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Even in the era of Big Data there are many real-world problems where the number of input features has about the some order of magnitude than the number of samples. Often many of those input features are irrelevant and thus inferring the relevant ones is an important problem in order to prevent over-fitting. Automatic Relevance Determination solves this problem by applying Bayesian techniques.

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はじパタ6章前半

はじパタ6章前半

Visual Explanation of Ridge Regression and LASSO

Visual Explanation of Ridge Regression and LASSO

[DL輪読会]Deep Learning 第16章 深層学習のための構造化確率モデル

[DL輪読会]Deep Learning 第16章 深層学習のための構造化確率モデル

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Visual Explanation of Ridge Regression and LASSO

Ridge regression and LASSO are regularization techniques used to address overfitting in regression analysis. Ridge regression minimizes residuals while also penalizing large coefficients, resulting in all coefficients remaining in the model. LASSO also minimizes residuals while penalizing large coefficients, but performs continuous variable selection by driving some coefficients to exactly zero. Both techniques involve a tuning parameter that controls the strength of regularization. Cross-validation is commonly used to select the optimal tuning parameter value.

[DL輪読会]Deep Learning 第16章 深層学習のための構造化確率モデル

2017/12/18
Deep Learning JP:
http://deeplearning.jp/seminar-2/

2013.12.26 prml勉強会 線形回帰モデル3.2~3.4

PRML勉強会2013冬
第7回 3章線形回帰モデルの３．２〜３．４です

VBAで数値計算 10 逆行列と疑似逆行列

VBAを使って数値計算の解説を行ったスライドシリーズです。

PRML 第4章

パターン認識と機械学習 第４章
Pattern Recognition and Machine Learning Chapter 4

zkStudyClub: HyperPlonk (Binyi Chen, Benedikt Bünz)

Paper: https://eprint.iacr.org/2022/1355
Plonk is a widely used succinct non-interactive proof system that uses univariate polynomial commitments. Plonk is quite flexible: it supports circuits with low-degree ``custom'' gates as well as circuits with lookup gates (a lookup gate ensures that its input is contained in a predefined table). For large circuits, the bottleneck in generating a Plonk proof is the need for computing a large FFT.
In this work, the authors present HyperPlonk, an adaptation of Plonk to the boolean hypercube, using multilinear polynomial commitments. HyperPlonk retains the flexibility of Plonk but provides several additional benefits. First, it avoids the need for an FFT during proof generation. Second, and more importantly, it supports custom gates of much higher degree than Plonk without harming the running time of the prover. Both of these can dramatically speed up the prover's running time. Since HyperPlonk relies on multilinear polynomial commitments, the authors revisit two elegant constructions: one from Orion and one from Virgo. The authors also show how to reduce the Orion opening proof size to less than 10kb (an almost factor 1000 improvement) and show how to make the Virgo FRI-based opening proof simpler and shorter.

線形代数の視覚的理解のためのノート

線形代数の視覚的理解のためのノート
Graphic Notes on Prof. Gilbert Strang's "Linear Algebra for Everyone"
Gilbert 先生の線形代数の講義に惚れて作りました。
コンタクト頂ければ、PowerPoint 版をお配りします。
また、マトリックスワールド、も是非ご覧ください。
https://qiita.com/kenjihiranabe/items/854dbe1f6d9fca9df85c

PRML第９章「混合モデルとEM」

パターン認識と機械学習(PRML)の第９章「混合モデルとEM」について説明したスライドです。文字多め。
潜在変数を持つモデルの最適化を行うことができるEMアルゴリズムについて、最初は具体的でイメージしやすいk-meansクラスタリングから説明し、最後は数式を詳細に見ていきその意味を考察します
9.1 K-meansクラスタリング
9.2 混合ガウス分布
9.3 EMアルゴリズムのもう１つの解釈
9.4 一般のEMアルゴリズム

[DL輪読会]Deep Learning 第11章 実用的な方法論

2017/12/04
Deep Learning JP:
http://deeplearning.jp/seminar-2/

StanとRでベイズ統計モデリング 11章 離散値をとるパラメータ

StanとRでベイズ統計モデリング 11章 離散値をとるパラメータ

Approximation algorithms

Concepts of approximation algorithm , approximation ratio, examples, travelling salesman problem analysis, it's application, pros and cons.

グラフィカル Lasso を用いた異常検知

多変数の間の直接相関をスパース推定する手法であるグラフィカル Lasso と、その異常検知への応用についてまとめました。

データ解析14 ナイーブベイズ

ナイーブベイズ

パターン認識と機械学習 13章 系列データ

パターン認識と機械学習(PRML) 13章 の発表内容です

Prml3.5 エビデンス近似〜

PRML3.5章 大阪PRML（予定）

SMO徹底入門 - SVMをちゃんと実装する

2013-05-05 PRML復々習レーン#10 で発表

はじパタ6章前半

はじパタ6章前半

Visual Explanation of Ridge Regression and LASSO

Visual Explanation of Ridge Regression and LASSO

[DL輪読会]Deep Learning 第16章 深層学習のための構造化確率モデル

[DL輪読会]Deep Learning 第16章 深層学習のための構造化確率モデル

2013.12.26 prml勉強会 線形回帰モデル3.2~3.4

2013.12.26 prml勉強会 線形回帰モデル3.2~3.4

VBAで数値計算 10 逆行列と疑似逆行列

VBAで数値計算 10 逆行列と疑似逆行列

パターン認識 第12章 正則化とパス追跡アルゴリズム

パターン認識 第12章 正則化とパス追跡アルゴリズム

PRML 第4章

PRML 第4章

zkStudyClub: HyperPlonk (Binyi Chen, Benedikt Bünz)

zkStudyClub: HyperPlonk (Binyi Chen, Benedikt Bünz)

線形代数の視覚的理解のためのノート

線形代数の視覚的理解のためのノート

PRML第９章「混合モデルとEM」

PRML第９章「混合モデルとEM」

[DL輪読会]Deep Learning 第11章 実用的な方法論

[DL輪読会]Deep Learning 第11章 実用的な方法論

StanとRでベイズ統計モデリング 11章 離散値をとるパラメータ

StanとRでベイズ統計モデリング 11章 離散値をとるパラメータ

Approximation algorithms

Approximation algorithms

PRML輪読#3

PRML輪読#3

グラフィカル Lasso を用いた異常検知

グラフィカル Lasso を用いた異常検知

データ解析14 ナイーブベイズ

データ解析14 ナイーブベイズ

パターン認識と機械学習 13章 系列データ

パターン認識と機械学習 13章 系列データ

Prml3.5 エビデンス近似〜

Prml3.5 エビデンス近似〜

Prml 4.1.1

Prml 4.1.1

SMO徹底入門 - SVMをちゃんと実装する

SMO徹底入門 - SVMをちゃんと実装する

t4_20110728_IGARSS11_tsutomuy.pdf

1) Satellite images detected the disappearance of a large supraglacial lake on Tshojo Glacier in Bhutan between March and July 2009, coinciding with a glacier outburst flood event.
2) The lake volume was estimated at 1.47 million cubic meters based on a post-flood DEM, suggesting it was an adequate source for the flood waters.
3) Subsequent images showed the lake beginning to refill in late 2010, indicating its behavior should continue to be monitored to predict future flood risks.

