24,761 research outputs found
Algorithmic Identification of Probabilities
TThe problem is to identify a probability associated with a set of natural
numbers, given an infinite data sequence of elements from the set. If the given
sequence is drawn i.i.d. and the probability mass function involved (the
target) belongs to a computably enumerable (c.e.) or co-computably enumerable
(co-c.e.) set of computable probability mass functions, then there is an
algorithm to almost surely identify the target in the limit. The technical tool
is the strong law of large numbers. If the set is finite and the elements of
the sequence are dependent while the sequence is typical in the sense of
Martin-L\"of for at least one measure belonging to a c.e. or co-c.e. set of
computable measures, then there is an algorithm to identify in the limit a
computable measure for which the sequence is typical (there may be more than
one such measure). The technical tool is the theory of Kolmogorov complexity.
We give the algorithms and consider the associated predictions.Comment: 19 pages LaTeX.Corrected errors and rewrote the entire paper. arXiv
admin note: text overlap with arXiv:1208.500
Algorithmic and Statistical Perspectives on Large-Scale Data Analysis
In recent years, ideas from statistics and scientific computing have begun to
interact in increasingly sophisticated and fruitful ways with ideas from
computer science and the theory of algorithms to aid in the development of
improved worst-case algorithms that are useful for large-scale scientific and
Internet data analysis problems. In this chapter, I will describe two recent
examples---one having to do with selecting good columns or features from a (DNA
Single Nucleotide Polymorphism) data matrix, and the other having to do with
selecting good clusters or communities from a data graph (representing a social
or information network)---that drew on ideas from both areas and that may serve
as a model for exploiting complementary algorithmic and statistical
perspectives in order to solve applied large-scale data analysis problems.Comment: 33 pages. To appear in Uwe Naumann and Olaf Schenk, editors,
"Combinatorial Scientific Computing," Chapman and Hall/CRC Press, 201
Model Selection and Adaptive Markov chain Monte Carlo for Bayesian Cointegrated VAR model
This paper develops a matrix-variate adaptive Markov chain Monte Carlo (MCMC)
methodology for Bayesian Cointegrated Vector Auto Regressions (CVAR). We
replace the popular approach to sampling Bayesian CVAR models, involving griddy
Gibbs, with an automated efficient alternative, based on the Adaptive
Metropolis algorithm of Roberts and Rosenthal, (2009). Developing the adaptive
MCMC framework for Bayesian CVAR models allows for efficient estimation of
posterior parameters in significantly higher dimensional CVAR series than
previously possible with existing griddy Gibbs samplers. For a n-dimensional
CVAR series, the matrix-variate posterior is in dimension , with
significant correlation present between the blocks of matrix random variables.
We also treat the rank of the CVAR model as a random variable and perform joint
inference on the rank and model parameters. This is achieved with a Bayesian
posterior distribution defined over both the rank and the CVAR model
parameters, and inference is made via Bayes Factor analysis of rank.
Practically the adaptive sampler also aids in the development of automated
Bayesian cointegration models for algorithmic trading systems considering
instruments made up of several assets, such as currency baskets. Previously the
literature on financial applications of CVAR trading models typically only
considers pairs trading (n=2) due to the computational cost of the griddy
Gibbs. We are able to extend under our adaptive framework to and
demonstrate an example with n = 10, resulting in a posterior distribution with
parameters up to dimension 310. By also considering the rank as a random
quantity we can ensure our resulting trading models are able to adjust to
potentially time varying market conditions in a coherent statistical framework.Comment: to appear journal Bayesian Analysi
Stationary Algorithmic Probability
Kolmogorov complexity and algorithmic probability are defined only up to an
additive resp. multiplicative constant, since their actual values depend on the
choice of the universal reference computer. In this paper, we analyze a natural
approach to eliminate this machine-dependence.
Our method is to assign algorithmic probabilities to the different computers
themselves, based on the idea that "unnatural" computers should be hard to
emulate. Therefore, we study the Markov process of universal computers randomly
emulating each other. The corresponding stationary distribution, if it existed,
would give a natural and machine-independent probability measure on the
computers, and also on the binary strings.
Unfortunately, we show that no stationary distribution exists on the set of
all computers; thus, this method cannot eliminate machine-dependence. Moreover,
we show that the reason for failure has a clear and interesting physical
interpretation, suggesting that every other conceivable attempt to get rid of
those additive constants must fail in principle, too.
However, we show that restricting to some subclass of computers might help to
get rid of some amount of machine-dependence in some situations, and the
resulting stationary computer and string probabilities have beautiful
properties.Comment: 13 pages, 5 figures. Added an example of a positive recurrent
computer se
Misplaced Trust: Measuring the Interference of Machine Learning in Human Decision-Making
ML decision-aid systems are increasingly common on the web, but their
successful integration relies on people trusting them appropriately: they
should use the system to fill in gaps in their ability, but recognize signals
that the system might be incorrect. We measured how people's trust in ML
recommendations differs by expertise and with more system information through a
task-based study of 175 adults. We used two tasks that are difficult for
humans: comparing large crowd sizes and identifying similar-looking animals.
Our results provide three key insights: (1) People trust incorrect ML
recommendations for tasks that they perform correctly the majority of the time,
even if they have high prior knowledge about ML or are given information
indicating the system is not confident in its prediction; (2) Four different
types of system information all increased people's trust in recommendations;
and (3) Math and logic skills may be as important as ML for decision-makers
working with ML recommendations.Comment: 10 page
Causal inference using the algorithmic Markov condition
Inferring the causal structure that links n observables is usually based upon
detecting statistical dependences and choosing simple graphs that make the
joint measure Markovian. Here we argue why causal inference is also possible
when only single observations are present.
We develop a theory how to generate causal graphs explaining similarities
between single objects. To this end, we replace the notion of conditional
stochastic independence in the causal Markov condition with the vanishing of
conditional algorithmic mutual information and describe the corresponding
causal inference rules.
We explain why a consistent reformulation of causal inference in terms of
algorithmic complexity implies a new inference principle that takes into
account also the complexity of conditional probability densities, making it
possible to select among Markov equivalent causal graphs. This insight provides
a theoretical foundation of a heuristic principle proposed in earlier work.
We also discuss how to replace Kolmogorov complexity with decidable
complexity criteria. This can be seen as an algorithmic analog of replacing the
empirically undecidable question of statistical independence with practical
independence tests that are based on implicit or explicit assumptions on the
underlying distribution.Comment: 16 figure
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