Search is not available for this dataset
query
stringlengths 1
13.4k
| pos
stringlengths 1
61k
| neg
stringlengths 1
63.9k
| query_lang
stringclasses 147
values | __index_level_0__
int64 0
3.11M
|
|---|---|---|---|---|
Neural-Symbolic Learning and Reasoning: Contributions and Challenges
|
Extracting Tree-Structured Representations of Trained Networks
|
Sparse Coding From a Bayesian Perspective
|
eng_Latn
| 34,200 |
On Contrastive Divergence Learning
|
probabilistic inference using markov chain monte carlo methods .
|
Prostaglandin F2α promotes ovulation in prepubertal heifers
|
eng_Latn
| 34,201 |
Wrapper Induction for Information Extraction
|
A theory of the learnable
|
How to construct random functions
|
eng_Latn
| 34,202 |
Predicting student success based on prior performance
|
Dynamic Bayesian Networks: Representation, Inference and Learning
|
A proposal to evaluate ontology content
|
eng_Latn
| 34,203 |
On learning causal models from relational data
|
7 probabilistic entity - relationship models , prms , and plate models .
|
Object-Oriented Bayesian Networks
|
eng_Latn
| 34,204 |
Mixed membership stochastic blockmodels
|
Gene Ontology: tool for the unification of biology
|
Mean Field Theory for Sigmoid Belief Networks
|
eng_Latn
| 34,205 |
A differential approach to inference in Bayesian networks
|
A Logical Approach to Factoring Belief Networks
|
Exploiting Causal Independence in Bayesian Network Inference
|
eng_Latn
| 34,206 |
Loopy Belief Propagation for Approximate Inference: An Empirical Study
|
A Tractable Inference Algorithm for Diagnosing Multiple Diseases
|
Effectiveness of Green Infrastructure for Improvement of Air Quality in Urban Street Canyons
|
kor_Hang
| 34,207 |
Bayesian Estimation of Beta Mixture Models with Variational Inference
|
Pattern recognition and machine learning
|
Pathogen effectors target Arabidopsis EDS1 and alter its interactions with immune regulators.
|
eng_Latn
| 34,208 |
Robust Calibration of Financial Models Using Bayesian Estimators
|
Pricing with a Smile
|
Comparing parallel performance of Go and C++ TBB on a direct acyclic task graph using a dynamic programming problem
|
eng_Latn
| 34,209 |
Bayesian Inference for a Covariance Matrix
|
The No-U-Turn Sampler: Adaptively Setting Path Lengths in Hamiltonian Monte Carlo
|
Two Is Bigger (and Better) Than One: the Wikipedia Bitaxonomy Project
|
eng_Latn
| 34,210 |
Bayesian regression and Bitcoin
|
A Latent Source Model for Nonparametric Time Series Classification
|
Dissecting Robotics - historical overview and future perspectives
|
eng_Latn
| 34,211 |
Optimal Proposal Distributions and Adaptive MCMC
|
Markov chains for exploring posterior distributions
|
Enabling the implementation of evidence based practice: a conceptual framework
|
eng_Latn
| 34,212 |
Variational Inference for Beta-Bernoulli Dirichlet Process Mixture Models
|
Variational inference for Dirichlet process mixtures
|
Evolutionary Theory of Cancer
|
eng_Latn
| 34,213 |
Beating the Bookies : Predicting the Outcome of Soccer Games
|
Analysis of sports data by using bivariate Poisson models
|
Learning Bayesian networks: The combination of knowledge and statistical data
|
eng_Latn
| 34,214 |
Learning physical parameters from dynamic scenes
|
An Introduction to Variational Methods for Graphical Models
|
probabilistic inference using markov chain monte carlo methods .
|
eng_Latn
| 34,215 |
Matchbox: large scale online bayesian recommendations
|
a family of algorithms for approximate bayesian inference .
|
Modeling primary school pre-service teachers' Technological Pedagogical Content Knowledge (TPACK) for meaningful learning with information and communication technology (ICT)
|
eng_Latn
| 34,216 |
Understanding Probabilistic Sparse Gaussian Process Approximations
|
A unifying view of sparse approximate Gaussian process regression
|
JBOSS: The Evolution of Professional Open Source Software
|
eng_Latn
| 34,217 |
Expectation Particle Belief Propagation
|
smoothing algorithms for state – space models .
|
Augmented Wnt Signaling in a Mammalian Model of Accelerated Aging
|
eng_Latn
| 34,218 |
Bayesian visual analytics: BaVA
|
Probabilistic Principal Component Analysis
|
Nonlinear Component Analysis as a Kernel Eigenvalue Problem
|
eng_Latn
| 34,219 |
Machine Learning Methods for Data-Driven Turbulence Modeling
|
Gaussian Processes for Machine Learning (GPML) Toolbox
|
Effectiveness of web-based social sensing in health information dissemination-A review
|
eng_Latn
| 34,220 |
Variational Bayesian Inference for Big Data Marketing Models 1
|
Pattern recognition and machine learning
|
Genetic dissection of mammalian fertility pathways
|
kor_Hang
| 34,221 |
Church: a language for generative models
|
Markov logic networks
|
A Bayesian Analysis of Some Nonparametric Problems
|
eng_Latn
| 34,222 |
Survey of Crime Analysis and Prediction.
|
A novel serial crime prediction model based on Bayesian learning theory
|
Predicting Serial Killers' Home Base Using a Decision Support System
|
kor_Hang
| 34,223 |
Hierarchical Bayesian Neural Networks for Personalized Classification
|
Weight Uncertainty in Neural Network
|
A Survey of Parallel Programming Models and Tools in the Multi and Many-Core Era
|
eng_Latn
| 34,224 |
Bayesian Grammar Learning for Inverse Procedural Modeling
|
Building reconstruction using manhattan-world grammars
|
GCD Computation of n Integers
|
eng_Latn
| 34,225 |
Online inference for time-varying temporal dependency discovery from time series
|
a bayesian networks approach for predicting protein - protein interactions from genomic data .
|
Learning graphical model structure using L1-regularization paths
|
eng_Latn
| 34,226 |
Exact Inference Techniques for the Dynamic Analysis of Bayesian Attack Graphs
|
Reverend Bayes on inference engines: a distributed hierarchical approach
|
The linkage of allergic rhinitis and obstructive sleep apnea.
|
kor_Hang
| 34,227 |
Bayesian recommender systems : models and algorithms
|
Advances in Prospect Theory: Cumulative Representation of Uncertainty
|
Comparison of continuous thoracic epidural analgesia with bilateral erector spinae plane block for perioperative pain management in cardiac surgery
|
eng_Latn
| 34,228 |
Boosting Variational Inference
|
two problems with variational expectation maximisation for time - series models .
