Other than the odds form of Bayes’s theorem, this chapter is not specifically Bayesian. But Bayesian analysis is all about distributions, so it is important to understand the concept of a distribution well. From a computational point of view, a distribution is any data structure that represents a set of values (possible outcomes of a random process) and their probabilities. We have seen two representations of distributions: Pmfs and Cdfs. These representations are equivalent in the sense that they contain the same infor- mation, so you can convert from one to the other. The primary difference between them is performance: some operations are faster and easier with a Pmf; others are faster with a Cdf.
The premise of this book, and the other books in the Think X series, is that if you know how to program, you can use that skill to learn other topics. Most books on Bayesianstatistics use mathematical notation and present ideas in terms of mathematical concepts like calculus. This book uses Python code instead of math, and discrete approximations instead of continuous math- ematics. As a result, what would be an integral in a math book becomes a summation, and most operations on probability distributions are simple loops. I think this presentation is easier to understand, at least for people with pro- gramming skills. It is also more general, because when we make modeling decisions, we can choose the most appropriate model without worrying too much about whether the model lends itself to conventional analysis.
Thus, Naive Bayes assigns a probability to every possible value in the target range. The resulting distribution is then condensed into a single prediction. In categorical problems, the optimal prediction under zero-one loss is the most likely value—the mode of the underlying distribution. However, in numeric problems the optimal prediction is either the mean or the median, depending on the loss function. These two statistics are far more sensitive to the underlying distribution than the most likely value: they almost always change when the underlying distribution changes, even by a small amount. For this reason NB is not as stable in regression as in classification problems. Some researchers have previously applied Naive Bayes to regression problems .
CAPES – Coordination for the Improvement of Higher Level Personnel CNPq - National Council for Scientifi c and Technological Development INCTMat – National Institute on Science and Technology on Mathematics IME-USP – Institute of Mathematics and Statistics – University of Sao Paulo ABJ - Brazilian Jurimetrics Association
An Indian buffet process (IBP) is a stochastic process that provides a probability distri- bution over equivalence classes of binary matrices of bounded rows and potentially infinite columns. Adoption of the IBP as a prior distribution in Bayesian modeling has led to suc- cessful Bayesian nonparametric models of human cognition including models of latent feature learning (Austerweil & Griffiths, 2009), causal induction (Wood, Griffiths, & Ghahramani, 2006), similarity judgements (Navarro & Griffiths, 2008), and the acquisition of multisensory representations (Yildirim & Jacobs, 2012).
Abstract—Random forests works by averaging several predictions of de-correlated trees. We show a conceptually radical approach to generate a random forest: random sampling of many trees from a prior distribution, and subsequently performing a weighted ensemble of predictive probabilities. Our approach uses priors that allow sampling of decision trees even before looking at the data, and a power likelihood that explores the space spanned by combination of decision trees. While each tree performs Bayesian inference to compute its predictions, our aggregation procedure uses the power likelihood rather than the likelihood and is therefore strictly speaking not Bayesian. Nonetheless, we refer to it as a Bayesian random forest but with a built-in safety. The safeness comes as it has good predictive performance even if the underlying probabilistic model is wrong. We demonstrate empirically that our Safe-Bayesian random forest outperforms MCMC or SMC based Bayesian decision trees in term of speed and accuracy, and achieves competitive performance to entropy or Gini optimised random forest, yet is very simple to construct.
The ABC method generates a sample, hence any statistics based on the posterior can be ap- proximated. Given a posterior mean obtained by one of the kernel methods, however, we may only obtain expectations of functions in the RKHS, meaning that certain statistics (such as confidence intervals) are not straightforward to compute. In Section 5.2, we present an experimental evaluation of the trade-off between computation time and accuracy for ABC and KBR.
The random density considered in the paper can be extended to multivariate problems introducing analogous partitions of d-dimensional Euclidean space. Also, as an alternative to the empirical approach used in Section 5, we can specify full priors for the hyperparameters. Although more computationally challenging, this choice de- fines a more flexible model with random dimension for which the density estimates are no longer simple densi- ties. More general random partitions can also be considered.
Gordon N.J., Salmond N.J., Smith A.F.M. 1993. Novel approach to nonlinear/non- Gaussian Bayesian state estimation, IEE Proceedings-F, Vol.140, No.2, pp.107-113. Handeby G., Karlsson R., Gustafsson F. 2010. The Rao-Blackwellized Particle Filter: A Filter Bank Implementation, EURASIP Journal on Advances in Signal Processing, Article ID 724087, pp.10.
