and basic data visualization. The concepts and techniques you will find in this course will serve as building blocks for the inference and modeling courses in the Specialization.

The Beta distribution is a continuous distribution that is often dubbed as the Probability Distribution of Probabilities. This is because it can only take on values between 0 and 1. It is used to infer the probability of an event when we have some information about the volumes of successes and failures. The primary use of the Beta distribution is being the conjugate prior in Bayesian statistics to the Binomial and Bernoulli distributions. In one of my next articles, we will dive into what this exactly means, however here we will just gain some intuition behind the Beta distribution.

It's no secret that artificial intelligence (AI)-enabled devices are listening in or observing what we're doing, collecting and digitising massive amounts of data. What underpins this entire enterprise is the work under the hood – maintaining data lakes and warehouses that store the data, performing data engineering tasks to establish and enhance the structure, using business intelligence and statistical analysis to make sense of it and, finally, training an AI program to make predictions that yield more "intelligent" decisions. For example, many of us are familiar with virtual assistant technologies that take in information, digitise the data, put it into data pipelines, and analyse it so the AI algorithms become better in near real-time, fine-tuning it specifically to one's world. What makes an AI algorithm potent is when it can start making connections with other data sets and data points. This results in the creation of a profile that encompasses your shopping behaviour, what you're doing at home, what music you like listening to, what you like eating, and even where you live – in suburbia or in the city.

The extensive bibliography provides entry points for further study for the motivated reader. The algorithm descriptions are clear and intuitive in a way that will give you a leg up to implement pre-existing algorithm and even develop your own variations.

This specialization program is especially dedicated to statistics. In this program, you will learn basic and intermediate concepts of statistical analysis using the Python programming language. In this program, you will learn the following topics- where data come from, what types of data can be collected, study data design, data management, and how to effectively carry out data exploration and visualization. Along with that, you will work on a variety of assignments that will help you to check your knowledge and ability. This specialization program is a 3-course series. Let's see the details of the courses-

Many of us (myself included) have felt discouraged from using Bayesian statistics for analysis. Supposedly, Bayesian statistics has a bad reputation: it is difficult and heavily dependent on math. Also, because of its relevance to many fields, Data Science included, writers and professionals, want to get a head start by publishing articles on how the formula works. I believe data professionals, academics, existing books, and online courses are responsible for creating the negative stereotype of Bayes' hard work. We can all agree that not everyone is attracted to mathematical formulas.

In machine learning and statistics, computing the accuracy, or loss, of a model is crucial for understanding the quality of the model and what improvements can be made to increase accuracy. Typically, researchers choose a loss function de- pending on their task, and this loss function runs over their test set of data, after training. However, in many cases, researchers want an estimation of their loss either before they test it or in cases when testing data is not yet available. This estimation is known as expected loss, or risk, and is usually utilized in order to assess how precarious an action or event will be. The foundations of Bayesian statistics are rooted in Bayes' Theorem, a theorem developed by Thomas Bayes who was an English mathematician and theologian during the 1700s.