|
Backward Simulation in Bayesian Networks
|
eng_Latn
| 34,229 |
Describing Visual Scenes using Transformed Dirichlet Processes
|
Bayesian Density Estimation and Inference Using Mixtures
|
Single-Fed Low Profile Broadband Circularly Polarized Stacked Patch Antenna
|
eng_Latn
| 34,230 |
Lifted Tree-Reweighted Variational Inference
|
Lifted probabilistic inference with counting formulas
|
Cumulative Attribute Space for Age and Crowd Density Estimation
|
eng_Latn
| 34,231 |
Modeling Human Understanding of Complex Intentional Action with a Bayesian Nonparametric Subgoal Model
|
Adaptor Grammars: A Framework for Specifying Compositional Nonparametric Bayesian Models
|
Surgical management of tricuspid stenosis
|
eng_Latn
| 34,232 |
A Tutorial on Learning With Bayesian Networks
|
Learning Bayesian Networks with Discrete Variables from Data
|
Continuous-Time Trajectory Estimation for Event-based Vision Sensors
|
eng_Latn
| 34,233 |
Learning from time series: Supervised Aggregative Feature Extraction
|
Bayesian Online Multitask Learning of Gaussian Processes
|
Predictive control of fractional state-space model
|
eng_Latn
| 34,234 |
Learning Optimal Bayesian Networks: A Shortest Path Perspective
|
The max-min hill-climbing Bayesian network structure learning algorithm
|
Pathwise coordinate optimization
|
kor_Hang
| 34,235 |
On Approximate Inference for Generalized Gaussian Process Models
|
Bayesian classification with Gaussian processes
|
A Scaled Conjugate Gradient Algorithm for Fast Supervised Learning
|
eng_Latn
| 34,236 |
A Tutorial on Learning With Bayesian Networks
|
Learning Bayesian Networks with Discrete Variables from Data
|
Squirrel: scatter hoarding VM image contents on IaaS compute nodes
|
eng_Latn
| 34,237 |
Graphical models for probabilistic and causal reasoning
|
Counterfactual Probabilities: Computational Methods, Bounds and Applications
|
Bayesian Optimization in a Billion Dimensions via Random Embeddings
|
eng_Latn
| 34,238 |
Batched Large-scale Bayesian Optimization in High-dimensional Spaces
|
Entropy Search for Information-Efficient Global Optimization
|
Normalization of EMG Signals: To Normalize or Not to Normalize and What to Normalize to?
|
eng_Latn
| 34,239 |
Bayesian optimization with robust Bayesian neural networks
|
Deep Residual Learning for Image Recognition
|
The Variational Gaussian Approximation Revisited
|
eng_Latn
| 34,240 |
the time has come bayesian methods for data analysis in the organizational sciences .
|
What to believe : Bayesian methods for data analysis
|
Minimizing air consumption of pneumatic actuators in mobile robots
|
eng_Latn
| 34,241 |
Idealised Bayesian Neural Networks Cannot Have Adversarial Examples: Theoretical and Empirical Study
|
Are Generative Classifiers More Robust to Adversarial Attacks?
|
Circuits for an RF cochlea
|
eng_Latn
| 34,242 |
Bayesian network classifiers. an application to remote sensing image classification
|
Expert Systems and Probabilistic Network Models
|
Collaborative Filtering with User-Item Co-Autoregressive Models
|
eng_Latn
| 34,243 |
Bayesian Grammar Learning for Inverse Procedural Modeling
|
Irregular lattices for complex shape grammar facade parsing
|
Learning an Integrated Distance Metric for Comparing Structure of Complex Networks
|
eng_Latn
| 34,244 |
Temporal Rules Discovery for Web Data Cleaning
|
A Bayesian Approach to Discovering Truth from Conflicting Sources for Data Integration
|
Parallel evolution of virulence in pathogenic Escherichia coli
|
kor_Hang
| 34,245 |
Model Selection Based on the Variational Bayes
|
SMEM Algorithm for Mixture Models
|
Probabilistic Principal Component Analysis
|
eng_Latn
| 34,246 |
A constrained parameter evolutionary learning algorithm for Bayesian network under incomplete and small data
|
Operations for Learning with Graphical Models
|
Blocking and Binding Folate Receptor Alpha Autoantibodies Identify Novel Autism Spectrum Disorder Subgroups
|
eng_Latn
| 34,247 |
Building fast Bayesian computing machines out of intentionally stochastic, digital parts
|
Deep Boltzmann machines
|
graphical evolutionary game for information diffusion over social networks .
|
eng_Latn
| 34,248 |
Slice Sampling for Probabilistic Programming
|
Exploring Network Structure, Dynamics, and Function using NetworkX
|
Compositional Vector Space Models for Knowledge Base Inference.
|
eng_Latn
| 34,249 |
Markov logic networks with numerical constraints
|
introduction to statistical relational learning ( adaptive computation and machine learning ) .
|
Markov logic networks
|
eng_Latn
| 34,250 |
a convenient category for higher - order probability theory .
|
Church: a language for generative models
|
Particle Markov chain Monte Carlo methods
|
eng_Latn
| 34,251 |
Discretizing Continuous Attributes While Learning Bayesian Networks
|
autoclass : a bayesian classification system .
|
On the handling of continuous-valued attributes in decision tree generation
|
eng_Latn
| 34,252 |
Learning a manifold of fonts
|
Probabilistic Non-linear Principal Component Analysis with Gaussian Process Latent Variable Models
|
Micro-crowdfunding : achieving a sustainable society through economic and social incentives in micro-level crowdfunding
|
eng_Latn
| 34,253 |
BoostMap: A method for efficient approximate similarity rankings
|
An Index Structure for Data Mining and Clustering
|
Gaussian mixture models for affordance learning using Bayesian Networks
|
eng_Latn
| 34,254 |
Deep Unsupervised Learning using Nonequilibrium Thermodynamics
|
Stochastic Backpropagation and Approximate Inference in Deep Generative Models
|
Probabilistic reasoning in intelligent systems: Networks of plausible inference
|
eng_Latn
| 34,255 |
Riemann manifold Langevin and Hamiltonian Monte Carlo methods
|
Bayesian Learning via Stochastic Dynamics
|
Incentivizing Blockchain Forks via Whale Transactions
|
eng_Latn
| 34,256 |
Trajectory analysis and semantic region modeling using a nonparametric Bayesian model
|
A Bayesian Analysis of Some Nonparametric Problems
|
Rule discovery from time series
|
eng_Latn
| 34,257 |
Fast Nonparametric Clustering of Structured Time-Series
|
A Split-Merge Markov Chain Monte Carlo Procedure for the Dirichlet Process Mixture Model
|
the case for banning killer robots : point .
|
eng_Latn
| 34,258 |
Predicting sports events from past results Towards effective betting on football matches
|
Predicting and Retrospective Analysis of Soccer Matches in a League
|
Learning Bayesian networks: The combination of knowledge and statistical data
|
eng_Latn
| 34,259 |
Tighter Linear Program Relaxations for High Order Graphical Models
|
Probabilistic reasoning in intelligent systems: Networks of plausible inference
|
deep learning and its application in silent sound technology .
