Questions tagged [bayes-optimal-classifier]

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Posteriori Probability, Parameter Vectors and the explicit form of Bayes Classifier

I'm working on some linear discriminant analysis problems to develop a better understanding of the underlying mechanisms for machine learning algorithms. I'd like to develop a better understanding of ...
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Expected value of given random variables [closed]

Consider $f(X,Y):=\underset{x \sim \mathcal{D}}{\mathbb{E}}_X \left[\underset{y \sim \mathcal{D}}{\mathbb{E}}_{Y | x}\left[\mathbb{\mathbf{1}}_{h(x) \neq y} | X = x\right]\right]$ where $\mathbf{1}$ ...
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Derive the criterion for minimizing the expected loss when there is a general loss matrix and general prior probabilities for the classes

In the book "Pattern Recognition and Machine Learning" I am trying to do exercise 1.23 (p.63): Derive the criterion for minimizing the expected loss when there is a general loss matrix and ...
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What is the relation between Linear Classifier and Linear Decission Boundary (or Non Linear Decision Boundary)?

As we know (Wikipedia Definition): Linear Classifier makes a classification decision based on the linear combination of the feature vectors. Mathematically : $y = f(\sum w_i x_i)$ So , $f$ is our ...
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How would you find a p threshold for a binary classification prediction?

Lets say that there's a binary classification problem where $X$ ∈ $R_p$ and $Y ∈ \{0,1\} $ and $Pr(Y = 1 | X = x) = p$ for $p$ in $[0,1]$. There is a loss function $L_{falseneg} > 0$ for false ...
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1answer
114 views

This decision is the best we can do if we have no prior information about the probabilities of the two classes?

I am currently studying the textbook Learning with kernels: support vector machines, regularization, optimization and beyond by Schölkopf and Smola. Chapter 1.2 A Simple Pattern Recognition Algorithm ...
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1answer
189 views

How is this a "Bayes classifier"?

I am currently studying the textbook Learning with kernels: support vector machines, regularization, optimization and beyond by Schölkopf and Smola. Chapter 1.2 A Simple Pattern Recognition Algorithm ...
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1answer
53 views

How $\dfrac{P_t(\mathbf{\mathrm{x}}, y)}{P_s(\mathbf{\mathrm{x}}, y)} = \dfrac{P_t(\mathbf{\mathrm{x}})}{P_s(\mathbf{\mathrm{x})}}$?

Related: Empirical Risk Minimization: Rewriting the expected loss using Bayes' rule and the definition of expectation I am currently studying Transfer Learning by Qiang Yang, Yu Zhang, Wenyuan Dai,...
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202 views

Empirical Risk Minimization: Rewriting the expected loss using Bayes' rule and the definition of expectation

I am currently studying Transfer Learning by Qiang Yang, Yu Zhang, Wenyuan Dai, and Sinno Jialin Pan. Chapter 2.2 Instance-Based Noninductive Transfer Learning says the following: As mentioned ...
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390 views

Improve Adaboost that using weighted logistic regression instead of decision trees

I implemented Adaboost that using weighted logistic regression instead of decision trees and I managed to get to 0.5% error, I'm trying to improve it for days with no success and I know it possible to ...
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1answer
55 views

Use different Naive Bayes classifiers to target different data

I am practicing using the Naive Bayes classifier to predict whether people get a stroke or not, but, I am confused with two classifiers. One is categorical Naive Bayes, another is Gaussian Naive Bayes....
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How to proof that the bayes optimal classifier is optimal for a continuous domain

Exercise 3.7 from the book »Understanding Machine Learning: From Theory to Algorithms«, Shalev-Shwartz and Ben-David, states the following: The Bayes optimal predictor: Show that for every ...
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1answer
152 views

Bayes Optimal Classifier for multinomial classification

I understand the meaning and how to deduce a Bayes optimal classifier in binary classification, but I am not sure how to derive this in the context of multinomial classification. Do we use the naive ...
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1answer
573 views

LDA and Fisher LDA - are their weight vectors always equivalent?

