程序代写 CS代考

Linear regression We are ready to consider our first machine-learning problem: linear regression. Suppose that we are interested in the values of a function y(x): Rd → R, where x is a d-dimensional vector-valued 􏰫 􏰬N input. We assume that we have made some observations of this mapping, D = (xi, yi) i=1, to serve […]

Bayesian model selection Consider the regression problem, where we want to predict the values of an unknown function y: Rd R given examples D xi,yiNi1 to serve as training data. In Bayesian linear regression, we made the following assumption about yx: yx xw x, 1 where x is a now explicitlywritten feature expansion of x.

Bayesian Monte Carlo Carl Edward Rasmussen and Zoubin Ghahramani Gatsby Computational Neuroscience Unit University College London 17 Queen Square, London WC1N 3AR, England edward,zoubin@gatsby.ucl.ac.uk http://www.gatsby.ucl.ac.uk Abstract We investigate Bayesian alternatives to classical Monte Carlo methods for evaluating integrals. Bayesian Monte Carlo (BMC) allows the in- corporation of prior knowledge, such as smoothness of the integrand,

CSE 515T (Fall 2019) Project A portion of your grade this semester will be based on a significant project investigating Bayesian methods in depth. You will have two possible paths to satisfy this project requirement. The main goal of the project is to give you hands-on experience applying Bayesian methods to a real-world dataset. The

CSE 515T (Fall 2019) Midterm • This will be due 11:59 PM Monday, 25 November. Late submissions will not be accepted. • You will upload your submission to Gradescope. • Please do not discuss the questions with other members of the class. • Please post any questions as a private message to the instructors on

Practical Bayesian Optimization of Machine Learning Algorithms Jasper Snoek Department of Computer Science University of Toronto jaspercs.toronto.edu Hugo Larochelle Department of Computer Science University of Sherbrooke hugo.larochelleusherbrooke.edu Ryan P. Adams School of Engineering and Applied Sciences Harvard University rpaseas.harvard.edu Abstract The use of machine learning algorithms frequently involves careful tuning of learning parameters and model

0.8 0.6 0.4 0.2 μ = 0, σ = 1 μ = 1, σ = 1/2 μ = 0, σ = 2 0 −4 −3 −2 −1 0 1 2 3 4 x Figure 1: Examples of univariate Gaussian pdfs N (x; μ, σ2). The Gaussian distribution Probably the most-important distribution in all of statistics

Until now we have always worked with likelihoods and prior distributions that were conjugate to each other, allowing the computation of the posterior distribution to be done in closed form. Unfortunately, there are numerous situations where this will not be the case, forcing us to approxi- mate the posterior and related quantities (such as the

The Kernel Trick Consider the assumption of linear regression with an explicit feature expansion φ : x 􏰳→ φ(x): y(x) = φ(x)⊤w + ε(x). Given training data D = 􏰫(x, y)􏰬 = (X, y), recall we modeled the residuals ε(x) as zero-mean independent, identically distributed Gaussians with variance σ2: p(ε) = N (ε; 0, σ2I),

Coin flipping Suppose there is a coin that may be biased – this coin has unknown probability θ of giving a “heads.” If we repeatedly flip this coin and observe the outcomes, how can we maintain our belief about θ? Note that the coin-flipping problem can be seen as a simplification of the survey problem