|
eng_Latn
| 34,260 |
Split Hamiltonian Monte Carlo
|
Equation of state calculations by fast computing machines
|
A tutorial on adaptive MCMC
|
eng_Latn
| 34,261 |
On the Optimality of the Simple Bayesian Classifier under Zero-One Loss
|
Syskill & Webert: Identifying Interesting Web Sites
|
Characteristics in information processing approaches
|
eng_Latn
| 34,262 |
Lossless Data Compression
|
A Bayesian Analysis of Some Nonparametric Problems
|
Scrap your boilerplate: a practical design pattern for generic programming
|
kor_Hang
| 34,263 |
Deep Knowledge Tracing and Dynamic Student Classification for Knowledge Tracing
|
More Accurate Student Modeling Through Contextual Estimation of Slip and Guess Probabilities in Bayesian Knowledge Tracing
|
A Survey of Wireless Path Loss Prediction and Coverage Mapping Methods
|
eng_Latn
| 34,264 |
Disintegration and Bayesian Inversion, Both Abstractly and Concretely
|
Causal Theories: A Categorical Perspective on Bayesian Networks
|
Categories for the working mathematician: making the impossible possible
|
eng_Latn
| 34,265 |
Reconstructing constructivism: Causal models, Bayesian learning mechanisms, and the theory theory.
|
probabilistic models in human sensorimotor control .
|
Feature Selection for Maximizing the Area Under the ROC Curve
|
eng_Latn
| 34,266 |
BayesOpt: A Bayesian Optimization Library for Nonlinear Optimization, Experimental Design and Bandits
|
The BOBYQA algorithm for bound constrained optimization without derivatives
|
Interpretable Graph Convolutional Neural Networks for Inference on Noisy Knowledge Graphs
|
eng_Latn
| 34,267 |
learning bayesian networks from data : an information theory based approach .
|
Artificial intelligence: A modern approach
|
Residual Policy Learning
|
eng_Latn
| 34,268 |
Coverage directed test generation for functional verification using Bayesian networks
|
Learning Dynamic Bayesian Networks
|
High-Performance 2D Rhenium Disulfide (ReS2) Transistors and Photodetectors by Oxygen Plasma Treatment
|
eng_Latn
| 34,269 |
Labeled directed acyclic graphs: a generalization of context-specific independence in directed graphical models
|
Exploiting Contextual Independence In Probabilistic Inference
|
Beyond"How may I help you?": Assisting Customer Service Agents with Proactive Responses
|
eng_Latn
| 34,270 |
Model Selection Based on the Variational Bayes
|
A Practical Bayesian Framework for Backpropagation Networks
|
Ubii: Towards Seamless Interaction between Digital and Physical Worlds
|
eng_Latn
| 34,271 |
A probabilistic model for component-based shape synthesis
|
Bayesian Classification (AutoClass): Theory and Results
|
Understanding Web Archiving Services and Their (Mis)Use on Social Media
|
eng_Latn
| 34,272 |
Probable networks and plausible predictions - a review of practical Bayesian methods for supervised neural networks
|
Fast Exact Multiplication by the Hessian
|
Subcarrier-index modulation OFDM
|
eng_Latn
| 34,273 |
Bayesian Nonparametric Poisson Factorization for Recommendation Systems
|
Stochastic Variational Inference
|
color transfer between images .
|
eng_Latn
| 34,274 |
On the application of stochastic control in population management
|
The problem of population management of herd animals is discussed. A discrete stochastic model is described and can be successfully optimized by the use of modern control theory.
|
We introduce a latent process model for time series of attributed random graphs for characterizing multiple modes of association among a collection of actors over time. Two mathematically tractable approximations are derived, and we examine the performance of a class of test statistics for an illustrative change-point detection problem and demonstrate that the analysis through approximation can provide valuable information regarding inference properties.
|
eng_Latn
| 34,275 |
I have taken a handful of statistics/ data science-oriented courses. And to this day, I feel like I grasp the underlying concept but I do not comprehend the in-depth understanding. I am asking this because it's been more of a formulaic relationship, and since it is so applicable and important, I was wondering if anybody had any readings or mental breakthroughs to what Bayes Theorem is!
|
I've been trying to develop an intuition based understanding of Bayes' theorem in terms of the prior, posterior, likelihood and marginal probability. For that I use the following equation: $$P(B|A) = \frac{P(A|B)P(B)}{P(A)}$$ where $A$ represents a hypothesis or belief and $B$ represents data or evidence. I've understood the concept of the posterior - it's a unifying entity that combines the prior belief and the likelihood of an event. What I don't understand is what does the likelihood signify? And why is the marginal probability in the denominator? After reviewing a couple of resources I came across this quote: The likelihood is the weight of event $B$ given by the occurrence of $A$ ... $P(B|A)$ is the posterior probability of event $B$ , given that event $A$ has occurred. The above 2 statements seem identical to me, just written in different ways. Can anyone please explain the difference between the two?
|
The entire site is blank right now. The header and footer are shown, but no questions.
|
eng_Latn
| 34,276 |
Bishop: EM algorithm, expectation step I am reading the Bishop Pattern Recognition and Machine Learning and am on section 9.3 An Alternative View on EM: I am confused as to how Bishop obtains the expression $Q(\theta, \theta^{old}) = \sum_{Z}p(Z | C, \theta^{old}) lnp(X , Z | \theta)$ Thank you for insight! ANSWER: In the method of maximum likelihood, we're generally interested in p(X|θ), or equivalently logp(X|θ). We seek a θ that makes this quantity most likely. However, in our model, in addition to observations X we have latent variable(s) Z which we haven't observed but we have some idea of what it could be (we've elsewhere described a probability distribution guessing Z). So, we're now interested in p(X,Z|θ), or equivalently logp(X,Z|θ), instead of just choosing a θ to maximize p(X|θ). Yet it's hard to find the θ that maximizes logp(X,Z|θ) if we don't even know what Z is. Instead, we seek the θ that maximizes logp(X,Z|θ) weighted over our distribution of what Z might be.
|
Derivation of E step in EM algorithm While im going through the derivation of E step in EM algorithm for pLSA, i came across the following derivation . Could anyone explain me how the following step is derived. $\sum_z q(z) log \frac{P(X|z,\theta)P(z|\theta)}{q(z)} = \sum_z q(z) log \frac{P(z|X,\theta)P(X,\theta)}{q(z)} $
|
Finding the maximum likelihood estimator (Theoretical statistics) Let $X_1, X_2, ..X_n$ represent a random sample from this pdf: $$f(x|\theta)= \frac{3x^2}{\theta^3}, 0\leq x\leq \theta$$ (with 0 elsewhere) Could someone explain how I would find the maximum likelihood estimator (MLE) of this pdf? I know I have to get the likelihood function L, then the log likelihood function, then differentiate, but I am getting lost in the calculations. Thanks!
|
eng_Latn
| 34,277 |
I've been studying Bayesian Statistics lately, and just came across the Metropolis-Hastings Algorithm. I understand that the goal is to sample from an intractable posterior - but I'm not really able to understand how the algorithm achieves what it sets out to achieve. Why and how does it work? What's the intuition behind the algorithm? To clarify the parts I've problems with, in particular, I've attached the algorithm above. How is the $q$ distribution (the proposal) related to the intractable posterior? I don't see how $q$ popped out of nowhere. Why is the acceptance ratio calculated the way it is? It doesn't make intuitive sense to me - it'd be great if someone could explain that better. In Step 3, we accept the $X$ we sampled from the $q$ distribution with some probability - why is that? How does that get me something closer to the intractable posterior, which is our goal? (right?) Please help me out here. Thanks!
|
Maybe the concept, why it's used, and an example.
