Misplaced Pages

#662337

63-466: AODE seeks to estimate the probability of each class y given a specified set of features x 1 , ... x n , P( y | x 1 , ... x n ). To do so it uses the formula where P ^ ( ⋅ ) {\displaystyle {\hat {P}}(\cdot )} denotes an estimate of P ( ⋅ ) {\displaystyle P(\cdot )} , F ( ⋅ ) {\displaystyle F(\cdot )}

126-417: A i , k + − a i , k − ) x i + ( a i , k + + a i , k − ) } k ) {\displaystyle {\text{softmax}}\left(\left\{\ln p(Y=k)+{\frac {1}{2}}\sum _{i}(a_{i,k}^{+}-a_{i,k}^{-})x_{i}+(a_{i,k}^{+}+a_{i,k}^{-})\right\}_{k}\right)} where

189-407: A i , s + = ln ⁡ p ( X i = + 1 ∣ Y = s ) ; a i , s − = ln ⁡ p ( X i = − 1 ∣ Y = s ) {\displaystyle a_{i,s}^{+}=\ln p(X_{i}=+1\mid Y=s);\quad a_{i,s}^{-}=\ln p(X_{i}=-1\mid Y=s)} This

252-875: A Bayes classifier , is the function that assigns a class label y ^ = C k {\displaystyle {\hat {y}}=C_{k}} for some k as follows: y ^ = argmax k ∈ { 1 , … , K }   p ( C k ) ∏ i = 1 n p ( x i ∣ C k ) . {\displaystyle {\hat {y}}={\underset {k\in \{1,\ldots ,K\}}{\operatorname {argmax} }}\ p(C_{k})\displaystyle \prod _{i=1}^{n}p(x_{i}\mid C_{k}).} A class's prior may be calculated by assuming equiprobable classes, i.e., p ( C k ) = 1 K {\displaystyle p(C_{k})={\frac {1}{K}}} , or by calculating an estimate for

315-513: A generative-discriminative pair with multinomial logistic regression classifiers: each naive Bayes classifier can be considered a way of fitting a probability model that optimizes the joint likelihood p ( C , x ) {\displaystyle p(C,\mathbf {x} )} , while logistic regression fits the same probability model to optimize the conditional p ( C ∣ x ) {\displaystyle p(C\mid \mathbf {x} )} . More formally, we have

378-1152: A linear classifier when expressed in log-space: log ⁡ p ( C k ∣ x ) ∝ log ⁡ ( p ( C k ) ∏ i = 1 n p k i x i ) = log ⁡ p ( C k ) + ∑ i = 1 n x i ⋅ log ⁡ p k i = b + w k ⊤ x {\displaystyle {\begin{aligned}\log p(C_{k}\mid \mathbf {x} )&\varpropto \log \left(p(C_{k})\prod _{i=1}^{n}{p_{ki}}^{x_{i}}\right)\\&=\log p(C_{k})+\sum _{i=1}^{n}x_{i}\cdot \log p_{ki}\\&=b+\mathbf {w} _{k}^{\top }\mathbf {x} \end{aligned}}} where b = log ⁡ p ( C k ) {\displaystyle b=\log p(C_{k})} and w k i = log ⁡ p k i {\displaystyle w_{ki}=\log p_{ki}} . Estimating

441-465: A collection of visualization tools and algorithms for data analysis and predictive modeling , together with graphical user interfaces for easy access to these functions. The original non-Java version of Weka was a Tcl / Tk front-end to (mostly third-party) modeling algorithms implemented in other programming languages, plus data preprocessing utilities in C , and a makefile -based system for running machine learning experiments. This original version

504-402: A common principle: all naive Bayes classifiers assume that the value of a particular feature is independent of the value of any other feature, given the class variable. For example, a fruit may be considered to be an apple if it is red, round, and about 10 cm in diameter. A naive Bayes classifier considers each of these features to contribute independently to the probability that this fruit

567-468: A constant if the values of the feature variables are known. The discussion so far has derived the independent feature model, that is, the naive Bayes probability model . The naive Bayes classifier combines this model with a decision rule . One common rule is to pick the hypothesis that is most probable so as to minimize the probability of misclassification; this is known as the maximum a posteriori or MAP decision rule. The corresponding classifier,

