I'm trying to modify an standard kNN algorithm to obtain the probability of belonging to a class instead of just the usual classification. I haven't found much information about Probabilistic kNN, but as far as I understand, it works similar to kNN, with the difference that it calculates the percentage of examples of every class inside the given radius.
So I wonder, what's the difference then between Naive Bayes and Probabilistic kNN? I just can spot that Naive Bayes takes into consideration the prior possibility, while PkNN does not. Am I getting it wrong?
Thanks in advance!
To be honest there is nearly no similarity.
Naive bayes assumes that each class is distributed according to a simple distribution, independent on feature basis. For contiuous case - It will fit a radial Normal distribution to your whole class (each of them) and then make a decision through argmax_y N(m_y, Sigma_y)
KNN on the other hand is not a probabilistic model. Modification that you are refering to is simply a "smooth" version of the original idea, where you return ratio of each class in the nearest neighbours set (and this is not really any "probabilistic kNN", it is just regular kNN which rough estimate of probability). This assumes nothing about data distribution (besides being localy smooth). In particular - it is a nonparametric model which, given enough training samples, will fit perfectly to any dataset. Naive Bayes will fit perfectly only to K gaussians (where K is number of classes).
(I don't know how to format math formulas. For more details and clear representations, please see this.)
I would like to propose an opposite view that KNN is a kind of simplified Naive Bayes (NB) by viewing KNN as a mean of density estimation.
To perform density estimation, we attempt to estimate p(x) = k/NV, where k is the number of samples lying in a region R, N is the total sample number, and V is the volume of the region R. Usually, there are two ways to estimate it: (1) fixing V, calculate k, which is known as kernel density estimation or Parzen window; (2) fixing k, calculate V, which is the KNN-based density estimation. The latter one is much less famous than the former one due to its many drawbacks.
Yet, we can use KNN-based density estimation to connect KNN and NB. Given total N samples, Ni samples for class ci, we can write the NB in the form of KNN-based density estimation by considering a region contain x:
P(ci|x) = P(x|ci)P(ci)/P(x) = (ki/NiV)(Ni/N)/(k/NV) = ki/k,
where ki is the sample number of class ci lying in the region. The final form ki/k is actually the KNN classifier.
Related
Consider a parametric binary classifier (such as Logistic Regression, SVM etc.) trained on a dataset (say containing two features for e.g. Blood Pressure and Cholesterol level). The dataset is thrown away and the trained model can only be used as a black box (no tweaks and inside information can be gathered from the trained model). Only a set of data points can be provided and their labels predicted.
Is it possible to get information about the mean and/or standard deviation and/or range of the features of the dataset on which this model was trained? If yes, how so? and If no, then why can't we?
Thank you for your response! :)
SVM does not provide any information about the data statistics, it is a maximum margin classifier and it finds the best separating hyperplane between two datasets in the feature space, as a linear combination of "support vectors". If you use kernel functions, then this combination is in the kernel space, it is not even in the original feature space. SVM does not have a straightforward probabilistic interpretation whatsoever.
Logistic regression is a discriminative classifer and models the conditional probability p (y|x,w) where y is your label, x is your data and w are the features. After maximum likelihood training you are left with w and it is again a discriminator (hyperplane) in the feature space, so you don't have the features again.
The following can be considered. Use a Gaussian classifier. Assume that your class is produced by the prior class probability p (y). Then a class conditional density p (x|y,w) produces your data. Then by the Bayes rule, you will have: p (y|x,w) = (p (y)p (x|y,w))/p (x). If you define the class conditional density p (x|y,w) as Gaussian, its parameter set w will consists of the mean vector m and covariance matrix C of x, assuming it is being produced by the class y. But remember that, this will work only based on the assumption that the current data vector belongs to a specific class. Conditioned on w, a better option would be for mean vector: E [x|w]. This the expectation of x with respect to p (x|w). It comes down to a weighted average of mean vectors for the class y=0 and y=1, with respect to their prior class probabilities. Same should work for covariance as well, but it needs to be derived properly, I am not %100 sure right now.
Can someone give me a clear and simple definition of Maximum entropy classification? It would be very helpful if someone can provide a clear analogy, as I am struggling to understand.
"Maximum Entropy" is synonymous with "Least Informative". You wouldn't want a classifier that was least informative. It is in reference to how the priors are established. Frankly, "Maximum Entropy Classification" is an example of using buzz words.
