Bayesian approaches formulate the problem differently. Instead of saying the parameter simply has one (unknown) true value, a Bayesian method says the parameter's value is fixed but has been chosen from some probability distribution -- known as the prior probability distribution.
Confessions of a moderate Bayesian, part 4 Bayesian statistics by and for non-statisticians Read part 1: How to Get Started with Bayesian Statistics Read part 2: Frequentist Probability vs Bayesian Probability Read part 3: How Bayesian Inference Works in the Context of Science Predictive distributions A predictive distribution is a distribution that we expect for future observations. In other ...
One of the continuous and occasionally contentious debates surrounding Bayesian statistics is the interpretation of probability. I am going to present both interpretations.
The Bayesian interpretation of probability as a measure of belief is unfalsifiable. Only if there exists a real-life mechanism by which we can sample values of $\theta$ can a probability distribution for $\theta$ be verified. In such settings probability statements about $\theta$ would have a purely frequentist interpretation.
@Xi'an's answer (below) helped me - clarifying that the Dirichlet distribution is A prior for the multinomial, not THE prior. It's chosen because it is a conjugate prior that works well to describe certain systems such as documents in NLP.
The Bayesian Choice for details.) In an interesting twist, some researchers outside the Bayesian perspective have been developing procedures called confidence distributions that are probability distributions on the parameter space, constructed by inversion from frequency-based procedures without an explicit prior structure or even a dominating ...
This is a very simple question but I can't find the derivation anywhere on the internet or in a book. I would like to see the derivation of how one Bayesian updates a multivariate normal distribut...
The concept is invoked in all sorts of places, and it is especially useful in Bayesian contexts because in those settings we have a prior distribution (our knowledge of the distribution of urns on the table) and we have a likelihood running around (a model which loosely represents the sampling procedure from a given, fixed, urn).
The debate about non-informative priors has been going on for ages, at least since the end of the 19th century with criticism by Bertrand and de Morgan about the lack of invariance of Laplace's uniform priors (the same criticism reported by in the above comments). This lack of invariance sounded like a death stroke for the Bayesian approach and, while some Bayesians were desperately trying to ...