630-424: A family of linear " probabilistic classifiers " which assumes that the features are conditionally independent, given the target class. The strength (naivety) of this assumption is what gives the classifier its name. These classifiers are among the simplest Bayesian network models. Naive Bayes classifiers are highly scalable, requiring a number of parameters linear in the number of variables (features/predictors) in

693-560: A given person is a male or a female based on the measured features. The features include height, weight, and foot size. Although with NB classifier we treat them as independent, they are not in reality. Example training set below. The classifier created from the training set using a Gaussian distribution assumption would be (given variances are unbiased sample variances ): The following example assumes equiprobable classes so that P(male)= P(female) = 0.5. This prior probability distribution might be based on prior knowledge of frequencies in

SECTION 10

#1732854835663

756-426: A learning problem. Maximum-likelihood training can be done by evaluating a closed-form expression , which takes linear time , rather than by expensive iterative approximation as used for many other types of classifiers. In the statistics literature, naive Bayes models are known under a variety of names, including simple Bayes and independence Bayes . All these names reference the use of Bayes' theorem in

819-629: A model on probability tables is infeasible. The model must therefore be reformulated to make it more tractable. Using Bayes' theorem , the conditional probability can be decomposed as: p ( C k ∣ x ) = p ( C k )   p ( x ∣ C k ) p ( x ) {\displaystyle p(C_{k}\mid \mathbf {x} )={\frac {p(C_{k})\ p(\mathbf {x} \mid C_{k})}{p(\mathbf {x} )}}\,} In plain English, using Bayesian probability terminology,

882-520: A more realistic estimate of the marginal densities of each class. This method, which was introduced by John and Langley, can boost the accuracy of the classifier considerably. With a multinomial event model, samples (feature vectors) represent the frequencies with which certain events have been generated by a multinomial ( p 1 , … , p n ) {\displaystyle (p_{1},\dots ,p_{n})} where p i {\displaystyle p_{i}}

945-927: A particular instance. This is the event model typically used for document classification, with events representing the occurrence of a word in a single document (see bag of words assumption). The likelihood of observing a histogram x is given by: p ( x ∣ C k ) = ( ∑ i = 1 n x i ) ! ∏ i = 1 n x i ! ∏ i = 1 n p k i x i {\displaystyle p(\mathbf {x} \mid C_{k})={\frac {(\sum _{i=1}^{n}x_{i})!}{\prod _{i=1}^{n}x_{i}!}}\prod _{i=1}^{n}{p_{ki}}^{x_{i}}} where p k i := p ( i ∣ C k ) {\displaystyle p_{ki}:=p(i\mid C_{k})} . The multinomial naive Bayes classifier becomes

1008-429: A problem instance to be classified, represented by a vector x = ( x 1 , … , x n ) {\displaystyle \mathbf {x} =(x_{1},\ldots ,x_{n})} encoding some n features (independent variables). The problem with the above formulation is that if the number of features n is large or if a feature can take on a large number of values, then basing such

1071-475: A small amount of training data to estimate the parameters necessary for classification. Abstractly, naive Bayes is a conditional probability model: it assigns probabilities p ( C k ∣ x 1 , … , x n ) {\displaystyle p(C_{k}\mid x_{1},\ldots ,x_{n})} for each of the K possible outcomes or classes C k {\displaystyle C_{k}} given

1134-588: A special form of One Dependence Estimator (ODE), a variant of the naive Bayes classifier that makes the above independence assumption that is weaker (and hence potentially less harmful) than the naive Bayes' independence assumption. In consequence, each ODE should create a less biased estimator than naive Bayes. However, because the base probability estimates are each conditioned by two variables rather than one, they are formed from less data (the training examples that satisfy both variables) and hence are likely to have more variance. AODE reduces this variance by averaging

1197-498: A typical assumption is that the continuous values associated with each class are distributed according to a normal (or Gaussian) distribution. For example, suppose the training data contains a continuous attribute, x {\displaystyle x} . The data is first segmented by the class, and then the mean and variance of x {\displaystyle x} is computed in each class. Let μ k {\displaystyle \mu _{k}} be

1260-495: A way to train a naive Bayes classifier from labeled data, it's possible to construct a semi-supervised training algorithm that can learn from a combination of labeled and unlabeled data by running the supervised learning algorithm in a loop: Convergence is determined based on improvement to the model likelihood P ( D ∣ θ ) {\displaystyle P(D\mid \theta )} , where θ {\displaystyle \theta } denotes