Prigogine, i. tan solo una_ilusion

Este documento discute la relación entre la ciencia, la filosofía y la literatura en relación con la noción del tiempo. Argumenta que aunque la ciencia tradicionalmente ha adoptado una visión atemporal del universo, excluyendo así a los seres humanos, los descubrimientos recientes en física están redescubriendo la importancia del tiempo a todos los niveles. Esto abre posibilidades para un nuevo diálogo interdisciplinario fructífero entre las ciencias naturales y humanas sobre cómo entender la naturaleza y la experiencia

Inference for stochastic differential equations via approximate Bayesian comp...

Despite the title the methods are appropriate for more general dynamical models (including state-space models). Presentation given at Nordstat 2012, Umeå. Relevant research paper at http://arxiv.org/abs/1204.5459 and software code at https://sourceforge.net/projects/abc-sde/

Hirsch s.w., smale s. differential equations, dynamical systems and linear ...

The document discusses the history and development of a new technology called blockchain. Blockchain first emerged with bitcoin as a way to track transactions without a central authority. It has since expanded and many now see potential for blockchain to disrupt industries like finance, healthcare, and government services by making transactions more secure and transparent through distributed ledger systems.

Francisco

El documento presenta un avance del trabajo de tesis sobre la implementación de un libro de ecuaciones diferenciales centrado en la modelación. Los objetivos son analizar el uso de gráficas para apoyar la comprensión de estudiantes sobre las condiciones iniciales en modelos de ecuaciones diferenciales y observar cambios cualitativos. Se revisará literatura sobre enfoques de modelación con ecuaciones diferenciales y el uso de gráficas.

Libro de Mancil

La Revolución Francesa comenzó en 1789 cuando la Asamblea Nacional se autoproclamó como el Tercer Estado y finalizó con el golpe de estado de Napoleón en 1799. Aunque Francia osciló entre diferentes formas de gobierno en los 71 años siguientes, la revolución marcó el fin del absolutismo y dio poder a la burguesía. Los principios de libertad, igualdad y fraternidad surgieron de la Declaración de los Derechos del Hombre y del Ciudadano de 1789 y definieron el carácter de las naciones dependiendo de su orden

Un Juego Diferencial Estocástico para Reaseguro

Basado en [Zen10]. Muestra un juego diferencial estocástico entre dos compañías de seguros que usan una estrategia de reaseguro para reducir el riesgo de exposición.
[Zen10] Zeng, X. (2010). A stochastic differential reinsurance game. Journal of Applied Probability, 47(2), 335-349.

Prigogine esayo

El documento resume las teorías del físico Ilya Prigogine. Prigogine cuestionó las concepciones establecidas sobre el equilibrio, el orden y el tiempo en la ciencia. Propuso que los estados de no equilibrio pueden dar lugar a nuevos órdenes complejos y que el tiempo surge a través de procesos irreversibles. Sus teorías sobre estructuras disipativas y el caos tuvieron aplicaciones en campos como la química, la física, la biología y la cosmología.

Calculo diferencial e integral2

Este documento presenta un resumen de tres capítulos sobre cálculo diferencial e integral de funciones de una variable. El documento introduce los conceptos básicos de números reales, funciones elementales, números complejos y funciones continuas. Explica temas como los axiomas de los números reales, funciones polinómicas, trigonométricas y exponenciales, operaciones con números complejos, y las propiedades de continuidad y límites funcionales. El documento proporciona definiciones, teoremas, ejemplos y ejercicios para cada uno

MODELACIÓN MATEMÁTICA A TRAVÉS DE LAS ECUACIONES EN DIFERENCIA

Este documento presenta un modelo didáctico para enseñar modelación matemática a través de ecuaciones en diferencia. Explica que la modelación matemática es importante para el desarrollo del pensamiento científico de los ingenieros pero que tradicionalmente se ha enseñado de manera aislada. Luego revisa investigaciones relacionadas y el marco teórico sobre modelos didácticos. Finalmente, propone una metodología para construir el modelo didáctico basada en principios de aprendizaje activo y el uso de ecu

Financial Markets with Stochastic Volatilities - markov modelling

The document summarizes the research of Anatoliy Swishchuk on stochastic volatility models and their applications in financial mathematics. Specifically, it discusses:
1. Random evolutions (REs), which are abstract dynamical systems with random components that can model stochastic processes.
2. Applications of REs, including modeling traffic, storage, risk, and biological processes. In finance, REs can model markets with stochastic volatility.
3. Pricing of derivatives like variance swaps, volatility swaps, and swing options under stochastic volatility models like Heston. Numerical examples are provided based on S&P60 Canada index data.

Calculo diferencial e_integral_en_la_vida_cotidiana (2)

El documento discute las aplicaciones del cálculo diferencial e integral en la vida cotidiana y profesional. Explica cómo se usa el cálculo diferencial para analizar gastos variables, velocidad y aceleración. También cómo el cálculo integral se aplica en áreas como geometría, física, economía y biología para calcular momentos de inercia, trabajo y calor. Finalmente, proporciona ejemplos del uso de integrales en máquinas simples y vigas curvas.

Calculo diferencial e integral

Este documento presenta problemas de cálculo diferencial e integral resueltos por profesores de la Universidad Autónoma Metropolitana. Está dividido en dos partes, la primera parte contiene problemas de cálculo diferencial e integral I y evaluaciones resueltas por los profesores Cutberto Romero y José Becerril. La segunda parte contiene problemas de cálculo diferencial e integral II y evaluaciones resueltas por los profesores Judith Omaña y Cutberto Romero. Finalmente, incluye una miscelánea de problemas de aplicación presentada por el profes

Calculo integral

El cálculo integral se utiliza para calcular áreas y volúmenes y es útil en ingeniería ambiental para determinar el caudal de ríos. También se emplea en estadística para calcular funciones de probabilidad y en administración para minimizar costos.

Solucionario de matematicas de g. mancill.

Este documento es el segundo volumen de una obra llamada Mancil. Trata sobre temas relacionados con la historia y cultura de una región. Contiene información detallada sobre eventos pasados y costumbres tradicionales de la zona.