|
The new Top-Bar does not show reputation changes from Area 51.
|
eng_Latn
| 34,278 |
Book Bayesian statistics I write here to ask for a suggestion about a graduate level Bayesian statistics book. I have a bachelor degree in statistics but despite having a fairly solid background on frequentist and non parametric statistics, I do not know much about Bayesian statistics. In particular I am undecided between these two books: Doing Bayesian Data Analysis Statistical Rethinking Which one is right for me? I know R very well, so the presence of examples on R is an added value.
|
What is the best introductory Bayesian statistics textbook? Which is the best introductory textbook for Bayesian statistics? One book per answer, please.
|
Why is it necessary to sample from the posterior distribution if we already KNOW the posterior distribution? My understanding is that when using a Bayesian approach to estimate parameter values: The posterior distribution is the combination of the prior distribution and the likelihood distribution. We simulate this by generating a sample from the posterior distribution (e.g., using a Metropolis-Hasting algorithm to generate values, and accept them if they are above a certain threshold of probability to belong to the posterior distribution). Once we have generated this sample, we use it to approximate the posterior distribution, and things like its mean. But, I feel like I must be misunderstanding something. It sounds like we have a posterior distribution and then sample from it, and then use that sample as an approximation of the posterior distribution. But if we have the posterior distribution to begin with why do we need to sample from it to approximate it?
|
eng_Latn
| 34,279 |
Analytical solution to marginal likelihood/normalizing constant I have a sample with data and I would like to compare the result I get from Monte Carlo integration and by analytically computing the marginal likelihood. Now I know that in many cases this is a challenging computation, but in this case apparently it is possible. I have the code for doing it, but I don't understand how it is done, and the code doesn't make me wiser. So, let $Y \sim N(\theta,1)$ and the prior on $\theta \sim N(0,1)$. How would I analytically compute the marginal likelihood $p(y)$ $$p(y) = \int p(y|\theta)\,p(\theta)\, d\theta$$
|
How to compute the marginal likelihood given two multivariate Normal distribution? Given that $a$~$\mathcal{N}(\mu, \Sigma)$ and $\mu$~$\mathcal{N}(0, \sigma^2 I)$, Is $p(a|\Sigma, \sigma)=\int p(a|\mu, \Sigma)p(\mu|\sigma)d\mu$ again a Normal distribution? And how to estimate the mean and variance respectively? Thanks a lot!
|
Discouraged by simple Bayesian Question My university is testing an intro Machine Learning (ML) course for us undergrads, and having been interested in ML since the beginning to understand what it was, I jumped at the chance to take it. But being someone who has not been back in school for a long time and much busier than I was in my first years long ago, I am finding that I need more info and examples than the class in its current form has. I am struggling to figure out what the attached question is asking. I am searching online for help with this particular topic, but I am also finding that everyone seems to have their own way of phrasing similar questions in such a way that I am finding it hard to link the info. My understanding from stats way back when is that $p(x|\omega_1)$ is stating the probability of $x$ given $\omega_1$ for $0\le x \le 2$ and zero other wise. Same obviously if given $\omega_2$. If I understand, the priors are the probability of $\omega_1$ and $\omega_2$ which are somehow obtained. And honestly that is as far as I am really comprehending. I am not sure what (a) and (b) are truly asking for or how to get them. If anyone is able to help me with some pointed info on the topic it would be greatly appreciated.
|
eng_Latn
| 34,280 |
Over time I've learned that many (most?) methods used in classical statistics can be interpreted as evaluating a Bayesian model in some plausible way while I find the standard explanations much less intuitive. So I was wondering whether there are any resources which explain standard stats methods in a Bayesian manner?
|
I'm a simple minded Bayesian who feels comfortable in the cosy world of Bayes. However, due to malevolent forces outside my control, I now have to do introductory graduate courses about the exotic and weird world of frequentist statistics. Some of these concepts seem very weird to me, and my teachers are not versed in Bayes, so I thought I'd get some help on the internet from those who understand both. How would you explain the different concepts in frequentist statistics to a Bayesian who finds frequentism weird and uncomfortable? For example, some things I already understand: The maximum likelihood estimator $\text{argmax}_\theta \;p(D|\theta)$ is equal to the maximum posterior estimator $\text{argmax}_\theta \;p(\theta |D)$, if $p(\theta)$ is flat. (not entirely sure about this one). If a certain estimator $\hat \theta$ is a sufficient statistic for a parameter $\theta$, and $p(\theta)$ is flat, then $p(\hat \theta|\theta)=c_1\cdot p(D|\theta)=c_1\cdot c_2\cdot p(\theta|D)$, i.e. the sampling distribution is equal to the likelihood function, and therefore equal to the posterior of the parameter given a flat prior. Those are examples of explaining frequentist concepts to someone who understands Bayesian ones. How would you similarly explain the other central concepts of frequentist statistics in terms a Bayesian can understand? Specifically, I'm interested in the following questions: What is the role of Mean Square Error? How does it relate to Bayesian loss functions? How does the criterion of "unbiasedness" relate to Bayesian criteria? I know that a Bayesian will not demand that its estimators are unbiased, but at the same time, a Bayesian would probably agree that an unbiased frequentist estimator is generally more desirable than a biased frequentist one (even though he would consider both to be inferior to the Bayesian estimator). So how does a Bayesian understand unbiasedness? If we have flat priors, do frequentist confidence intervals somehow coincide with Bayesian ones? What in the name of Laplace is going on with specification tests like the $F$ test? Is this some degenerate special case of a Bayesian update on the distribution over model space? More generally: Is there some resource that explains frequentism to Bayesians? Most of the books run the other way around: they explain Bayesianism to people who are experienced in frequentist statistics. ps. I have looked, and while there are a lot of questions already about the difference between Bayesian and Frequentism, none explicitly explain Frequentism from the perspective of a Bayesian. is related, but is not specifically about explaining Frequentist concepts to a Bayesian (more about justifying frequentist thinking in general). Also, my point is not to bash frequentism. I really do want to understand it better
|
Please use UK pre-uni methods only (at least at first). Thank you.
|
eng_Latn
| 34,281 |
I know there are a lot of questions here about ignoring the denominator in a Bayesian approach, but I don't think mine is a duplicate of any of them. I am reading the book "Pattern recognition and machine learning" by Cristopher Bishop. Imagine we have a set of N observations of a (single) variable, which we collect in a vector $\mathbf{x} \in \mathcal{R}^N$. We would like to find the mean $\mu$ of the probbility density function that generated that data, using a Bayesian approach. Thus, we first need to find the posterior probability $p(\mu|\mathbf{x})$ We can write: $p(\mu|\mathbf{x}) = p(\mathbf{x}|\mu) \cdot \dfrac{p(\mu)}{p(\mathbf{x})}$ Now as the book says, we can ignore the denominator because it is just a normalizing factor $p(\mu|\mathbf{x}) \propto p(\mathbf{x}|\mu) \cdot p(\mu) = p(\mathbf{x}, \mu) $ where the last equation follows from the product rule, or the defition of conditional density for $p(\mathbf{x}|\mu)$ if you want. So we are approximating a conditional distribution with a joint distribution? How is that even possible? For one, $p(\mu|\mathbf{x})$ whould be a function of $\mathbf{x}$, while $p(\mathbf{x}, \mu) $ whould be a function of both $\mathbf{x}$ and $\mu$, right?