1323-536: Is an apple, regardless of any possible correlations between the color, roundness, and diameter features. In many practical applications, parameter estimation for naive Bayes models uses the method of maximum likelihood ; in other words, one can work with the naive Bayes model without accepting Bayesian probability or using any Bayesian methods. Despite their naive design and apparently oversimplified assumptions, naive Bayes classifiers have worked quite well in many complex real-world situations. In 2004, an analysis of

SECTION 20

#1732854835663

1386-1107: Is exactly a logistic regression classifier. The link between the two can be seen by observing that the decision function for naive Bayes (in the binary case) can be rewritten as "predict class C 1 {\displaystyle C_{1}} if the odds of p ( C 1 ∣ x ) {\displaystyle p(C_{1}\mid \mathbf {x} )} exceed those of p ( C 2 ∣ x ) {\displaystyle p(C_{2}\mid \mathbf {x} )} ". Expressing this in log-space gives: log ⁡ p ( C 1 ∣ x ) p ( C 2 ∣ x ) = log ⁡ p ( C 1 ∣ x ) − log ⁡ p ( C 2 ∣ x ) > 0 {\displaystyle \log {\frac {p(C_{1}\mid \mathbf {x} )}{p(C_{2}\mid \mathbf {x} )}}=\log p(C_{1}\mid \mathbf {x} )-\log p(C_{2}\mid \mathbf {x} )>0} The left-hand side of this equation

1449-569: Is expected to be formatted according the Attribute-Relational File Format and with the filename bearing the .arff extension. All of Weka's techniques are predicated on the assumption that the data is available as one flat file or relation, where each data point is described by a fixed number of attributes (normally, numeric or nominal attributes, but some other attribute types are also supported). Weka provides access to SQL databases using Java Database Connectivity and can process

1512-589: Is greater in the female case, the prediction is that the sample is female. Weka (machine learning) Waikato Environment for Knowledge Analysis ( Weka ) is a collection of machine learning and data analysis free software licensed under the GNU General Public License . It was developed at the University of Waikato , New Zealand and is the companion software to the book "Data Mining: Practical Machine Learning Tools and Techniques". Weka contains

1575-423: Is sequence modeling. In version 3.7.2, a package manager was added to allow the easier installation of extension packages. Some functionality that used to be included with Weka prior to this version has since been moved into such extension packages, but this change also makes it easier for others to contribute extensions to Weka and to maintain the software, as this modular architecture allows independent updates of

1638-566: Is the frequency with which the argument appears in the sample data and m is a user specified minimum frequency with which a term must appear in order to be used in the outer summation. In recent practice m is usually set at 1. We seek to estimate P( y | x 1 , ... x n ). By the definition of conditional probability For any 1 ≤ i ≤ n {\displaystyle 1\leq i\leq n} , Under an assumption that x 1 , ... x n are independent given y and x i , it follows that This formula defines

1701-417: Is the log-odds, or logit , the quantity predicted by the linear model that underlies logistic regression. Since naive Bayes is also a linear model for the two "discrete" event models, it can be reparametrised as a linear function b + w ⊤ x > 0 {\displaystyle b+\mathbf {w} ^{\top }x>0} . Obtaining the probabilities is then a matter of applying

1764-491: Is the number of features, l is the number of training examples and k is the number of classes. This makes it infeasible for application to high-dimensional data. However, within that limitation, it is linear with respect to the number of training examples and hence can efficiently process large numbers of training examples. The free Weka machine learning suite includes an implementation of AODE. Naive Bayes classifier In statistics , naive Bayes classifiers are

1827-466: Is the probability of class C k {\displaystyle C_{k}} generating the term x i {\displaystyle x_{i}} . This event model is especially popular for classifying short texts. It has the benefit of explicitly modelling the absence of terms. Note that a naive Bayes classifier with a Bernoulli event model is not the same as a multinomial NB classifier with frequency counts truncated to one. Given

1890-410: Is the probability that event i occurs (or K such multinomials in the multiclass case). A feature vector x = ( x 1 , … , x n ) {\displaystyle \mathbf {x} =(x_{1},\dots ,x_{n})} is then a histogram , with x i {\displaystyle x_{i}} counting the number of times event i was observed in

1953-421: Is to use binning to discretize the feature values and obtain a new set of Bernoulli-distributed features. Some literature suggests that this is required in order to use naive Bayes, but it is not true, as the discretization may throw away discriminative information . Sometimes the distribution of class-conditional marginal densities is far from normal. In these cases, kernel density estimation can be used for