Fractales y Teoría del Caos

El documento presenta información sobre fractales y la teoría del caos. Explica conceptos clave como dimensión, recursividad y autosemejanza para entender los fractales. También describe tipos de fractales como lineales, generados por iteración de funciones y caóticos. Finalmente, menciona algunas aplicaciones prácticas de los fractales en campos como la medicina, la música, la arquitectura y la computación.

Ejercicios algebra superior hall y knight

El libro tiene las soluciones del libro ALGEBRA SUPERIOR de Hall y Knight Edit. UTEHA
El libro se usa en Algebra I de 1er semestre de la Facultad

Algebra Elemental Moderna

Este documento no contiene información relevante. Consiste en una serie de líneas en blanco sin texto.

Algebra proschle

The document discusses the history of chocolate production in Europe and the Americas. It details how chocolate was first cultivated and used by Mesoamerican cultures before being introduced to Europe in the 16th century. Cacao beans then became a popular commodity traded between European colonial powers and their colonies in places like West Africa, the Caribbean, and South America. Chocolate production has since expanded globally and become a multi-billion dollar industry.

Algebra arrayan

Este documento describe los detalles de un proyecto de construcción de una carretera. Explica que la carretera tendrá 6 carriles y medirá 50 kilómetros de largo. También incluirá 3 intercambiadores y se espera que cueste $200 millones de dólares. La construcción tomará aproximadamente 2 años y creará muchos puestos de trabajo.

t4_20110728_IGARSS11_tsutomuy.pdf

t4_20110728_IGARSS11_tsutomuy.pdf

Prigogine, i. tan solo una_ilusion

Prigogine, i. tan solo una_ilusion

Inference for stochastic differential equations via approximate Bayesian comp...

Inference for stochastic differential equations via approximate Bayesian comp...

Hirsch s.w., smale s. differential equations, dynamical systems and linear ...

Hirsch s.w., smale s. differential equations, dynamical systems and linear ...

Francisco

Francisco

Libro de Mancil

Libro de Mancil

Un Juego Diferencial Estocástico para Reaseguro

Un Juego Diferencial Estocástico para Reaseguro

Prigogine esayo

Prigogine esayo

Calculo diferencial e integral2

Calculo diferencial e integral2

MODELACIÓN MATEMÁTICA A TRAVÉS DE LAS ECUACIONES EN DIFERENCIA

MODELACIÓN MATEMÁTICA A TRAVÉS DE LAS ECUACIONES EN DIFERENCIA

Financial Markets with Stochastic Volatilities - markov modelling

Financial Markets with Stochastic Volatilities - markov modelling

Calculo diferencial e_integral_en_la_vida_cotidiana (2)

Calculo diferencial e_integral_en_la_vida_cotidiana (2)

Calculo diferencial e integral

Calculo diferencial e integral

Calculo integral

Calculo integral

Solucionario de matematicas de g. mancill.

Solucionario de matematicas de g. mancill.

Fractales y Teoría del Caos

Fractales y Teoría del Caos

Ejercicios algebra superior hall y knight

Ejercicios algebra superior hall y knight

Algebra Elemental Moderna

Algebra Elemental Moderna

Algebra proschle

Algebra proschle

Algebra arrayan

Algebra arrayan

Koh_Liang_ICML2017

1) The paper introduces the influence function for interpreting black-box machine learning models. The influence function traces a model's predictions back to the training data by examining how the model's parameters would change if a particular training point was removed or perturbed.
2) The influence function approximates this change in parameters by assuming a quadratic approximation to the empirical risk function around the learned parameters and taking a single Newton step. It shows the parameter change due to removing a point is approximated by the influence function.
3) The paper demonstrates how the influence function can be used to understand model behavior, find adversarial examples, debug issues, and correct errors, among other applications. It also proposes practical methods to compute the influence function for

Robot, Learning From Data

Robot, Learning from Data
1. Direct Policy Learning in RKHS with learning theory
2. Inverse Reinforcement Learning Methods
Sungjoon Choi (sungjoon.choi@cpslab.snu.ac.kr)

Generalised Statistical Convergence For Double Sequences

Recently, the concept of 𝛽-statistical Convergence was introduced considering a sequence of infinite
matrices 𝛽 = (𝑏𝑛𝑘 𝑖 ). Later, it was used to define and study 𝛽-statistical limit point, 𝛽-statistical cluster point,
𝑠𝑡𝛽 − 𝑙𝑖𝑚𝑖𝑡 inferior and 𝑠𝑡𝛽 − 𝑙𝑖𝑚𝑖𝑡 superior. In this paper we analogously define and study 2𝛽-statistical
limit, 2𝛽-statistical cluster point, 𝑠𝑡2𝛽 − 𝑙𝑖𝑚𝑖𝑡 inferior and 𝑠𝑡2𝛽 − 𝑙𝑖𝑚𝑖𝑡 superior for double sequences.

Boundness of a neural network weights using the notion of a limit of a sequence

feed forward neural network with backpropagation le
arning algorithm is considered as a black box
learning classifier since there is no certain inter
pretation or anticipation of the behavior of a neur
al
network weights. The weights of a neural network ar
e considered as the learning tool of the classifier
, and
the learning task is performed by the repetition mo
dification of those weights. This modification is
performed using the delta rule which is mainly used
in the gradient descent technique. In this article
a
proof is provided that helps to understand and expl
ain the behavior of the weights in a feed forward n
eural
network with backpropagation learning algorithm. Al
so, it illustrates why a feed forward neural networ
k is
not always guaranteed to converge in a global minim
um. Moreover, the proof shows that the weights in t
he
neural network are upper bounded (i.e. they do not
approach infinity). Data Mining, Delta

STLtalk about statistical analysis and its application

The document provides an introduction to statistical learning theory. It describes the supervised learning setting, where the goal is to select a predictor from a set of candidates that minimizes the expected loss on new data, given training data, candidates, and a loss function. It discusses how empirical risk minimization (ERM), such as by minimizing error on the training set, can approximate this goal. One sufficient condition for ERM consistency is uniform convergence of the empirical risk to the true risk as more data is observed.

MM - KBAC: Using mixed models to adjust for population structure in a rare-va...

Confounding from population structure, extended families and inbreeding can be a significant issue for burden and kernel association tests on rare variants from next generation DNA sequencing. An obvious solution is to combine the power of a mixed model regression analysis with the ability to assess the rare variant burden using methods such as KBAC or CMC. Recent approaches have adjusted burden and kernel tests using linear regression models; this method adjusts for the relatedness of samples and includes that directly into a logistic regression model.
This webcast will focus on the details of bringing Mixed Model Regression and KBAC together, including: deriving an optimal logistic mixed model algorithm for calculating the reduced model score, how the kinship or random effects matrix should be specified, and how it all comes together into one algorithm. Results from applying the method to variants from the 1000 Genomes project will also be presented and compared to famSKAT.