Linear Discriminant Analysis (LDA) and Fisher Linear Discriminant Analysis (FLDA) both project high-dimensional observations to univariate classification scores using different rationals and ...
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386 views

How to find the optimal classifier for a given loss function?

For a binary classification problem (labels being 0 and 1) and a classifier $g$ we consider the loss function $L(g)=P[Y\neq g(X)]$. It is known that the optimal classifier $g^*$ that minimizes $L$ is $...
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1answer
37 views

Penalization term for unfairness

I am reading [1], where the researchers do a logistic regression, but add to the loss function the following penalization term for fairness $ R^{AVD}_{FP}(\theta; S) = \left\lvert \dfrac{\sum\limits_{...
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Question about using Bayesian rule as a classification for continuous data set

Please note that my question is not about coding. I am now learning Bayesian classification and I think I understand it in a discrete case. I have trouble understanding it for multivariate continuous ...
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1answer
517 views

Combining Classifiers with different Precision and Recall values

Suppose I have two binary classifiers, A and B. Both are trained on the same set of data, and produce predictions on a different (but same for both classifiers) set of data. The precision for A is ...
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Can we increase the accuracy of a classifier using sketches?

I am using a sketch technique to improve the memory of a standard classifier (naive Bayes) with data streams. The sketch technique is composed of a sketch table (hash table) means the true values can ...
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1answer
177 views

What is D in Optimal Bayes Classification (Machine Learning by Tom Mitchell Ed2)?

I'm reading chapter 6 from "Machine Learning" by Tom Mitchell, 2nd edition. It seems like the author changes in each paragraph what "D" is without saying anything, but it becomes really confussing at ...
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54 views

Bayes classifier in terms of generative and discriminant analysis approaches

For simplicity I will only discuss binary classification. If $p_k(x) = P(X \mid Y = k)$ for $k = 0,1$, then Bayes classifier $h$ minimizes the risk $P(h(x) \neq Y)$. It is well known that $$h(x) = 1\{...
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Is the Bayes optimal classifier well defined?

The Bayes optimal classifier (BOC) is defined as follows. When data $D$ is given, the classifier returns the value $$\text{argmax}_{y\in Y} \sum_{h} P(y\mid h) P(h\mid D)\text{,}$$ where the $Y$ is a ...
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540 views

Finding the error probability of an optimal bayes classifier analytically

I have two classes $\omega_1,\omega_2$ with equal prior probability $P(\omega_1)=P(\omega_2)=0.5$. And the points in 2D are distributed $\mathcal{N}(\mu_i,\Sigma), \mu_1=(0,0)^T, \mu_1=(4,4)^T, \Sigma=...
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1answer
184 views

error probability of decision function

If I have a binary calssification task with prior probability $p(0) = 0.6$, and I make two decisions. 1) solely based on the prior probability i.e. I make prediction 0 60% of the time and prediction ...
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Bayes optimal classifier vs Likelihood Ratio

I am getting slightly confused by all the probabilistic classifiers. The bayes optimal classifier is given as $ max (p(x|C)p(C)) $ and if all classes have equal prior then it reduces to $ max (p(x|C))...
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551 views

Can the Bayes Optimal Predictor be generalized?

I'm reading Understanding Machine Learning by Shai and Shai. In it, the Bayes Optimal Predictor is defined as $$f_{\mathcal{D}}(x) = \mathbb{1}[\mathbb{P}[y = 1 | x] \geq 1/2]$$ Where $\mathcal{D}$ ...
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1answer
461 views

Why classifiers report the class with maximum posterior probability as the predicted class?

When we train a classifier to predict $y \in \{1, \dots, K\}$ given an input $x$, classification is done by reporting the class with the highest posterior probability as the prediction; that is: $$ \...
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Help with a proof of Bayes classifier optimality

I have a class assignment to provide a proof that Bayes classifier for the two label version is optimal in that it's error rate is always ${\le}$ any other classifier. I've never worked through a ...
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4answers
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Calculating the error of Bayes classifier analytically

If two classes $w_1$ and $w_2$ have normal distribution with known parameters ($M_1$, $M_2$ as their means and $\Sigma_1$,$\Sigma_2$ are their covariances) how we can calculate error of the Bayes ...