Averaged one-dependence estimators - Misplaced Pages Continue

2016-461: Is true regardless of whether the probability estimate is slightly, or even grossly inaccurate. In this manner, the overall classifier can be robust enough to ignore serious deficiencies in its underlying naive probability model. Other reasons for the observed success of the naive Bayes classifier are discussed in the literature cited below. In the case of discrete inputs (indicator or frequency features for discrete events), naive Bayes classifiers form

2079-2006: The chain rule for repeated applications of the definition of conditional probability : p ( C k , x 1 , … , x n ) = p ( x 1 , … , x n , C k ) = p ( x 1 ∣ x 2 , … , x n , C k )   p ( x 2 , … , x n , C k ) = p ( x 1 ∣ x 2 , … , x n , C k )   p ( x 2 ∣ x 3 , … , x n , C k )   p ( x 3 , … , x n , C k ) = ⋯ = p ( x 1 ∣ x 2 , … , x n , C k )   p ( x 2 ∣ x 3 , … , x n , C k ) ⋯ p ( x n − 1 ∣ x n , C k )   p ( x n ∣ C k )   p ( C k ) {\displaystyle {\begin{aligned}p(C_{k},x_{1},\ldots ,x_{n})&=p(x_{1},\ldots ,x_{n},C_{k})\\&=p(x_{1}\mid x_{2},\ldots ,x_{n},C_{k})\ p(x_{2},\ldots ,x_{n},C_{k})\\&=p(x_{1}\mid x_{2},\ldots ,x_{n},C_{k})\ p(x_{2}\mid x_{3},\ldots ,x_{n},C_{k})\ p(x_{3},\ldots ,x_{n},C_{k})\\&=\cdots \\&=p(x_{1}\mid x_{2},\ldots ,x_{n},C_{k})\ p(x_{2}\mid x_{3},\ldots ,x_{n},C_{k})\cdots p(x_{n-1}\mid x_{n},C_{k})\ p(x_{n}\mid C_{k})\ p(C_{k})\\\end{aligned}}} Now

2142-444: The curse of dimensionality , such as the need for data sets that scale exponentially with the number of features. While naive Bayes often fails to produce a good estimate for the correct class probabilities, this may not be a requirement for many applications. For example, the naive Bayes classifier will make the correct MAP decision rule classification so long as the correct class is predicted as more probable than any other class. This

2205-600: The i 'th term from the vocabulary, then the likelihood of a document given a class C k {\displaystyle C_{k}} is given by: p ( x ∣ C k ) = ∏ i = 1 n p k i x i ( 1 − p k i ) ( 1 − x i ) {\displaystyle p(\mathbf {x} \mid C_{k})=\prod _{i=1}^{n}p_{ki}^{x_{i}}(1-p_{ki})^{(1-x_{i})}} where p k i {\displaystyle p_{ki}}

2268-467: The logistic function to b + w ⊤ x {\displaystyle b+\mathbf {w} ^{\top }x} , or in the multiclass case, the softmax function . Discriminative classifiers have lower asymptotic error than generative ones; however, research by Ng and Jordan has shown that in some practical cases naive Bayes can outperform logistic regression because it reaches its asymptotic error faster. Problem: classify whether

2331-621: The "naive" conditional independence assumptions come into play: assume that all features in x {\displaystyle \mathbf {x} } are mutually independent , conditional on the category C k {\displaystyle C_{k}} . Under this assumption, p ( x i ∣ x i + 1 , … , x n , C k ) = p ( x i ∣ C k ) . {\displaystyle p(x_{i}\mid x_{i+1},\ldots ,x_{n},C_{k})=p(x_{i}\mid C_{k})\,.} Thus,

2394-455: The Bayesian classification problem showed that there are sound theoretical reasons for the apparently implausible efficacy of naive Bayes classifiers. Still, a comprehensive comparison with other classification algorithms in 2006 showed that Bayes classification is outperformed by other approaches, such as boosted trees or random forests . An advantage of naive Bayes is that it only requires

2457-421: The above equation can be written as posterior = prior × likelihood evidence {\displaystyle {\text{posterior}}={\frac {{\text{prior}}\times {\text{likelihood}}}{\text{evidence}}}\,} In practice, there is interest only in the numerator of that fraction, because the denominator does not depend on C {\displaystyle C} and