Machine Learning 1

This document provides an overview of neural networks and machine learning concepts. It discusses how neural networks mimic the brain and simulate networks of neurons. It then covers perceptrons and their limitations in solving XOR problems. Next, it introduces multi-layer neural networks, backpropagation for training networks, and regularization to address overfitting. Key concepts are explained through examples, including computing gradients, error minimization, and determining optimal hidden unit numbers.

Mncs 16-09-4주-변승규-introduction to the machine learning

Introduction to the machine learning
Reference. Machine Learning: The art and science of algorithms that make sense of data

Symbolic Computation via Gröbner Basis

The purpose of this paper is to find the orthogonal projection of a rational parametric curve onto a rational parametric surface in 3-space. We show that the orthogonal projection problem can be reduced to the problem of finding elimination ideals via Gröbnerbasis. We provide a computational algorithm to find the orthogonal projection, and include a few illustrative examples. The presented method is effective and potentially useful for many applications related to the design of surfaces and other industrial and research fields.

Intro to Quant Trading Strategies (Lecture 2 of 10)

This document provides an introduction to hidden Markov models for algorithmic trading strategies. It discusses key concepts like Bayes' theorem, Markov chains, and the Markov property. It then covers the three main problems in hidden Markov models: likelihood, decoding, and learning. It presents solutions to these problems, including the forward-backward, Viterbi, and Baum-Welch algorithms. It also discusses extensions to non-discrete distributions and trading ideas using hidden Markov models.

A machine learning method for efficient design optimization in nano-optics

The slideshow contains a brief explanation of Gaussian process regression and Bayesian optimization. For two optimization problems, benchmarks against other local gradient-based and global heuristic optimization methods are included. They show, that Bayesian optimization can identify better designs in exceptionally short computation times.

Anti-differentiating Approximation Algorithms: PageRank and MinCut

We study how Google's PageRank method relates to mincut and a particular type of electrical flow in a network. We also explain the details of how the "push method" for computing PageRank helps to accelerate it. This has implications for semi-supervised learning and machine learning, as well as social network analysis.

Machine learning ppt and presentation code

Principal Component Analysis (PCA) is a technique for dimensionality reduction that projects high-dimensional data onto a lower-dimensional space in a way that maximizes variance. It works by finding the directions (principal components) along which the variance of the data is highest. These principal components become the new axes of the reduced space. PCA involves computing the covariance matrix of the data, performing eigendecomposition on the covariance matrix to obtain its eigenvectors, and projecting the data onto the top K eigenvectors corresponding to the largest eigenvalues, where K is the target dimensionality. This projection both reduces dimensionality and maximizes retained variance.

Learning group em - 20171025 - copy

The EM algorithm is an iterative method to find maximum likelihood estimates of parameters in probabilistic models with latent variables. It has two steps: E-step, where expectations of the latent variables are computed based on current estimates, and M-step, where parameters are re-estimated to maximize the expected complete-data log-likelihood found in the E-step. As an example, the EM algorithm is applied to estimate the parameters of a Gaussian mixture model, where the latent variables indicate component membership of each data point.

07 Machine Learning - Expectation Maximization

This document provides an introduction to the Expectation Maximization (EM) algorithm. EM is used to estimate parameters in statistical models when data is incomplete or has missing values. It is a two-step process: 1) Expectation step (E-step), where the expected value of the log likelihood is computed using the current estimate of parameters; 2) Maximization step (M-step), where the parameters are re-estimated to maximize the expected log likelihood found in the E-step. EM is commonly used for problems like clustering with mixture models and hidden Markov models. Applications of EM discussed include clustering data using mixture of Gaussian distributions, and training hidden Markov models for natural language processing tasks. The derivation of the EM algorithm and

Ck31369376

The document describes the (G'/G)-expansion method for finding traveling wave solutions of nonlinear partial differential equations (PDEs) arising in mathematical physics. The method involves expressing the solution as a polynomial in (G'/G), where G satisfies a second order linear ordinary differential equation. The method is demonstrated by using it to find traveling wave solutions of the variable coefficients KdV (vcKdV) equation, the modified dispersive water wave (MDWW) equations, and the symmetrically coupled KdV equations. These solutions are expressed in terms of hyperbolic, trigonometric, and rational functions. The method provides a simple way to obtain exact solutions to important nonlinear PDEs.

Kernel Bayes Rule

1) Kernel Bayes' rule provides a nonparametric approach to Bayesian inference using positive definite kernels. It represents probabilities as elements in a reproducing kernel Hilbert space.
2) Using kernel mean embeddings, kernel Bayes' rule computes the posterior kernel mean directly from covariance operators without needing to compute integrals or approximations.
3) Given samples from the joint distribution and the prior kernel mean, kernel Bayes' rule computes the posterior kernel mean as a weighted sum of prior sample kernel embeddings, providing a nonparametric realization of Bayesian inference.

PRML Chapter 6

1) Gaussian processes provide a distribution over functions and can be used for regression and classification problems. They define a prior directly over functions, rather than parameters as in linear regression.
2) To make predictions in Gaussian processes, we compute the posterior distribution p(tN+1|tN) which depends on the kernel function K(x,x'). The predictive distribution has a closed form Gaussian distribution.
3) Kernel functions define the similarity between data points and should satisfy certain properties. Common kernels include the Gaussian and polynomial kernels. Kernel hyperparameters can be estimated through maximum likelihood.

A PROBABILISTIC ALGORITHM OF COMPUTING THE POLYNOMIAL GREATEST COMMON DIVISOR...

In the earlier work, subresultant algorithm was proposed to decrease the coefficient growth in the Euclidean algorithm of polynomials. However, the output polynomial remainders may have a small factor which can be removed to satisfy our needs. Then later, an improved subresultant algorithm was given by representing the subresultant algorithm in another way, where we add a variant called 𝜏 to express the small factor. There was a way to compute the variant proposed by Brown, who worked at IBM. Nevertheless, the way failed to determine each𝜏 correctly.

PRML Chapter 8

This document summarizes key concepts from Chapter 8 of the book "Pattern Recognition and Machine Learning" regarding probabilistic graphical models. It introduces directed and undirected graphical models as visualization tools for probabilistic relationships between random variables. It provides examples of Bayesian networks and conditional independence. Key points covered include using graphs to factorize joint probabilities, the d-separation criteria for identifying conditional independence based on a graph, and applying these concepts to linear Gaussian models and discrete variable models.

Koh_Liang_ICML2017

Koh_Liang_ICML2017

Robot, Learning From Data

Robot, Learning From Data

Generalised Statistical Convergence For Double Sequences

Generalised Statistical Convergence For Double Sequences

Boundness of a neural network weights using the notion of a limit of a sequence

Boundness of a neural network weights using the notion of a limit of a sequence

STLtalk about statistical analysis and its application

STLtalk about statistical analysis and its application

MM - KBAC: Using mixed models to adjust for population structure in a rare-va...

