This article is a brief personal account of the past, present, and future of
algorithmic randomness, emphasizing its role in inductive inference and
artificial intelligence. It is written for a general audience interested in
science and philosophy. Intuitively, randomness is a lack of order or
predictability. If randomness is the opposite of determinism, then algorithmic
randomness is the opposite of computability. Besides many other things, these
concepts have been used to quantify Ockham's razor, solve the induction
problem, and define intelligence.

Hutter (2007) recently introduced the loss rank principle (LoRP) as a
generalpurpose principle for model selection. The LoRP enjoys many attractive
properties and deserves further investigations. The LoRP has been well-studied
for regression framework in Hutter and Tran (2010). In this paper, we study the
LoRP for classification framework, and develop it further for model selection
problems in unsupervised learning where the main interest is to describe the
associations between input measurements, like cluster analysis or graphical
modelling.

Increasingly encompassing models have been suggested for our world. Theories
range from generally accepted to increasingly speculative to apparently bogus.
The progression of theories from ego- to geo- to helio-centric models to
universe and multiverse theories and beyond was accompanied by a dramatic
increase in the sizes of the postulated worlds, with humans being expelled from
their center to ever more remote and random locations.

We are studying long term sequence prediction (forecasting). We approach this
by investigating criteria for choosing a compact useful state representation.
The state is supposed to summarize useful information from the history. We want
a method that is asymptotically consistent in the sense it will provably
eventually only choose between alternatives that satisfy an optimality property
related to the used criterion.

A key issue in statistics and machine learning is to automatically select the
"right" model complexity, e.g., the number of neighbors to be averaged over in
k nearest neighbor (kNN) regression or the polynomial degree in regression with
polynomials. We suggest a novel principle - the Loss Rank Principle (LoRP) -
for model selection in regression and classification. It is based on the loss
rank, which counts how many other (fictitious) data would be fitted better.
LoRP selects the model that has minimal loss rank.

While statistics focusses on hypothesis testing and on estimating (properties
of) the true sampling distribution, in machine learning the performance of
learning algorithms on future data is the primary issue. In this paper we
bridge the gap with a general principle (PHI) that identifies hypotheses with
best predictive performance. This includes predictive point and interval
estimation, simple and composite hypothesis testing, (mixture) model selection,
and others as special cases. For concrete instantiations we will recover
well-known methods, variations thereof, and new ones.

The Minimum Description Length (MDL) principle selects the model that has the
shortest code for data plus model. We show that for a countable class of
models, MDL predictions are close to the true distribution in a strong sense.
The result is completely general. No independence, ergodicity, stationarity,
identifiability, or other assumption on the model class need to be made. More
formally, we show that for any countable class of models, the distributions
selected by MDL (or MAP) asymptotically predict (merge with) the true measure
in the class in total variation distance.