|
When I studied Bayesian statistics, a question about the notation of Bayes' Theorem came to my mind. Below is the density function version of Bayes' Theorem, where $y$ is data vector and $\theta$ is the parameter vector: $$ p(\theta|y)=\frac{p(y|\theta)p(\theta)}{p(y)} $$ The numerator on the right handside can be written as: $$ p(y,\theta) $$ which is the joint probability distribution of $y$ and $\theta$, then Bayes' theorem could be written as: $$ p(\theta|y)=\frac{p(y,\theta)}{p(y)} $$ Furthermore, $$ p(\theta|y)\propto p(y,\theta) $$ Am I right on this? I think it does not look right. Because the posterior is the proportional to the joint density function. But where is the mistake?
|
Prove that there is no retraction (i.e. continuous function constant on the codomain) $r: M \rightarrow S^1 = \partial M$ where $M$ is the Möbius strip. I've tried to find a contradiction using $r_*$ homomorphism between the fundamental groups, but they are both $\mathbb{Z}$ and nothing seems to go wrong...
|
eng_Latn
| 34,282 |
Definition of Likelihood in Bayesian Statistics Can the likelihood be defined as the probability of the rate parameter given a range of data. Or as the probability of the data, given a range of rate parameters?
|
Is there any difference between Frequentist and Bayesian on the definition of Likelihood? Some sources say likelihood function is not conditional probability, some say it is. This is very confusing to me. According to most sources I have seen, the likelihood of a distribution with parameter $\theta$, should be a product of probability mass functions given $n$ samples of $x_i$: $$L(\theta) = L(x_1,x_2,...,x_n;\theta) = \prod_{i=1}^n p(x_i;\theta)$$ For example in Logistic Regression, we use an optimization algorithm to maximize the likelihood function (Maximum Likelihood Estimation) to obtain the optimal parameters and therefore the final LR model. Given the $n$ training samples, which we assume to be independent from each other, we want to maximize the product of probabilities (or the joint probability mass functions). This seems quite obvious to me. According to , "likelihood is not a probability and it is not a conditional probability". It also mentioned, "likelihood is a conditional probability only in Bayesian understanding of likelihood, i.e., if you assume that $\theta$ is a random variable." I read about the different perspectives of treating a learning problem between frequentist and Bayesian. According to a source, for Bayesian inference, we have a priori $P(\theta)$, likelihood $P(X|\theta)$, and we want to obtain the posterior $P(\theta|X)$, using Bayesian theorem: $$P(\theta|X)=\dfrac{P(X|\theta) \times P(\theta)}{P(X)}$$ I'm not familiar with Bayesian Inference. How come $P(X|\theta)$ which is the distribution of the observed data conditional on its parameters, is also termed the likelihood? In , it says sometimes it is written $L(\theta|X)=p(X|\theta)$. What does this mean? is there a difference between Frequentist and Bayesian's definitions on likelihood?? Thanks. EDIT: There are different ways of interpreting Bayes' theorem - Bayesian interpretation and Frequentist interpretation (See: ).
|
Why are these estimates to the German tank problem different? Suppose that I observe $k=4$ tanks with serial numbers $2,6,7,14$. What is the best estimate for the total number of tanks $n$? I assume the observations are drawn from a discrete uniform distribution with the interval $[1,n]$. I know that for a $[0,1]$ interval the expected maximum draw $m$ for $k$ draws is $1 - (1/(1+k))$. So I estimate $\frac {k}{k+1}$$(n-1)≈$ $m$, rearranged so $n≈$ $\frac {k+1 }{k}$$m+1$. But the frequentist estimate from is defined as: $n ≈ m-1 + $$\frac {m}{k}$ I suspect there is some flaw in the way I have extrapolated from one interval to another, but I would welcome an explanation of why I have gone wrong!
|
eng_Latn
| 34,283 |
Can I use seq to go from 001 to 999? Can I use seq to go from 001 to 999?
|
How to create a sequence with leading zeroes using brace expansion When I use the following, I get a result as expected: $ echo {8..10} 8 9 10 How can I use this brace expansion in an easy way, to get the following output? $ echo {8..10} 08 09 10 I now that this may be obtained using seq (didn't try), but that is not what I am looking for. Useful info may be that I am restricted to this bash version. (If you have a zsh solution, but no bash solution, please share as well) $ bash -version GNU bash, version 3.2.51(1)-release (x86_64-suse-linux-gnu)
|
Sequential Update of Bayesian I am currently reading Murphy's ML: A Probabilistic Perspective. In CH 3 he explains that a batch update of the posterior is equivalent to a sequential update of the posterior, and I am trying to understand this in the context of his example. Suppose $D_a$ and $D_b$ are two data sets and $\theta$ is the parameter to our model. We are trying to update the posterior $P(\theta \mid D_a, D_b)$. In a sequential update, he states that, $$ (1) \ \ \ \ \ \ \ \ P(\theta \mid D_{a}, D_{b}) \propto P(D_b \mid \theta) P(\theta \mid D_a) $$ However, I am slightly confused as to how he got this mathematically. Conceptually, I understand that he is saying the posterior $P(\theta \mid D_a)$ is now a prior used to update the new posterior, which includes the new data $D_b$, and is multiplying this prior with the likelihood $P(D_b \mid \theta)$. Expanding the last statement out, I have, $$ P(D_b \mid \theta) P(\theta \mid D_a) = P(D_b \mid \theta) P(D_a \mid \theta) P(\theta) $$ but are we allowed to say $P(D_a \mid \theta) P(D_b \mid \theta) = P(D_a, D_b \mid \theta)$ in order to make the connection in (1)?
|
eng_Latn
| 34,284 |
Simple Bayesian classification has me discouraged My university is testing an intro Machine Learning (ML) course for us undergrads, and having been interested in ML since the beginning to understand what it was, I jumped at the chance to take it. But being someone who has not been back in school for a long time and much busier than I was in my first years long ago, I am finding that I need more info and examples than the class in its current form has. I am struggling to figure out what the attached question is asking. I am searching online for help with this particular topic, but I am also finding that everyone seems to have their own way of phrasing similar questions in such a way that I am finding it hard to link the info. My understanding from stats way back when is that p(x|ω1) p ( x | ω 1 ) is stating the probability of x x given ω1 ω 1 for 0≤x≤2 0 ≤ x ≤ 2 and zero other wise. Same obviously if given ω2 ω 2 . If I understand, the priors are the probability of ω1 ω 1 and ω2 ω 2 which are somehow obtained. And honestly that is as far as I am really comprehending. I am not sure what (a) and (b) are truly asking for or how to get them. If anyone is able to help me with some pointed info on the topic it would be greatly appreciated.