MM - KBAC: Using mixed models to adjust for population structure in a rare-va...

Machine Learning 1

Machine Learning 1

Mncs 16-09-4주-변승규-introduction to the machine learning

Mncs 16-09-4주-변승규-introduction to the machine learning

Symbolic Computation via Gröbner Basis

Symbolic Computation via Gröbner Basis

Intro to Quant Trading Strategies (Lecture 2 of 10)

Intro to Quant Trading Strategies (Lecture 2 of 10)

A machine learning method for efficient design optimization in nano-optics

A machine learning method for efficient design optimization in nano-optics

Anti-differentiating Approximation Algorithms: PageRank and MinCut

Anti-differentiating Approximation Algorithms: PageRank and MinCut

Machine learning ppt and presentation code

Machine learning ppt and presentation code

Learning group em - 20171025 - copy

Learning group em - 20171025 - copy

07 Machine Learning - Expectation Maximization

07 Machine Learning - Expectation Maximization

Ck31369376

Ck31369376

Kernel Bayes Rule

Kernel Bayes Rule

PRML Chapter 6

PRML Chapter 6

A PROBABILISTIC ALGORITHM OF COMPUTING THE POLYNOMIAL GREATEST COMMON DIVISOR...

A PROBABILISTIC ALGORITHM OF COMPUTING THE POLYNOMIAL GREATEST COMMON DIVISOR...

PRML Chapter 8

PRML Chapter 8

Streamlining Python Development: A Guide to a Modern Project Setup

Designed for beginners, this presentation demystifies Python project management using Hatch and delves into pyproject.toml for efficient configuration. We'll guide you through organizing directories, implementing unit testing for code reliability, and using mypy for type checking to enhance code quality. The session concludes with insights into ruff, a modern linter for maintaining Python standards, which is replacing black, isort, flake8. This talk is a comprehensive toolkit for anyone eager to learn and apply the latest practices in Python development.
The talk was given at PyConDE / PyData Berlin 2024. More details here: https://pretalx.com/pyconde-pydata-2024/talk/CBVTEG/

Unlocking the Power of Integer Programming

In this presentation, you will be introduced to the concept of Integer Programming and its application in conference scheduling. We will delve into the fundamentals of Integer Programming and its practical utilization in optimizing the allocation of talks to specific time slots and rooms within a conference program. By the conclusion of the talk, attendees will gain a clearer comprehension of the potential of this powerful tool in creating a conference schedule that is both efficient and effective, ultimately maximizing attendee satisfaction. Whether you are involved in conference organization or simply curious about optimization algorithms, this presentation is tailored to meet your interests.

WALD: A Modern & Sustainable Analytics Stack

This document discusses the WALD stack, a modern and sustainable analytics stack combining Snowflake, Airbyte, dbt, Lightdash, and optionally Streamlit. It provides an example of analyzing Formula 1 racing data using these tools, with Airbyte ingesting data into Snowflake, dbt for transformations, Lightdash for visualization, and dbt with Snowpark for machine learning predictions. The speaker argues the WALD stack is flexible for many use cases from analytics to full data products at scale.

Forget about AI and do Mathematical Modelling instead!

- The document discusses the limitations of artificial intelligence and machine learning approaches like deep learning, such as a lack of interpretability, robustness, and ability to generalize.
- It promotes mathematical modelling as an alternative that allows incorporating domain knowledge, providing more interpretable and trustworthy models, and gaining insights from data.
- Mathematical modelling involves translating real-world problems into mathematical formulations and using algorithms designed specifically for the problem to provide useful insights and guidance.

An Interpretable Model for Collaborative Filtering Using an Extended Latent D...

With the increasing use of AI and ML-based systems, interpretability is becoming an increasingly important issue to ensure user trust and safety. This also applies to the area of recommender systems, where methods based on matrix factorization (MF) are among the most popular methods for collaborative filtering tasks with implicit feedback. Despite their simplicity, the latent factors of users and items lack interpretability in the case of the effective, unconstrained MF-based methods. In this work, we propose an extended latent Dirichlet Allocation model (LDAext) that has interpretable parameters such as user cohorts of item preferences and the affiliation of a user with different cohorts. We prove a theorem on how to transform the factors of an unconstrained MF model into the parameters of LDAext. Using this theoretical connection, we train an MF model on different real-world data sets, transform the latent factors into the parameters of LDAext and test their interpretation in several experiments for plausibility. Our experiments confirm the interpretability of the transformed parameters and thus demonstrate the usefulness of our proposed approach.

Honey I Shrunk the Target Variable! Common pitfalls when transforming the tar...

The talk addresses the consequences of transforming the target variable on a conceptual but also a mathematical level. Still, the emphasis is on conveying the notion behind the interplay of your chosen error measure and the transformation of your target variable, so that you get some practical gain from it. Thus, everything will also be demonstrated on some use-case using a Jupyter notebook.

Matrix Factorization for Collaborative Filtering Is Just Solving an Adjoint L...

Matrix factorization-based methods are among the most popular methods for collaborative filtering tasks with implicit feedback. The most effective of these methods do not apply sign constraints, such as non-negativity, to their factors. Despite their simplicity, the latent factors for users and items lack interpretability, which is becoming an increasingly important requirement. In this work, we provide a theoretical link between unconstrained and the interpretable non-negative matrix factorization in terms of the personalized ranking induced by these methods. We also introduce a novel, latent Dirichlet allocation-inspired model for recommenders and extend our theoretical link to also allow the interpretation of an unconstrained matrix factorization as an adjoint formulation of our new model. Our experiments indicate that this novel approach represents the unknown processes of implicit user-item interactions in the real world much better than unconstrained matrix factorization while being interpretable.
This talk was presented at 15th ACM Conference on Recommender Systems in Amsterdam (RecSys 2021). Find more information under https://dl.acm.org/doi/fullHtml/10.1145/3460231.3474266

Uncertainty Quantification in AI

With the advent of Deep Learning (DL), the field of AI made a giant leap forward and it is nowadays applied in many industrial use-cases. Especially critical systems like autonomous driving, require that DL methods not only produce a prediction but also state the certainty about the prediction in order to assess risks and failure.
In my talk, I will give an introduction to different kinds of uncertainty, i.e. epistemic and aleatoric. To have a baseline for comparison, the classical method of Gaussian Processes for regression problems is presented. I then elaborate on different DL methods for uncertainty quantification like Quantile Regression, Monte-Carlo Dropout, and Deep Ensembles. The talk is concluded with a comparison of these techniques to Gaussian Processes and the current state of the art.