For an example of an uninformative prior, consider given a six-sided object. The probability that any given face will appear if the object is tossed is 1/6. This would be your starting prior. It's the least informative. You really wouldn't want to start with anything else or you will bias later calculations. Of course, if you have knowledge that one side will appear more often you should incorporate that into your priors.
The Bayes formula is P(H|E) = P(E|H)P(H)/P(D)
where P(H) is the prior for the hypothesis and P(D) is the sum of all possible numerators.
For text classification where a missing word is to be inserted, E is some given document and H is the given word. IOW, the hypothesis is that H is the word which should be selected and P(H) is the weight given to the word.
Maximum Entropy Text classification means: start with least informative weights (priors) and optimize to find weights that maximize the likelihood of the data, the P(D). Essentially, it's the EM algorithm.
A simple Naive Bayes classifier would assume the prior weights would be proportional to the number of times the word appears in the document. However,this ignore correlations between words.
The so-called MaxEnt classifier, takes the correlations into account.
I can't think of a simple example to illustrate this but I can think of some correlations. For example, "the missing" in English should give higher weights to nouns but a Naive Bayes classifier might give equal weight to a verb if its relative frequency were the same as a given noun. A MaxEnt classifier considering missing would give more weight to nouns because they would be more likely in context.
I may also advise HIDDEN MARKOV AND
MAXIMUM ENTROPY
MODELS from the Department of Computer Science, Johns Hopkins. Specifically, take a look at chapter 6.6. This book explains the Maximum Entropy on the example of PoS tagging and compare MaxEnt application in MEMM with Hidden Markov Model. There are also explanation what is exactly MaxEnt with math behind.
(Taken from UNDERSTANDING DEEP LEARNING
GENERALIZATION
BY
MAXIMUM
ENTROPY (Zheng et al., 2017):
(Original Maximum Entropy Model) Supposing the dataset has input X and label
Y, the task is to find a good prediction of Y using X. The prediction Yˆ needs to maximize the
conditional entropy H(Yˆ |X) while preserving the same distribution with data (X, Y ). This is
formulated as:
min −H(Yˆ |X) (1)
s.t. P(X, Y ) = P(X, Yˆ ),
\sum(Yˆ) P(Yˆ |X) = 1
Berger et al., 1996 solves this with lagrange multipliers ωi as an exponential form:
Pω(Yˆ = y|X = x) = 1/Zω(x) exp (\sum(i) ωifi(x, y))
Okay I have a lot of confusion in regards to the way likelihood functions are defined in the context of different machine learning algorithms. For the context of this discussion, I will reference Andrew Ng 229 lecture notes.
Here is my understanding thus far.
In the context of classification, we have two different types of algorithms: discriminative and generative. The goal in both of these cases is to determine the posterior probability, that is p(C_k|x;w), where w is parameter vector and x is feature vector and C_k is kth class. The approaches are different as in discriminative we are trying to solve for the posterior probability directly given x. And in the generative case, we are determining the conditional distributions p(x|C_k), and prior classes p(C_k), and using Bayes theorem to determine P(C_k|x;w).
From my understanding Bayes theorem takes the form: p(parameters|data) = p(data|parameters)p(parameters)/p(data) where the likelihood function is p(data|parameters), posterior is p(parameters|data) and prior is p(parameters).
Now in the context of linear regression, we have the likelihood function:
p(y|X;w) where y is the vector of target values, X is design matrix.
This makes sense in according to how we defined the likelihood function above.
Now moving over to classification, the likelihood is defined still as p(y|X;w). Will the likelihood always be defined as such ?
The posterior probability we want is p(y_i|x;w) for each class which is very weird since this is apparently the likelihood function as well.
When reading through a text, it just seems the likelihood is always defined to different ways, which just confuses me profusely. Is there a difference in how the likelihood function should be interpreted for regression vs classification or say generative vs discriminative. I.e the way the likelihood is defined in Gaussian discriminant analysis looks very different.
If anyone can recommend resources that go over this in detail I would appreciate this.
A quick answer is that the likelihood function is a function proportional to the probability of seeing the data conditional on all the parameters in your model. As you said in linear regression it is p(y|X,w) where w is your vector of regression coefficients and X is your design matrix.
