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Questions tagged [bayes-optimal-classifier]

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Bayes Optimal vs Naive Bayes

I am a 3rd year CS student and I am currently studying machine learning. We have recently been covering Bayes classifiers and I am very confused by the Bayes Optimal Classifier. For Naive Bayes, we ...
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23 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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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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26 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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28 views

Training and test data set with Bayesian classfier

I have a simple question and try to make sure that I understand the classification process. Assume that I have a data set with three different classes (say iris ...
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48 views

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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61 views

Linearity of Bayes Optimal Classifer

Given a Bayes optimal classifier, that is, the classifier that decides: $$arg\,max_{c \in C} ~p(c|X)$$ What are sufficient condition for that classifier to be linear? My understanding is that using ...
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1answer
88 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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57 views

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
92 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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42 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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174 views

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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390 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
152 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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1answer
4k views

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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419 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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319 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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2k views

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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11k views

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 ...