2520-421: The class probability from the training set: prior for a given class = no. of samples in that class total no. of samples {\displaystyle {\text{prior for a given class}}={\frac {\text{no. of samples in that class}}{\text{total no. of samples}}}\,} To estimate the parameters for a feature's distribution, one must assume a distribution or generate nonparametric models for

2583-443: The classes of the classification problem. Despite the fact that the far-reaching independence assumptions are often inaccurate, the naive Bayes classifier has several properties that make it surprisingly useful in practice. In particular, the decoupling of the class conditional feature distributions means that each distribution can be independently estimated as a one-dimensional distribution. This helps alleviate problems stemming from

Averaged one-dependence estimators - Misplaced Pages Continue

2646-1454: The classification as female the posterior is given by posterior (female) = P ( female ) p ( height ∣ female ) p ( weight ∣ female ) p ( foot size ∣ female ) evidence {\displaystyle {\text{posterior (female)}}={\frac {P({\text{female}})\,p({\text{height}}\mid {\text{female}})\,p({\text{weight}}\mid {\text{female}})\,p({\text{foot size}}\mid {\text{female}})}{\text{evidence}}}} The evidence (also termed normalizing constant) may be calculated: evidence = P ( male ) p ( height ∣ male ) p ( weight ∣ male ) p ( foot size ∣ male ) + P ( female ) p ( height ∣ female ) p ( weight ∣ female ) p ( foot size ∣ female ) {\displaystyle {\begin{aligned}{\text{evidence}}=P({\text{male}})\,p({\text{height}}\mid {\text{male}})\,p({\text{weight}}\mid {\text{male}})\,p({\text{foot size}}\mid {\text{male}})\\+P({\text{female}})\,p({\text{height}}\mid {\text{female}})\,p({\text{weight}}\mid {\text{female}})\,p({\text{foot size}}\mid {\text{female}})\end{aligned}}} However, given

2709-411: The classifier's decision rule, but naive Bayes is not (necessarily) a Bayesian method. Naive Bayes is a simple technique for constructing classifiers: models that assign class labels to problem instances, represented as vectors of feature values, where the class labels are drawn from some finite set. There is not a single algorithm for training such classifiers, but a family of algorithms based on

2772-406: The confidence with which each classification can be made. Its probabilistic model can directly handle situations where some data are missing. AODE has computational complexity O ( l n 2 ) {\displaystyle O(ln^{2})} at training time and O ( k n 2 ) {\displaystyle O(kn^{2})} at classification time, where n

2835-685: The denominator p ( x ) {\displaystyle p(\mathbf {x} )} is omitted. This means that under the above independence assumptions, the conditional distribution over the class variable C {\displaystyle C} is: p ( C k ∣ x 1 , … , x n ) = 1 Z   p ( C k ) ∏ i = 1 n p ( x i ∣ C k ) {\displaystyle p(C_{k}\mid x_{1},\ldots ,x_{n})={\frac {1}{Z}}\ p(C_{k})\prod _{i=1}^{n}p(x_{i}\mid C_{k})} where

2898-418: The estimates of all such ODEs. Like naive Bayes, AODE does not perform model selection and does not use tuneable parameters. As a result, it has low variance. It supports incremental learning whereby the classifier can be updated efficiently with information from new examples as they become available. It predicts class probabilities rather than simply predicting a single class, allowing the user to determine

2961-443: The evidence Z = p ( x ) = ∑ k p ( C k )   p ( x ∣ C k ) {\displaystyle Z=p(\mathbf {x} )=\sum _{k}p(C_{k})\ p(\mathbf {x} \mid C_{k})} is a scaling factor dependent only on x 1 , … , x n {\displaystyle x_{1},\ldots ,x_{n}} , that is,

3024-413: The features from the training set. The assumptions on distributions of features are called the "event model" of the naive Bayes classifier. For discrete features like the ones encountered in document classification (include spam filtering), multinomial and Bernoulli distributions are popular. These assumptions lead to two distinct models, which are often confused. When dealing with continuous data,