Performance evaluation of GANs in a semisupervised OCR use case

This document discusses using generative adversarial networks (GANs) for a semi-supervised optical character recognition (OCR) use case involving vehicle identification numbers (VINs). It describes the text spotting pipeline, challenges with limited training data, data augmentation techniques, and implementing a GAN for character detection. Evaluation shows the semi-supervised GAN approach outperforms other methods, achieving over 99% accuracy on VIN detection and recognition from images using only 85 labeled examples. Key learnings include that custom solutions can outperform off-the-shelf tools for specific tasks, and GANs are well-suited for problems with limited labeled data when combined with data augmentation.

Bridging the Gap: from Data Science to Production

A recent but quite common observation in industry is that although there is an overall high adoption of data science, many companies struggle to get it into production. Huge teams of well-payed data scientists often present one fancy model after the other to their managers but their proof of concepts never manifest into something business relevant. The frustration grows on both sides, managers and data scientists.
In my talk I elaborate on the many reasons why data science to production is such a hard nut to crack. I start with a taxonomy of data use cases in order to easier assess technical requirements. Based thereon, my focus lies on overcoming the two-language-problem which is Python/R loved by data scientists vs. the enterprise-established Java/Scala. From my project experiences I present three different solutions, namely 1) migrating to a single language, 2) reimplementation and 3) usage of a framework. The advantages and disadvantages of each approach is presented and general advices based on the introduced taxonomy is given.
Additionally, my talk also addresses organisational as well as problems in quality assurance and deployment. Best practices and further references are presented on a high-level in order to cover all facets of data science to production.
With my talk I hope to convey the message that breakdowns on the road from data science to production are rather the rule than the exception, so you are not alone. At the end of my talk, you will have a better understanding of why your team and you are struggling and what to do about it.

How mobile.de brings Data Science to Production for a Personalized Web Experi...

Here are some tips to help you find the right car:
1. Narrow down your top 3 models based on your needs and budget. Research reviews.
2. Set up alerts on mobile.de for your top models within your search radius.
3. Schedule test drives for your top choices on the weekends to compare. Bring a mechanic if needed.
4. Negotiate the best price by getting competing offers from different dealerships in writing.
5. Consider financing options before making an offer to get the best rate.
6. Have the vehicle inspected by your mechanic before purchase.
Let me know if you have any other questions! I'm here to help you through the process.

Deep Learning-based Recommendations for Germany's Biggest Vehicle Marketplace

As presented at the Düsseldorf Data Science Meetup on March, 12th, the talk covers business as well as technical aspects of recommender systems based on deep learning. It is an extended version of the talk held at Bitkom A.I. Summit 2018 with the same title and covers more technical details in depth.

Deep Learning-based Recommendations for Germany's Biggest Online Vehicle Mark...

At mobile.de, Germany’s biggest car marketplace, a dedicated data team, supported by the IT project house inovex, is responsible for creating smart data products. One focus are personalised vehicle recommendations to improve the customer experience during browsing as well as finding the perfect offering.
As an introduction, we briefly mention the traditional approaches for recommendation engines, thereby motivating the need for more sophisticated approaches. We then illustrate how Deep Learning can be leveraged to capture the underlying non-linear correlations of features for personalised recommendations. In particular, we’ve customised Google Play’s algorithm for an online marketplace with a fast-changing inventory. Several variants of our adapted approach are evaluated against traditional methods as well as scalability aspects are addressed.
We conclude our talk by giving an outlook on the importance of personalised user experiences and the application of Deep Learning and AI at mobile.de.

Declarative Thinking and Programming

Declarative Programming is a programming paradigm that focuses on describing what should be computed in a problem domain without describing how it should be done. The talk starts by explaining differences between a declarative and imperative approach with the help of examples from everyday life. Having established a clear notion of declarative programming as well as pointed out some advantages, we transfer these concepts to programming in general. For example, the usage of control flow statements like loops over-determine the order of computation which impedes scalable execution as well as it often violates the single level of abstraction principle.

Which car fits my life? - PyData Berlin 2017

As Germany’s largest online vehicle marketplace mobile.de uses recommendations at scale to help users find the perfect car. We elaborate on collaborative & content-based filtering as well as a hybrid approach addressing the problem of a fast-changing inventory. We then dive into the technical implementation of the recommendation engine, outlining the various challenges faced and experiences made.

PyData Meetup Berlin 2017-04-19

In the field of machine learning and particularly in supervised learning, correlation is key in order to predict the target variable with the help of feature variables. Rarely do we think about causation and the actual effect of a single feature variable or covariate on the target or response respectively. Some even go so far saying that “correlation trumps causation” like in the book “Big Data: A Revolution That Will Transform How We Live, Work, and Think” by Viktor Mayer-Schönberger and Kenneth Cukier. Following their reasoning with Big Data there is no need anymore to think about causation since nonparametric models will do just fine using only correlation. For many practical use-cases this point of view seems to be acceptable but surely not for all. In my talk I will present the theory of causal inference and demonstrate it's application with the help of inverse probability of treatment weighting (IPTW) which is a propensity score method on a practical use-case.

Streamlining Python Development: A Guide to a Modern Project Setup

Streamlining Python Development: A Guide to a Modern Project Setup

Unlocking the Power of Integer Programming

Unlocking the Power of Integer Programming

WALD: A Modern & Sustainable Analytics Stack

WALD: A Modern & Sustainable Analytics Stack

Forget about AI and do Mathematical Modelling instead!

Forget about AI and do Mathematical Modelling instead!

An Interpretable Model for Collaborative Filtering Using an Extended Latent D...

An Interpretable Model for Collaborative Filtering Using an Extended Latent D...

Honey I Shrunk the Target Variable! Common pitfalls when transforming the tar...

Honey I Shrunk the Target Variable! Common pitfalls when transforming the tar...

Matrix Factorization for Collaborative Filtering Is Just Solving an Adjoint L...

Matrix Factorization for Collaborative Filtering Is Just Solving an Adjoint L...

Uncertainty Quantification in AI

Uncertainty Quantification in AI

Performance evaluation of GANs in a semisupervised OCR use case

Performance evaluation of GANs in a semisupervised OCR use case

Bridging the Gap: from Data Science to Production

Bridging the Gap: from Data Science to Production

How mobile.de brings Data Science to Production for a Personalized Web Experi...

How mobile.de brings Data Science to Production for a Personalized Web Experi...

Deep Learning-based Recommendations for Germany's Biggest Vehicle Marketplace

Deep Learning-based Recommendations for Germany's Biggest Vehicle Marketplace

Deep Learning-based Recommendations for Germany's Biggest Online Vehicle Mark...

Deep Learning-based Recommendations for Germany's Biggest Online Vehicle Mark...