|
Discouraged by simple Bayesian Question My university is testing an intro Machine Learning (ML) course for us undergrads, and having been interested in ML since the beginning to understand what it was, I jumped at the chance to take it. But being someone who has not been back in school for a long time and much busier than I was in my first years long ago, I am finding that I need more info and examples than the class in its current form has. I am struggling to figure out what the attached question is asking. I am searching online for help with this particular topic, but I am also finding that everyone seems to have their own way of phrasing similar questions in such a way that I am finding it hard to link the info. My understanding from stats way back when is that $p(x|\omega_1)$ is stating the probability of $x$ given $\omega_1$ for $0\le x \le 2$ and zero other wise. Same obviously if given $\omega_2$. If I understand, the priors are the probability of $\omega_1$ and $\omega_2$ which are somehow obtained. And honestly that is as far as I am really comprehending. I am not sure what (a) and (b) are truly asking for or how to get them. If anyone is able to help me with some pointed info on the topic it would be greatly appreciated.
|
Meaning of probability notations $P(z;d,w)$ and $P(z|d,w)$ What is the difference in meaning between the notation $P(z;d,w)$ and $P(z|d,w)$ which are commonly used in many books and papers?
|
eng_Latn
| 34,285 |
Where can I find materials for an Advanced Statistics course? I'm attending a course in Advanced Statistics and I want to find materials and exercises (like the MIT Open Courses) which cover the syllabus. We're using the book Stochastic Modeling and Mathematical Statistics. The topics are: Review of discrete and continuous random variables. Random vectors. Transformations of random variables and of random vectors. Simulation of random variables. Strong laws of large numbers and the central limit theorem. Parametric statistical models. Parameter estimation: maximum likelihood and Bayesian methods. Hypothesis testing. Regression models.
|
Advanced statistics books recommendation There are several threads on this site for book recommendations on and but I am looking for a text on advanced statistics including, in order of priority: maximum likelihood, generalized linear models, principal component analysis, non-linear models. I've tried by A.C. Davison but frankly I had to put it down after 2 chapters. The text is encyclopedic in its coverage and mathematical treats but, as a practitioner, I like to approach subjects by understanding the intuition first, and then delve into the mathematical background. These are some texts that I consider outstanding for their pedagogical value. I would like to find an equivalent for the more advanced subjects I mentioned. , D. Freedman, R. Pisani, R. Purves. , R. Hyndman et al. , T. Z. Keith , Rand R. Wilcox , Gareth James, Daniela Witten, Trevor Hastie and Robert Tibshirani , Hastie, Tibshirani and Friedman (2009)
|
Why is it necessary to sample from the posterior distribution if we already KNOW the posterior distribution? My understanding is that when using a Bayesian approach to estimate parameter values: The posterior distribution is the combination of the prior distribution and the likelihood distribution. We simulate this by generating a sample from the posterior distribution (e.g., using a Metropolis-Hasting algorithm to generate values, and accept them if they are above a certain threshold of probability to belong to the posterior distribution). Once we have generated this sample, we use it to approximate the posterior distribution, and things like its mean. But, I feel like I must be misunderstanding something. It sounds like we have a posterior distribution and then sample from it, and then use that sample as an approximation of the posterior distribution. But if we have the posterior distribution to begin with why do we need to sample from it to approximate it?
|
eng_Latn
| 34,286 |
Why do we only look at the sample from MCMC? (And ignore the sample points' posterior probabilities?)
|
When approximating a posterior using MCMC, why don't we save the posterior probabilities but use the parameter value frequencies afterwards?
|
When approximating a posterior using MCMC, why don't we save the posterior probabilities but use the parameter value frequencies afterwards?
|
eng_Latn
| 34,287 |
What do we mean when we say that an approach is "Bayesian"? I'm trying to explain to a nontechnical colleague of mine what a Bayesian approach is. I realized that despite having used Bayesian methods on more than one occasion in the past, I don't have an intuitive definition of what makes an approach Bayesian or not. Based on several definitions I've seen in textbooks and online resources, the term "Bayesian" seems to mean: Choosing a prior model, and then updating this prior model with new empirical data to obtain an improved posterior model. This comes from applying Bayes rule to the context of modeling: $P(Model\: |\: Data) \propto P(Data\: |\: Model) \:P(Model)$ But then isn't this just the definition of supervised machine learning in general? What makes an approach specifically Bayesian and not just supervised learning? Or is it the case that all supervised learning really boils down to an application of Bayes rule?
|
Study path to Bayesian thinking? I am six years into a business role and have a bachelor's in physics and applied math/stats. Sean Carroll's (Caltech physicist) opened me to the idea that Bayesian statistics is one useful way of thinking about anything - inevitably you hold a prior and you should update your credence as additional information becomes available. Is there a path to training your intuition to thinking this way? Critically, it would require repeated practice with verifiable answers through either a course, or self study that includes many problems and solutions. I do not believe simply reading will do. Possible resources, having read every related question on this site I could find: "Probability Theory" by Jaynes. Pro: analytic; intuitive explanation of bayesian statistics. Con: prerequisites; missing problems/solutions. "Doing Bayesian Data Analysis" by Kruschke. Pro: includes problems & solutions; requires only "algebra and rusty calculus". Con: works in R, which I think provides for less intuitive learning than the analytical (I may be wrong). If it is a multi-year path I need to take, starting elsewhere, I am happy to do so! Ideally, I would avoid the frequentist methods, as I have no use for them. My goal is not to be a scientist, but to leverage insight into how reality works to go above and beyond the established thinking in business. Many thanks for any suggestions!
|
Interpreting the residuals vs. fitted values plot for verifying the assumptions of a linear model Consider the following figure from Faraway's Linear Models with R (2005, p. 59). The first plot seems to indicate that the residuals and the fitted values are uncorrelated, as they should be in a homoscedastic linear model with normally distributed errors. Therefore, the second and third plots, which seem to indicate dependency between the residuals and the fitted values, suggest a different model. But why does the second plot suggest, as Faraway notes, a heteroscedastic linear model, while the third plot suggest a non-linear model? The second plot seems to indicate that the absolute value of the residuals is strongly positively correlated with the fitted values, whereas no such trend is evident in the third plot. So if it were the case that, theoretically speaking, in a heteroscedastic linear model with normally distributed errors $$ \mbox{Cor}\left(\mathbf{e},\hat{\mathbf{y}}\right) = \left[\begin{array}{ccc}1 & \cdots & 1 \\ \vdots & \ddots & \vdots \\ 1 & \cdots & 1\end{array}\right] $$ (where the expression on the left is the variance-covariance matrix between the residuals and the fitted values) this would explain why the second and third plots agree with Faraway's interpretations. But is this the case? If not, how else can Faraway's interpretations of the second and third plots be justified? Also, why does the third plot necessarily indicate non-linearity? Isn't it possible that it is linear, but that the errors are either not normally distributed, or else that they are normally distributed, but do not center around zero?
|
eng_Latn
| 34,288 |
Masters before Math PHD? I'm currently finishing up my last year of study as an undergraduate mathematics major at a top 2 public school. I've been interested in getting a phD in mathematics for some time now, but my GPA and work in some classes in my 1st and 2nd years leaves a lot to be desired. Let's just say I got pretty bad grades in important classes. Since then, I've worked my but off and currently have around a 3.3 GPA. Luckily, I was accepted into a Statistics/Computer Science related Masters program. In this program, I will have the opportunity to take electives and plan to take some higher level math classes. My question is, will doing very well in my masters program next year override my poor performance as an undergrad? Also, should I be really trying to get a research position this summer?