3087-920: The following: Theorem  —  Naive Bayes classifiers on binary features are subsumed by logistic regression classifiers. Consider a generic multiclass classification problem, with possible classes Y ∈ { 1 , . . . , n } {\displaystyle Y\in \{1,...,n\}} , then the (non-naive) Bayes classifier gives, by Bayes theorem: p ( Y ∣ X = x ) = softmax ( { ln ⁡ p ( Y = k ) + ln ⁡ p ( X = x ∣ Y = k ) } k ) {\displaystyle p(Y\mid X=x)={\text{softmax}}(\{\ln p(Y=k)+\ln p(X=x\mid Y=k)\}_{k})} The naive Bayes classifier gives softmax ( { ln ⁡ p ( Y = k ) + 1 2 ∑ i (

3150-1132: The joint model can be expressed as p ( C k ∣ x 1 , … , x n ) ∝   p ( C k , x 1 , … , x n ) = p ( C k )   p ( x 1 ∣ C k )   p ( x 2 ∣ C k )   p ( x 3 ∣ C k )   ⋯ = p ( C k ) ∏ i = 1 n p ( x i ∣ C k ) , {\displaystyle {\begin{aligned}p(C_{k}\mid x_{1},\ldots ,x_{n})\varpropto \ &p(C_{k},x_{1},\ldots ,x_{n})\\&=p(C_{k})\ p(x_{1}\mid C_{k})\ p(x_{2}\mid C_{k})\ p(x_{3}\mid C_{k})\ \cdots \\&=p(C_{k})\prod _{i=1}^{n}p(x_{i}\mid C_{k})\,,\end{aligned}}} where ∝ {\displaystyle \varpropto } denotes proportionality since

3213-740: The larger population or in the training set. Below is a sample to be classified as male or female. In order to classify the sample, one has to determine which posterior is greater, male or female. For the classification as male the posterior is given by posterior (male) = P ( male ) p ( height ∣ male ) p ( weight ∣ male ) p ( foot size ∣ male ) evidence {\displaystyle {\text{posterior (male)}}={\frac {P({\text{male}})\,p({\text{height}}\mid {\text{male}})\,p({\text{weight}}\mid {\text{male}})\,p({\text{foot size}}\mid {\text{male}})}{\text{evidence}}}} For

SECTION 50

#1732854835663

3276-589: The mean of the values in x {\displaystyle x} associated with class C k {\displaystyle C_{k}} , and let σ k 2 {\displaystyle \sigma _{k}^{2}} be the Bessel corrected variance of the values in x {\displaystyle x} associated with class C k {\displaystyle C_{k}} . Suppose one has collected some observation value v {\displaystyle v} . Then,

3339-408: The multivariate Bernoulli event model, features are independent Boolean variables ( binary variables ) describing inputs. Like the multinomial model, this model is popular for document classification tasks, where binary term occurrence features are used rather than term frequencies. If x i {\displaystyle x_{i}} is a Boolean expressing the occurrence or absence of

3402-418: The number of occurrences of a feature's value. This is problematic because it will wipe out all information in the other probabilities when they are multiplied. Therefore, it is often desirable to incorporate a small-sample correction, called pseudocount , in all probability estimates such that no probability is ever set to be exactly zero. This way of regularizing naive Bayes is called Laplace smoothing when

3465-401: The parameters in log space is advantageous since multiplying a large number of small values can lead to significant rounding error. Applying a log transform reduces the effect of this rounding error. If a given class and feature value never occur together in the training data, then the frequency-based probability estimate will be zero, because the probability estimate is directly proportional to

3528-2539: The parameters of normal distribution which have been previously determined from the training set. Note that a value greater than 1 is OK here – it is a probability density rather than a probability, because height is a continuous variable. p ( weight ∣ male ) = 1 2 π σ 2 exp ⁡ ( − ( 130 − μ ) 2 2 σ 2 ) = 5.9881 ⋅ 10 − 6 {\displaystyle p({\text{weight}}\mid {\text{male}})={\frac {1}{\sqrt {2\pi \sigma ^{2}}}}\exp \left({\frac {-(130-\mu )^{2}}{2\sigma ^{2}}}\right)=5.9881\cdot 10^{-6}} p ( foot size ∣ male ) = 1 2 π σ 2 exp ⁡ ( − ( 8 − μ ) 2 2 σ 2 ) = 1.3112 ⋅ 10 − 3 {\displaystyle p({\text{foot size}}\mid {\text{male}})={\frac {1}{\sqrt {2\pi \sigma ^{2}}}}\exp \left({\frac {-(8-\mu )^{2}}{2\sigma ^{2}}}\right)=1.3112\cdot 10^{-3}} posterior numerator (male) = their product = 6.1984 ⋅ 10 − 9 {\displaystyle {\text{posterior numerator (male)}}={\text{their product}}=6.1984\cdot 10^{-9}} P ( female ) = 0.5 {\displaystyle P({\text{female}})=0.5} p ( height ∣ female ) = 2.23 ⋅ 10 − 1 {\displaystyle p({\text{height}}\mid {\text{female}})=2.23\cdot 10^{-1}} p ( weight ∣ female ) = 1.6789 ⋅ 10 − 2 {\displaystyle p({\text{weight}}\mid {\text{female}})=1.6789\cdot 10^{-2}} p ( foot size ∣ female ) = 2.8669 ⋅ 10 − 1 {\displaystyle p({\text{foot size}}\mid {\text{female}})=2.8669\cdot 10^{-1}} posterior numerator (female) = their product = 5.3778 ⋅ 10 − 4 {\displaystyle {\text{posterior numerator (female)}}={\text{their product}}=5.3778\cdot 10^{-4}} Since posterior numerator

