Major concepts Gaussian, Multinomial, Bernoulli Distributions Joint vs. Conditional Distributions Marginalization Maximum Likelihood Risk Minimization Gradient Descent Feature Extraction, Kernel Methods
Some favorite distributions Bernoulli Multinomial Gaussian
Maximum Likelihood Identify the parameter values that yield the maximum likelihood of generating the observed data. Take the partial derivative of the likelihood function Set to zero Solve NB: maximum likelihood parameters are the same as maximum log likelihood parameters
Maximum Log Likelihood Why do we like the log function? It turns products (difficult to differentiate) and turns them into sums (easy to differentiate) log(xy) = log(x) + log(y) log(x c ) = c log(x)
Risk Minimization Pick a loss function – Squared loss – Linear loss – Perceptron (classification) loss Identify the parameters that minimize the loss function. – Take the partial derivative of the loss function – Set to zero – Solve
Frequentists v. Bayesians Point estimates vs. Posteriors Risk Minimization vs. Maximum Likelihood L2-Regularization – Frequentists: Add a constraint on the size of the weight vector – Bayesians: Introduce a zero-mean prior on the weight vector – Result is the same!
L2-Regularization Frequentists: – Introduce a cost on the size of the weights Bayesians: – Introduce a prior on the weights
Types of Classifiers Generative Models – Highest resource requirements. – Need to approximate the joint probability Discriminative Models – Moderate resource requirements. – Typically fewer parameters to approximate than generative models Discriminant Functions – Can be trained probabilistically, but the output does not include confidence information
Linear Regression Fit a line to a set of points
Linear Regression Extension to higher dimensions – Polynomial fitting – Arbitrary function fitting Wavelets Radial basis functions Classifier output
Logistic Regression Fit gaussians to data for each class The decision boundary is where the PDFs cross No “closed form” solution to the gradient. Gradient Descent
Graphical Models General way to describe the dependence relationships between variables. Junction Tree Algorithm allows us to efficiently calculate marginals over any variable.
Junction Tree Algorithm Moralization – “Marry the parents” – Make undirected Triangulation – Remove cycles >4 Junction Tree Construction – Identify separators such that the running intersection property holds Introduction of Evidence – Pass slices around the junction tree to generate marginals
Hidden Markov Models Sequential Modeling – Generative Model Relationship between observations and state (class) sequences
Perceptron Step function used for squashing. Classifier as Neuron metaphor.
Perceptron Loss Classification Error vs. Sigmoid Error – Loss is only calculated on Mistakes Perceptrons use strictly classification error
Neural Networks Interconnected Layers of Perceptrons or Logistic Regression “neurons”
Neural Networks There are many possible configurations of neural networks – Vary the number of layers – Size of layers
Support Vector Machines Maximum Margin Classification Small Margin Large Margin
Support Vector Machines Optimization Function Decision Function
Visualization of Support Vectors 25
Questions? Now would be a good time to ask questions about Supervised Techniques.
Clustering Identify discrete groups of similar data points Data points are unlabeled
Recall K-Means Algorithm – Select K – the desired number of clusters – Initialize K cluster centroids – For each point in the data set, assign it to the cluster with the closest centroid – Update the centroid based on the points assigned to each cluster – If any data point has changed clusters, repeat
Soft K-means In k-means, we force every data point to exist in exactly one cluster. This constraint can be relaxed. Minimizes the entropy of cluster assignment
Soft k-means example
Soft k-means We still define a cluster by a centroid, but we calculate the centroid as the weighted mean of all the data points Convergence is based on a stopping threshold rather than changed assignments
Gaussian Mixture Models Rather than identifying clusters by “nearest” centroids Fit a Set of k Gaussians to the data.
Gaussian Mixture Models Formally a Mixture Model is the weighted sum of a number of pdfs where the weights are determined by a distribution,
Graphical Models with unobserved variables What if you have variables in a Graphical model that are never observed? – Latent Variables Training latent variable models is an unsupervised learning application laughing amused sweating uncomfortable
Latent Variable HMMs We can cluster sequences using an HMM with unobserved state variables We will train the latent variable models using Expectation Maximization
Expectation Maximization Both the training of GMMs and Gaussian Models with latent variables are accomplished using Expectation Maximization – Step 1: Expectation (E-step) Evaluate the “responsibilities” of each cluster with the current parameters – Step 2: Maximization (M-step) Re-estimate parameters using the existing “responsibilities” Related to k-means
Questions One more time for questions on supervised learning…
Next Time Gaussian Mixture Models (GMMs) Expectation Maximization