|
Doing bad in undergraduate but good in a masters program Suppose you do bad in undergraduate school in say computer science. But you do very well in a masters program in computer science. If you want to apply to a PhD program in computer science, will the masters degree grades offset the undergraduate degree grades?
|
Discouraged by simple Bayesian Question My university is testing an intro Machine Learning (ML) course for us undergrads, and having been interested in ML since the beginning to understand what it was, I jumped at the chance to take it. But being someone who has not been back in school for a long time and much busier than I was in my first years long ago, I am finding that I need more info and examples than the class in its current form has. I am struggling to figure out what the attached question is asking. I am searching online for help with this particular topic, but I am also finding that everyone seems to have their own way of phrasing similar questions in such a way that I am finding it hard to link the info. My understanding from stats way back when is that $p(x|\omega_1)$ is stating the probability of $x$ given $\omega_1$ for $0\le x \le 2$ and zero other wise. Same obviously if given $\omega_2$. If I understand, the priors are the probability of $\omega_1$ and $\omega_2$ which are somehow obtained. And honestly that is as far as I am really comprehending. I am not sure what (a) and (b) are truly asking for or how to get them. If anyone is able to help me with some pointed info on the topic it would be greatly appreciated.
|
eng_Latn
| 34,289 |
Some bibliography of an introduction to Bayesian Data Analysis
|
Learn Bayesian inference applied to astronomy / astrophysics?
|
Apply empty style to the entire bibliography
|
kor_Hang
| 34,290 |
Consider the following "facts" about Bayes theorem and likelihood: Bayes theorem, written generically as $P(A|B) = \frac{ P(B|A) P(A) }{ P(B) }$ involves conditional and marginal probabilities. Focus on $P(B|A)$. Wiki Bayes theorem says this is a conditional probability (or conditional probability density in the continuous case). This seems quite clear in the alternate expression $P(B|A) P(A) = P(A|B) P(B)$. In the Bayes theorem, $P(B|A)$ is called the likelihood. The likelihood is $P(B|A)$ viewed as a function of $A$, not of $B$. It is not a conditional probability because it does not integrate to one. See , or Bishop Pattern Recognition & Machine Learing book p.22, "Note that the likelihood is not a probability distribution over w, and its integral with respect to w does not (necessarily) equal one." There is a problem here, one of these three facts must be wrong, or else I do not understand something. How can the likelihood in the Bayes theorem be a conditional probability, and also not a conditional probability? I am not sure (since I do not understand!), but perhaps an answer would be to explain how to view the Bayes equation in terms of what is variable and what is fixed, and how probabilities (and non-probabilities -- the likelihood) can combine in a "type consistent" way. For example, is it accurate to say that in the Bayes equation $P(A|B) = \cdots$ we should regard $B$ as fixed and $A$ as variable? And if $P(B|A)$, the likelihood, is not a conditional probability, then the form of Bayes is $$ \text{conditional probability} = \frac{ \text{other} \cdot \text{probability} }{ \text{probability} } $$ (or if dealing with continuous variables, $$ \text{conditional probability density} = \frac{ \text{other} \cdot \text{probability density} }{ \text{probability density} } $$ ) where "other" is the type of the likelihood (not a conditional probability density). Is there a rule that $$ \text{other} \cdot \text{probability} = \text{probability} $$ ? To me this seems wrong: multiplying a probability by a thing (likelihood) with arbitrarily large values will cause the result to not integrate to 1. Aside, I have tried to asked this question , but it was closed as a duplicate. However, I think the people deciding it was a duplicate did not understand the question, which is specifically about the likelihood appearing in Bayes theorem. I hope this version is more clear! Please read the whole question before marking it as a duplicate, thank you!
|
I have a simple question regarding "conditional probability" and "Likelihood". (I have already surveyed this question but to no avail.) It starts from the Wikipedia . They say this: The likelihood of a set of parameter values, $\theta$, given outcomes $x$, is equal to the probability of those observed outcomes given those parameter values, that is $$\mathcal{L}(\theta \mid x) = P(x \mid \theta)$$ Great! So in English, I read this as: "The likelihood of parameters equaling theta, given data X = x, (the left-hand-side), is equal to the probability of the data X being equal to x, given that the parameters are equal to theta". (Bold is mine for emphasis). However, no less than 3 lines later on the same page, the Wikipedia entry then goes on to say: Let $X$ be a random variable with a discrete probability distribution $p$ depending on a parameter $\theta$. Then the function $$\mathcal{L}(\theta \mid x) = p_\theta (x) = P_\theta (X=x), \, $$ considered as a function of $\theta$, is called the likelihood function (of $\theta$, given the outcome $x$ of the random variable $X$). Sometimes the probability of the value $x$ of $X$ for the parameter value $\theta$ is written as $P(X=x\mid\theta)$; often written as $P(X=x;\theta)$ to emphasize that this differs from $\mathcal{L}(\theta \mid x) $ which is not a conditional probability, because $\theta$ is a parameter and not a random variable. (Bold is mine for emphasis). So, in the first quote, we are literally told about a conditional probability of $P(x\mid\theta)$, but immediately afterwards, we are told that this is actually NOT a conditional probability, and should be in fact written as $P(X = x; \theta)$? So, which one is is? Does the likelihood actually connote a conditional probability ala the first quote? Or does it connote a simple probability ala the second quote? EDIT: Based on all the helpful and insightful answers I have received thus far, I have summarized my question - and my understanding thus far as so: In English, we say that: "The likelihood is a function of parameters, GIVEN the observed data." In math, we write it as: $L(\mathbf{\Theta}= \theta \mid \mathbf{X}=x)$. The likelihood is not a probability. The likelihood is not a probability distribution. The likelihood is not a probability mass. The likelihood is however, in English: "A product of probability distributions, (continuous case), or a product of probability masses, (discrete case), at where $\mathbf{X} = x$, and parameterized by $\mathbf{\Theta}= \theta$." In math, we then write it as such: $L(\mathbf{\Theta}= \theta \mid \mathbf{X}=x) = f(\mathbf{X}=x ; \mathbf{\Theta}= \theta) $ (continuous case, where $f$ is a PDF), and as $L(\mathbf{\Theta}= \theta \mid \mathbf{X}=x) = P(\mathbf{X}=x ; \mathbf{\Theta}= \theta) $ (discrete case, where $P$ is a probability mass). The takeaway here is that at no point here whatsoever is a conditional probability coming into play at all. In Bayes theorem, we have: $P(\mathbf{\Theta}= \theta \mid \mathbf{X}=x) = \frac{P(\mathbf{X}=x \mid \mathbf{\Theta}= \theta) \ P(\mathbf{\Theta}= \theta)}{P(\mathbf{X}=x)}$. Colloquially, we are told that "$P(\mathbf{X}=x \mid \mathbf{\Theta}= \theta)$ is a likelihood", however, this is not true, since $\mathbf{\Theta}$ might be an actual random variable. Therefore, what we can correctly say however, is that this term $P(\mathbf{X}=x \mid \mathbf{\Theta}= \theta)$ is simply "similar" to a likelihood. (?) [On this I am not sure.] EDIT II: Based on @amoebas answer, I have drawn his last comment. I think it's quite elucidating, and I think it clears up the main contention I was having. (Comments on the image). EDIT III: I extended @amoebas comments to the Bayesian case just now as well:
|
The new Top-Bar does not show reputation changes from Area 51.