Declarative Thinking and Programming

Declarative Thinking and Programming

Which car fits my life? - PyData Berlin 2017

Which car fits my life? - PyData Berlin 2017

PyData Meetup Berlin 2017-04-19

PyData Meetup Berlin 2017-04-19

Willis Tower //Sears Tower- Supertall Building .pdf

Sears Tower was the last supertall building constructed during the
International architecture period, and SOM's interpretation of the style is
remarkably bold and awe-inspiring.

OpenMetadata Spotlight - OpenMetadata @ Aspire by Vinol Joy Dsouza

The OpenMetadata Community Meeting was held on July 10th, 2024. In the Community Spotlight, Vinol Joy Dsouza from Aspire (https://aspireapp.com/) spoke about their journey with OpenMetadata: What problems they were trying to solve, how they chose OpenMetadata, and how they integrated it to manage more than 6.000 Data Quality tests!

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the potential of the development of the Ford–Fulkerson algorithm to solve the...

strengths, weaknesses, opportunities and threats of the Ford – Fulkerson algorithm to solve the problem

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Biometric Question Bank 2021 - 1 Soln-1.pdf

This contains questions for biometric

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Potential Uses of the Floyd-Warshall Algorithm as appropriate

Seeing the Floyd-Warshall Algorithm's point of view from strengths, weaknesses, opportunities and threats in various potentials

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the unexpected potential of Dijkstra's Algorithm

This explanation is related to the impact and solutions that will be provided by the algorithm

DataScienceConcept_Kanchana_Weerasinghe.pptx

I've developed a comprehensive handbook covering the core concepts of data science and the mathematics behind them. It's a culmination of my experience and knowledge in the field. I hope this will be a great starting point for beginners

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Data Preprocessing Cheatsheet for learners

Data pre processing in very simple and easy steps

Why_are_we_hypnotizing_ourselves-_ATeggin-1.pdf

I’m excited to finally share my research from last year on the hypnotic effects of mass media and digital platformization. This study explores how our attention is influenced through YouTube’s audio-visual content. Key points:
- **Objective:** Examine the hypnotic side effects of media on attention.
- **Focus:** Sound and visual experiences on YouTube.
- **Methodology:** Mixed digital approach with quantitative and qualitative analysis.
- **Findings:** Observations on techniques in attention-based economies and their cognitive impact.
- **Implications:** Considerations for future research in media and mind interactions, especially within OSINT-oriented communities.
Curious about the details? Check out my slide deck and let’s discuss the future possibilities.
#Research #AttentionEconomy #YouTube #DigitalMedia #MediaStudies #VisualNetworkAnalysis #HypnodelicMedia

Willis Tower //Sears Tower- Supertall Building .pdf

Willis Tower //Sears Tower- Supertall Building .pdf

OpenMetadata Spotlight - OpenMetadata @ Aspire by Vinol Joy Dsouza

OpenMetadata Spotlight - OpenMetadata @ Aspire by Vinol Joy Dsouza

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the unexpected potential of Dijkstra's Algorithm

the unexpected potential of Dijkstra's Algorithm

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Why_are_we_hypnotizing_ourselves-_ATeggin-1.pdf