3591-414: The parameters of the naive Bayes model. This training algorithm is an instance of the more general expectation–maximization algorithm (EM): the prediction step inside the loop is the E -step of EM, while the re-training of naive Bayes is the M -step. The algorithm is formally justified by the assumption that the data are generated by a mixture model , and the components of this mixture model are exactly

3654-1085: The probability density of v {\displaystyle v} given a class C k {\displaystyle C_{k}} , i.e., p ( x = v ∣ C k ) {\displaystyle p(x=v\mid C_{k})} , can be computed by plugging v {\displaystyle v} into the equation for a normal distribution parameterized by μ k {\displaystyle \mu _{k}} and σ k 2 {\displaystyle \sigma _{k}^{2}} . Formally, p ( x = v ∣ C k ) = 1 2 π σ k 2 e − ( v − μ k ) 2 2 σ k 2 {\displaystyle p(x=v\mid C_{k})={\frac {1}{\sqrt {2\pi \sigma _{k}^{2}}}}\,e^{-{\frac {(v-\mu _{k})^{2}}{2\sigma _{k}^{2}}}}} Another common technique for handling continuous values

3717-424: The pseudocount is one, and Lidstone smoothing in the general case. Rennie et al. discuss problems with the multinomial assumption in the context of document classification and possible ways to alleviate those problems, including the use of tf–idf weights instead of raw term frequencies and document length normalization, to produce a naive Bayes classifier that is competitive with support vector machines . In

3780-457: The result returned by a database query. Weka provides access to deep learning with Deeplearning4j . It is not capable of multi-relational data mining, but there is separate software for converting a collection of linked database tables into a single table that is suitable for processing using Weka. Another important area that is currently not covered by the algorithms included in the Weka distribution

3843-1049: The sample, the evidence is a constant and thus scales both posteriors equally. It therefore does not affect classification and can be ignored. The probability distribution for the sex of the sample can now be determined: P ( male ) = 0.5 {\displaystyle P({\text{male}})=0.5} p ( height ∣ male ) = 1 2 π σ 2 exp ⁡ ( − ( 6 − μ ) 2 2 σ 2 ) ≈ 1.5789 , {\displaystyle p({\text{height}}\mid {\text{male}})={\frac {1}{\sqrt {2\pi \sigma ^{2}}}}\exp \left({\frac {-(6-\mu )^{2}}{2\sigma ^{2}}}\right)\approx 1.5789,} where μ = 5.855 {\displaystyle \mu =5.855} and σ 2 = 3.5033 ⋅ 10 − 2 {\displaystyle \sigma ^{2}=3.5033\cdot 10^{-2}} are

SECTION 60

#1732854835663

3906-419: The values of the features x i {\displaystyle x_{i}} are given, so that the denominator is effectively constant. The numerator is equivalent to the joint probability model p ( C k , x 1 , … , x n ) {\displaystyle p(C_{k},x_{1},\ldots ,x_{n})\,} which can be rewritten as follows, using

3969-501: Was primarily designed as a tool for analyzing data from agricultural domains, but the more recent fully Java -based version (Weka 3), for which development started in 1997, is now used in many different application areas, in particular for educational purposes and research. Advantages of Weka include: Weka supports several standard data mining tasks, more specifically, data preprocessing, clustering , classification , regression , visualization , and feature selection . Input to Weka

#662337