In a classification context, your likelihood would be proportional to P(y|X,w) where y is your vector of observed class labels. You do not have a y_i for each class, because your training data was observed to be in one particular class. Given your model specification and your model parameters, for each observed data point you should be able to calculate the probability of seeing the observed class. This is your likelihood.
The posterior predictive distribution, p(y_new_i|X,y), is the probability you want in paragraph 4. This is distinct from the likelihood because it is the probability for some unobserved case, rather than the likelihood, which relates to your training data. Note that I removed w because typically you would want to marginalize over it rather than condition on it because there is still uncertainty in the estimate after training your model and you would want your predictions to marginalize over that rather than condition on one particular value.
As an aside, the goal of all classification methods is not to find a posterior distribution, only Bayesian methods are really concerned with a posterior and those methods are necessarily generative. There are plenty of non-Bayesian methods and plenty of non-probabilistic discriminative models out there.
Any function proportional to p(a|b) where a is fixed is a likelihood function for b. Note that p(a|b) might be called something else, depending on what's interesting at the moment. For example, p(a|b) can also be called the posterior for a given b. The names don't really matter.
Suppose I have a training set made by (x, y) samples.
To apply a generative algorithm, let's say the Gaussian discriminative, I must assume that
p(x|y) ~ Normal(mu, sigma) for every possible sigma
or I just need to I know if x ~ Normal(mu, sigma) given y?
How can I evaluate if p(x|y) follows a multivariate Normal distribution well enough (up to a threshold) to me to use generative algorithm?
That's a lot of questions.
To apply a generative algorithm, let's say the Gaussian
discriminative, I must assume that
p(x|y) ~ Normal(mu, sigma) for every possible sigma
No, you must assume that's true for some mu, sigma pair. In practice you won't know what mu and sigma is, so you'll need to either estimate it (frequentist, Max Likelihood/Max A Posteriori estimates), or even better incorporate uncertainty about your estimates of the parameters into predictions (Bayesian methodology).
How can I evaluate if p(x|y) follows a multivariate Normal distribution?
Classically, using a goodness of fit test. If the dimensionality of x is more than a handful, though, this won't work because standard tests involve the number of items in bins, and the number of bins you need in high dimensions is astronomical so you have very low expected counts.
A better idea is to say the following: what are my options for modelling the (conditional) distribution of x? You can compare between these options using model comparison techniques. Read up on model checking and comparison.
Finally, your last point:
well enough (up to a threshold) to me to use generative algorithm?
The paradox of many generative methods, including Fisher's Linear Discriminant Analysis for example, as well as the Naive Bayes classifier, is that the classifier can work very well even though the model is poor for the data. There's no particularly sound reason why this should be the case, but many have observed it to be empirically true. Whether it works can be checked much more easily than whether the assumed distribution explains the data very well: just split your data into training and testing and find out!
I have a set of images. I would like to learn a one class SVM (OC-SVM) to model the distribution of a particular class (positive) as I dont have enough examples to represent the other classes (negative). What I understood about OC-SVM is that it tries to separate the data from the origin or in other words it tries to learn a hyper sphere to fit the one class data.
My questions are,
If I want to use the output of the OC-SVM as a probability estimate, how can I do it?
What is the difference between the OC-SVM and any clustering algorithm (e.g. k-means)?
If you want a probability estimate, don't use a one-class SVM. This is not what they were designed for. You want something like kernel density estimation, which provides a non-parametric density estimate given some positive examples.
The difference between a one-class SVM and clustering is that in clustering, you're given points from several classes but you don't know which points correspond to which classes: this is the goal of inference (and you may also end up with density estimates for the classes and the marginal density over all of feature space too). The one-class SVM is given points only from one class, and expected to learn a separation between members of that class and anything else.
EDIT: Clustering is not the same as density estimation. Clustering is concerned with determining which instances belong to which classes (clusters), when the assignments are not given, and does not necessarily result in a similarity score between the supplied examples and any point in input space.
If the goal is to say, how similar is this new instance to the positive training examples I've seen, then what you do is fit a probability distribution to your training examples, then evaluate the density function at the new point. If this density falls below a threshold, you say the new point is outside of the class defined by the supplied examples.
You can build a parametric model of the class if you like, but this is usually tricky unless you either know something about the problem or are willing to take a standard distribution (multi-variate normal or Naive Bayes being the two obvious ones). So, the alternative is to use a non-parametric density estimate. This is the kernel density estimation I mentioned.