Why_are_we_hypnotizing_ourselves-_ATeggin-1.pdf

- 1. Dr. Florian Wilhelm March 13th 2016 PyData Amsterdam P 1
- 2. 2 What‘s the best model to describe our data? And what does „best“ actually mean?
- 3. 3
- 4. 4
- 5. 5 Simple model „Generality“ Complex model „Best Fit“ Occam‘s Razor: „It is vain to do with more what can be done with fewer“
- 6. 6 Simple Model ℋ1 Complex model ℋ2 Space of all possible datasets 𝐷
- 7. 7 Simple Model ℋ1 Complex model ℋ2 ℋ1 fits only a small subset of 𝐷 well
- 8. 8 Simple Model ℋ1 Complex model ℋ2 ℋ2 can fit large parts of 𝐷 well
- 9. 9 Prefer the model with high evidence for a given dataset Source: D. J. C. MacKay. Bayesian Interpolation. 1992
- 10. 1. Model fitting: Assume ℋ𝑖 is the right model and fit its parameters 𝒘 with Bayes: 𝑃 𝒘 𝐷, ℋ𝑖 = 𝑃 𝐷 𝒘, ℋ𝑖 𝑃(𝒘|ℋ𝑖) 𝑃(𝐷|ℋ𝑖) “Business as usual” 2. Model comparison: Compare different models with the help of their evidence 𝑃 𝐷 ℋ𝑖 and model prior 𝑃 ℋ𝑖 : 𝑃 ℋ𝑖 𝐷 ∝ 𝑃 𝐷 ℋ𝑖 𝑃 ℋ𝑖 “Occam‘s razor at work“ 10
- 11. Marginalize & approximate: 𝑃 𝐷 ℋ𝑖 = 𝑃 𝐷 𝒘, ℋ𝑖 𝑃 𝒘 ℋ𝑖 𝑑𝒘 𝑃 𝐷 ℋ𝑖 ≅ 𝑃 𝐷 𝒘 𝑀𝑃, ℋ𝑖 𝑃 𝒘 𝑀𝑃 ℋ𝑖 ∆𝒘 𝑒𝑣𝑖𝑑𝑒𝑛𝑐𝑒 ≅ 𝑏𝑒𝑠𝑡 𝑓𝑖𝑡 𝑙𝑖𝑘𝑒𝑙𝑖ℎ𝑜𝑜𝑑 × 𝑂𝑐𝑐𝑎𝑚 𝑓𝑎𝑐𝑡𝑜𝑟 11 Occam factor: ∆𝒘 ∆ 𝟎 𝒘 Source: D. J. C. MacKay. Bayesian Interpolation. 1992
- 12. 12
- 13. Given: Dataset 𝐷 = 𝑥 𝑛, 𝑡 𝑛 with 𝑛 = 1 … 𝑁 Set of (non-linear) functions Φ = {𝜙ℎ: 𝑥 ⟼ 𝜙(𝑥)} with ℎ = 1 … 𝑀 Assumption: 𝑦 𝒙; 𝒘 = ℎ=1 𝑀 𝑤ℎ 𝜙ℎ(𝒙) , 𝑡 𝑛 = 𝑦 𝒙; 𝒘 + 𝜐 𝑛, where 𝜐 𝑛 is an additive noise with 𝒩 0, 𝛼−1 Task: Find min 𝒘 ‖Φ𝒘 − 𝒕‖2 (Ordinary Least Squares) 13
- 14. 14 Problem: Having too many features leads to overfitting! Regularization Assumption: „Weights are small“ 𝑝 𝒘; 𝜆 ~𝒩(0, 𝜆−1 𝕀) Task: Given 𝛼, 𝜆 find min 𝒘 𝛼 Φ𝒘 − 𝒕 2 + 𝜆 𝒘 2
- 15. 15 Consider each 𝛼𝑖, 𝜆𝑖 defining a model ℋ𝑖 𝛼, 𝜆 . Yes! That means we can use our Bayesian Interpolation to find 𝒘, 𝜶, 𝝀 with the highest evidence! This is the idea behind BayesianRidge as found in sklearn.linear_model
- 16. Consider that each weight has an individual variance, so that 𝑝 𝒘 𝝀 ~𝒩 0, Λ−1 , where Λ = diag(𝜆1, … , 𝜆 𝐻), 𝜆ℎ ∈ ℝ+. Now, our minimization problem is: min 𝒘 𝛼 Φ𝒘 − 𝒕 2 + 𝒘 𝑡Λ𝒘 16 Pruning: If precision 𝜆ℎ of feature ℎ is high, its weight 𝑤ℎ is very likely to be close to zero and is therefore pruned. This is called Sparse Bayesian Learning or Automatic Relevance Determination. Found as ARDRegression under sklearn.linear_model.
- 17. Crossvalidation can be used for the estimation of hyperparmeters but suffers from the curse of dimensionality (inappropriate for low-statistics). 17 Source: Peter Ellerton, http://pactiss.org/2011/11/02/bayesian-inference-homo-bayesianis/
- 18. • Random 100 × 100 design matrix Φ with 100 samples and 100 features • Weights 𝑤𝑖, 𝑖 ∈ 𝐼 = 1, … , 100 , random subset J ⊂ 𝐼 with 𝐽 = 10, and 𝑤𝑖 = 0, 𝑖 ∈ 𝐼J 𝒩(𝑤𝑖; 0, 1 4), 𝑖 ∈ 𝐽 • Target 𝒕 = Φ𝒘 + 𝝂 with random noise 𝜈𝑖 ∼ 𝒩(0, 1 50) Task: Reconstruct the weights, especially the 10 non-zero weights! Source: http://scikit-learn.org/stable/auto_examples/linear_model/plot_ard.html#example-linear-model-plot-ard-py 18
- 19. 19
- 20. 20
- 21. 21
- 22. 22
- 23. We have to determine the parameters 𝑤, 𝜆, 𝛼 for 𝑃 𝒘, 𝝀, 𝛼 𝒕 = 𝑃 𝒘 𝒕, 𝝀, 𝛼 𝑃 𝝀, 𝛼 𝒕 1) Model fitting: For the first factor, we have 𝑃 𝒘 𝒕, 𝝀, 𝛼 ~𝒩(𝝁, Σ) with Σ = Λ + 𝛼Φ 𝑇 Φ −1 , 𝝁 = 𝛼ΣΦT 𝐭. 23
- 24. 2) Model comparison: For the second factor, we have 𝑃 𝝀, 𝛼 𝒕 ∝ 𝑃 𝒕 𝝀, 𝛼 𝑃 𝝀 𝑃 𝛼 , where 𝑃 𝝀 and 𝑃(𝛼) are hyperpriors which we assume uniform. Using marginalization, we have 𝑃 𝒕 𝝀, 𝛼 = 𝑃 𝒕 𝒘, 𝛼 𝑃 𝒘 𝝀 𝑑𝒘, i.e. marginal likelihood or the “evidence for the hyperparameter“. 24
- 25. Differentiation of the log marginal likelihood with respect to 𝜆𝑖 and 𝛼 as well as setting these to zero, we get 𝜆𝑖 = 𝛾𝑖 𝜇𝑖 2 , 𝛼 = 𝑁 − 𝑖 𝛾𝑖 𝒕 − Φ𝝁 2 , with 𝛾𝑖 = 1 − 𝜆𝑖Σ𝑖𝑖. These formulae are used to find the maximum points 𝝀 𝑀𝑃 and 𝛼 𝑀𝑃. 25
- 26. 1. Starting values 𝛼 = 𝜎−2(𝒕), 𝝀 = 𝟏 2. Calculate Σ = Λ + 𝛼Φ 𝑇Φ −1 and 𝒘 = 𝝁 = 𝛼ΣΦT 𝐭 3. Update 𝜆𝑖 = 𝛾 𝑖 𝜇 𝑖 2 and 𝛼 = 𝑁− 𝑖 𝛾 𝑖 𝒕−Φ𝝁 2 where 𝛾𝑖 = 1 − 𝜆𝑖Σ𝑖𝑖 4. Prune 𝜆𝑖 and 𝜙𝑖 if 𝜆𝑖 > 𝜆 𝑡ℎ𝑟𝑒𝑠ℎ𝑜𝑙𝑑 5. If not converged go to 2. Sklearn implementation: The parameters 𝛼1, 𝛼2 as well as 𝜆1, 𝜆2 are the hyperprior parameters for 𝛼 and 𝝀 with 𝑃 𝛼 ∼ Γ 𝛼1, 𝛼2 −1 , 𝑃 𝜆𝑖 ∼ Γ 𝜆1, 𝜆2 −1 . 𝐸 Γ 𝛼, 𝛽 = 𝛼 𝛽 and 𝑉 Γ 𝛼, 𝛽 = 𝛼 𝛽2. 26
- 27. Given a some new data 𝑥∗, a prediction for 𝑡∗ is made by 𝑃 𝑡∗ 𝒕, 𝝀 𝑀𝑃, 𝛼 𝑀𝑃 = 𝑃 𝑡∗ 𝒘, 𝛼 𝑀𝑃 𝑃 𝒘 𝒕, 𝝀 𝑀𝑃, 𝛼 𝑀𝑃 𝑑𝒘 = 𝒩 𝝁 𝑇 𝜙 𝑥∗ , 𝛼 𝑀𝑃 −1 + 𝜙 𝑥∗ 𝑡Σ𝜙 𝑥∗ . This is a good approximation of the predictive distribution 𝑃 𝑡∗ 𝒕 = 𝑃 𝑡∗ 𝒘, 𝝀, 𝛼 𝑃 𝒘, 𝝀, 𝛼 𝒕 𝑑𝒘 𝑑𝝀 𝑑α . 27
- 28. 1. D. J. C. MacKay. Bayesian Interpolation. 1992 (… to understand the overall idea) 2. M. E.Tipping. Sparse Bayesian learning and the RelevanceVector Machine. June, 2001 (… to understand the ARD algorithm) 3. T. Fletcher. RelevanceVector Machines Explained. October, 2010 (… to understand the ARD algorithm in detail) 4. D.Wipf. A NewView of Automatic Relevance Determination. 2008 (… not as good as the ones above) Graphs from slides 7 and 9 were taken from [1] and the awesome tutorials of Scikit-Learn were consulted many times. 28
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