|
eng_Latn
| 34,291 |
First, I am sorry if this is an obvious question, I am starting to study bayesian statistics (mainly for machine learning) and I was seeing the classic coin flip example using a Bernoulli distribution with parameter $q$. So I checked the math using an uniform prior ($P(q)=1$), and I was trying to see how the posterior would change after seeing for example $D=\{heads,tails\}$. So After computing the equations, I got that the posterior $P(q|D)=6(q-q^2)$. Then I did the same computation, but this time in 2 steps; that is, first I computed $P(q|D)$ but only with $D=\{heads\}$ which was equal to $2q$. Then I computed the posterior again with $D=\{tails\}$ but this time using the posterior that I just obtained ($P(q)=2q$). And then I obtained the same posterior as before. My question is, is this always the case if we assume iid samples or just works for this simple case? Is this how real life systems are programmed (using the previous posterior distribution as prior)? Thanks!
|
I am currently reading Murphy's ML: A Probabilistic Perspective. In CH 3 he explains that a batch update of the posterior is equivalent to a sequential update of the posterior, and I am trying to understand this in the context of his example. Suppose $D_a$ and $D_b$ are two data sets and $\theta$ is the parameter to our model. We are trying to update the posterior $P(\theta \mid D_a, D_b)$. In a sequential update, he states that, $$ (1) \ \ \ \ \ \ \ \ P(\theta \mid D_{a}, D_{b}) \propto P(D_b \mid \theta) P(\theta \mid D_a) $$ However, I am slightly confused as to how he got this mathematically. Conceptually, I understand that he is saying the posterior $P(\theta \mid D_a)$ is now a prior used to update the new posterior, which includes the new data $D_b$, and is multiplying this prior with the likelihood $P(D_b \mid \theta)$. Expanding the last statement out, I have, $$ P(D_b \mid \theta) P(\theta \mid D_a) = P(D_b \mid \theta) P(D_a \mid \theta) P(\theta) $$ but are we allowed to say $P(D_a \mid \theta) P(D_b \mid \theta) = P(D_a, D_b \mid \theta)$ in order to make the connection in (1)?
|
Prove that there are no integers $a,b \gt 2$ such that $a^2{\mid}(b^3 + 1)$ and $b^2{\mid}(a^3 + 1)$.
|
eng_Latn
| 34,292 |
Updating posterior distribution in online fashion First, I am sorry if this is an obvious question, I am starting to study bayesian statistics (mainly for machine learning) and I was seeing the classic coin flip example using a Bernoulli distribution with parameter $q$. So I checked the math using an uniform prior ($P(q)=1$), and I was trying to see how the posterior would change after seeing for example $D=\{heads,tails\}$. So After computing the equations, I got that the posterior $P(q|D)=6(q-q^2)$. Then I did the same computation, but this time in 2 steps; that is, first I computed $P(q|D)$ but only with $D=\{heads\}$ which was equal to $2q$. Then I computed the posterior again with $D=\{tails\}$ but this time using the posterior that I just obtained ($P(q)=2q$). And then I obtained the same posterior as before. My question is, is this always the case if we assume iid samples or just works for this simple case? Is this how real life systems are programmed (using the previous posterior distribution as prior)? Thanks!
|
Sequential Update of Bayesian I am currently reading Murphy's ML: A Probabilistic Perspective. In CH 3 he explains that a batch update of the posterior is equivalent to a sequential update of the posterior, and I am trying to understand this in the context of his example. Suppose $D_a$ and $D_b$ are two data sets and $\theta$ is the parameter to our model. We are trying to update the posterior $P(\theta \mid D_a, D_b)$. In a sequential update, he states that, $$ (1) \ \ \ \ \ \ \ \ P(\theta \mid D_{a}, D_{b}) \propto P(D_b \mid \theta) P(\theta \mid D_a) $$ However, I am slightly confused as to how he got this mathematically. Conceptually, I understand that he is saying the posterior $P(\theta \mid D_a)$ is now a prior used to update the new posterior, which includes the new data $D_b$, and is multiplying this prior with the likelihood $P(D_b \mid \theta)$. Expanding the last statement out, I have, $$ P(D_b \mid \theta) P(\theta \mid D_a) = P(D_b \mid \theta) P(D_a \mid \theta) P(\theta) $$ but are we allowed to say $P(D_a \mid \theta) P(D_b \mid \theta) = P(D_a, D_b \mid \theta)$ in order to make the connection in (1)?
|
How do I prove $F(a)=F(a^2)?$ Let $E$ be an extension field of $F$. If $a \in E$ has a minimal polynomial of odd degree over $F$, show that $F(a)=F(a^2)$. let $n$ be the degree of the minimal polynomial $p(x)$ of $a$ over $F$ and $k$ be the degree of the minimal polynomial $q(x)$ of $a^2$ over $F$. Since $a^2 \in F(a)$, We have $F(a^2) \subset F(a)$, then $k\le n$ In order to prove the converse: $q(a^2)=b_0+b_1a^2+b_2(a^2)^2\ldots+b_k(a^2)^k=0$ implies $q(a)=b_0+b_1a^2+b_2a^4\ldots+b_ka^{2k}=0$ Then $p(x)|q(x)$, because $p(x)$ is the minimal polynomial of $a$ over $F$. If I prove that $n|2k$ we done, since $k$ is odd, we have $n|k$ and $n\le k$ and finally $n=k$. So I almost finished the question I only need to know how to prove that $n|2k$ It should be only a detail, but I can't see, someone can help me please? Thanks
|
eng_Latn
| 34,293 |
Find the probability that an employee is a drug user
|
Bayes rule logic. Why we do use this?
|
Direct proof that nilpotent matrix has zero trace
|
eng_Latn
| 34,294 |
In MCMC simulation, how to deal with very small likelihood values that couldn't be represented by computer?
|
Computation of likelihood when $n$ is very large, so likelihood gets very small?
|
Every principal ideal domain satisfies ACCP.
|
eng_Latn
| 34,295 |
Nested Sequential Monte Carlo Methods
|
Novel approach to nonlinear/non-Gaussian Bayesian state estimation
|
Forward and inverse problems in the mechanics of soft filaments
|
eng_Latn
| 34,296 |
What is a Gaussian mixture model and why do we use it?
|
What is an intuitive explanation of Gaussian mixture models?
|
Can you give me 5 examples of homogeneous mixtures and substance and 5 heterogeneous mixtures?
|
eng_Latn
| 34,297 |
Bayesian inference with Stan: A tutorial on adding custom distributions
|
winbugs - a bayesian modelling framework : concepts , structure , and extensibility .
|
Strategies for quantum computing molecular energies using the unitary coupled cluster ansatz
|
eng_Latn
| 34,298 |
Inference for High-dimensional Exponential Family Graphical Models
|
On Poisson Graphical Models
|
Cutaneous sarcoidosis.
|
eng_Latn
| 34,299 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.