Don’t assume students know a word after defining it once: the word and its meaning need to be reinforced through many activities. Richardson et al. (2013a) found that tutors already had a sense of which terms students would stumble over, because the tutors are immersed in the language of statistics, have met tricky terms many times and have developed a deep sense of the meaning of a particular term. They can then use activities that expose students to tricky terms using two approaches: context-free exercises (“define significant”; Kaplan et al., 2009) or contextual (“What does ‘significant’ mean in this journal extract?”; Richardson et al., 2013b). A semester’s worth of activities can then expose students to appropriate use of a term. Richardson et al. (2013a) found, this type of reinforcement led to substantial gains in understanding after one semester.
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We investigated earthquake time distributions in the south- ern California earthquake catalog by the method of calcu- lation of integral deviation times (IDTs) relative to regular time markers. The main goal of the research was to quan- tify when the time distribution of earthquakes become closer to the random process. Together with IDT calculation, stan- dard methods of complex data analysis such as power spec- trum regression, Lempel and Ziv complexity, and recurrence quantification analysis, as well as multiscale entropy calcu- lations, have been used. Analysis was accomplished for dif- ferent time intervals and for different magnitude thresholds. Based on a simple statistical analysis result, we infer that the strongest earthquakes in southern California occur only in windows with rising values of IDTs and that the character of the temporal distributions of earthquakes in these windows is less random-like compared to the periods of decreased local seismic activity.
While there is a wide range of open-source fuzzy systems software available online, their features differ to those of SyFSeL. Most libraries aid in designing and analysing fuzzy logic systems, or solving specific problems in a given field. For example, FisPro  is a toolkit for creating type-1 fuzzy logic systems by generating fuzzy partitions and rules from data, Xfuzzy  provides tools to aid in describing, verifying and tuning a type-1 fuzzy logic systems, and Type2-FL  provides tools to generate interval type-2 fuzzy sets from interval-valued data and calculate fuzzy statistics. For a more comprehensive overview of fuzzy software in the literature and a list of libraries and toolkits, see  and . SyFSeL is unique in providing the ability to generate synthetic fuzzy sets that can be used to empirically test methods.
More recently, SMC methods (also known as particle filter) have become a popular tool for Bayesian computation in complex dynamic systems ([25, 19]). It can be used to analyze data that are ordered or sequential in time. So one is able to update the current information (via Bayes theorem) on some unknown quantity (e.g. parameters). The methods have been used to provide solutions to filtering problems in HMMs (), where for given the past observations, the posterior distribution of the current state is represented by a cloud of particles and weights. These approaches have been used, successfully, for a wide class of applications in engineering, statistics, physics and operations research (). There has also been many computational advances of SMC algorithms () and a wide variety of convergence results have been established in the past fifteen years ().
In the test class the programmer instantiates sample ob- jects for the target program and encapsulates in separate methods the code for each different set of tests. Running of the JPF program then opens a JPT console, and creates a simple GUI with a list of action buttons - one for each method in the test class that requires no arguments and re- turns void. A programmer can perform the desired tests in an arbitrary sequence by selecting the appropriate action button. Additionally, the JPF creates a graphics panel that can be used for display of graphics and images, and for the input of mouse events.
Today, nearly any company, from SMBs to the largest enterprise, can go global. Rapid and reliable transportation has effectively eliminated distance as an obstacle, and communications breakthroughs have made conducting and managing business fairly easy. With the emergence of the internet as a core business enabler and with its continuous evolution enabling new and better capabilities virtually every day, global businesses can always be open and customers’ experiences can be local.
The other main aim of the paper has been to show how multiproduct monopoly analysis is made simpler when consumer surplus is a homothetic function of quantities. Whether the …rm’s objective is pro…t or a Ramsey combination of pro…t and consumer surplus, it optimizes by selecting e¢cient (i.e., welfare-maximizing) quantities scaled back equipropor- tionately. The resulting optimal markups yield, for example, transparent results on mul- tiproduct cost passthrough, including instances of without cross-cost passthrough. With Ramsey-Cournot equivalence, these results extend to the Cournot oligopoly setting with symmetric …rms.