Pattern Recognition and Machine Learning

Pattern Recognition and Machine Learning
Chapter 1: Introduction

Example Handwritten Digit Recognition

Polynomial Curve Fitting

Sum-of-Squares Error Function

0th Order Polynomial

1st Order Polynomial

3rd Order Polynomial

9th Order Polynomial

Over-fitting Root-Mean-Square (RMS) Error:

Polynomial Coefficients

Data Set Size: 9th Order Polynomial

Regularization Penalize large coefficient values

Regularization:

Regularization: vs.

Polynomial Coefficients

Probability Theory Apples and Oranges

Probability Theory Marginal Probability Conditional Probability
Joint Probability

Probability Theory Sum Rule Product Rule

The Rules of Probability
Sum Rule Product Rule

Bayes’ Theorem posterior  likelihood × prior

Probability Densities

Transformed Densities

Expectations Conditional Expectation (discrete)
Approximate Expectation (discrete and continuous)

Variances and Covariances

The Gaussian Distribution

Gaussian Mean and Variance

The Multivariate Gaussian

Gaussian Parameter Estimation
Likelihood function

Maximum (Log) Likelihood

Properties of and

Curve Fitting Re-visited

Maximum Likelihood Determine by minimizing sum-of-squares error,

Predictive Distribution

MAP: A Step towards Bayes
Determine by minimizing regularized sum-of-squares error,

Bayesian Curve Fitting

Bayesian Predictive Distribution

Model Selection Cross-Validation

Curse of Dimensionality

Curse of Dimensionality
Polynomial curve fitting, M = 3 Gaussian Densities in higher dimensions

Decision Theory Inference step Determine either or . Decision step For given x, determine optimal t.

Minimum Misclassification Rate

Minimum Expected Loss Example: classify medical images as ‘cancer’ or ‘normal’ Decision Truth

Minimum Expected Loss Regions are chosen to minimize

Reject Option

Why Separate Inference and Decision?
Minimizing risk (loss matrix may change over time) Reject option Unbalanced class priors Combining models

Decision Theory for Regression
Inference step Determine . Decision step For given x, make optimal prediction, y(x), for t. Loss function:

The Squared Loss Function

Generative vs Discriminative
Generative approach: Model Use Bayes’ theorem Discriminative approach: Model directly

Entropy Important quantity in coding theory statistical physics
machine learning

Entropy Coding theory: x discrete with 8 possible states; how many bits to transmit the state of x? All states equally likely

Entropy

Entropy In how many ways can N identical objects be allocated M bins? Entropy maximized when

Entropy

Differential Entropy Put bins of width ¢ along the real line Differential entropy maximized (for fixed ) when in which case

Conditional Entropy

The Kullback-Leibler Divergence

Mutual Information

Chapter 2: Probability distributions

Parametric Distributions
Basic building blocks: Need to determine given Representation: or ? Recall Curve Fitting

Binary Variables (1) Coin flipping: heads=1, tails=0 Bernoulli Distribution

Binary Variables (2) N coin flips: Binomial Distribution

Binomial Distribution

Parameter Estimation (1)
ML for Bernoulli Given:

Parameter Estimation (2)
Example: Prediction: all future tosses will land heads up Overfitting to D

Beta Distribution Distribution over

Bayesian Bernoulli The Beta distribution provides the conjugate prior for the Bernoulli distribution.

Beta Distribution

Prior ∙ Likelihood = Posterior

Properties of the Posterior
As the size of the data set, N , increase

Prediction under the Posterior
What is the probability that the next coin toss will land heads up?

Multinomial Variables
1-of-K coding scheme:

ML Parameter estimation
Given: Ensure , use a Lagrange multiplier, ¸.

The Multinomial Distribution

The Dirichlet Distribution
Conjugate prior for the multinomial distribution.

Bayesian Multinomial (1)

Bayesian Multinomial (2)

The Gaussian Distribution

Central Limit Theorem The distribution of the sum of N i.i.d. random variables becomes increasingly Gaussian as N grows. Example: N uniform [0,1] random variables.

Geometry of the Multivariate Gaussian

Moments of the Multivariate Gaussian (1)
thanks to anti-symmetry of z

Moments of the Multivariate Gaussian (2)

Partitioned Gaussian Distributions

Partitioned Conditionals and Marginals

Bayes’ Theorem for Gaussian Variables
Given we have where

Maximum Likelihood for the Gaussian (1)
Given i.i.d. data , the log likeli-hood function is given by Sufficient statistics

Set the derivative of the log likelihood function to zero, and solve to obtain Similarly

Under the true distribution Hence define

Sequential Estimation
Contribution of the N th data point, xN correction given xN correction weight old estimate

The Robbins-Monro Algorithm (1)
Consider µ and z governed by p(z,µ) and define the regression function Seek µ? such that f(µ?) = 0.

Assume we are given samples from p(z,µ), one at the time.

Successive estimates of µ? are then given by Conditions on aN for convergence :

Robbins-Monro for Maximum Likelihood (1)
Regarding as a regression function, finding its root is equivalent to finding the maximum likelihood solution µML. Thus

Robbins-Monro for Maximum Likelihood (2)
Example: estimate the mean of a Gaussian. The distribution of z is Gaussian with mean ¹ { ¹ML. For the Robbins-Monro update equation, aN = ¾2=N.

Bayesian Inference for the Gaussian (1)
Assume ¾2 is known. Given i.i.d. data , the likelihood function for ¹ is given by This has a Gaussian shape as a function of ¹ (but it is not a distribution over ¹).

Combined with a Gaussian prior over ¹, this gives the posterior Completing the square over ¹, we see that

… where Note:

Example: for N = 0, 1, 2 and 10.

Sequential Estimation The posterior obtained after observing N { 1 data points becomes the prior when we observe the N th data point.

Now assume ¹ is known. The likelihood function for ¸ = 1/¾2 is given by This has a Gamma shape as a function of ¸.

The Gamma distribution

Now we combine a Gamma prior, , with the likelihood function for ¸ to obtain which we recognize as with

If both ¹ and ¸ are unknown, the joint likelihood function is given by We need a prior with the same functional dependence on ¹ and ¸.

The Gaussian-gamma distribution Quadratic in ¹. Linear in ¸. Gamma distribution over ¸. Independent of ¹.

The Gaussian-gamma distribution

Multivariate conjugate priors ¹ unknown, ¤ known: p(¹) Gaussian. ¤ unknown, ¹ known: p(¤) Wishart, ¤ and ¹ unknown: p(¹,¤) Gaussian-Wishart,

Student’s t-Distribution
where Infinite mixture of Gaussians.

Robustness to outliers: Gaussian vs t-distribution.

The D-variate case: where . Properties:

Periodic variables Examples: calendar time, direction, … We require

von Mises Distribution (1)
This requirement is satisfied by where is the 0th order modified Bessel function of the 1st kind.

von Mises Distribution (4)

Maximum Likelihood for von Mises
Given a data set, , the log likelihood function is given by Maximizing with respect to µ0 we directly obtain Similarly, maximizing with respect to m we get which can be solved numerically for mML.

Mixtures of Gaussians (1)
Old Faithful data set Single Gaussian Mixture of two Gaussians

Combine simple models into a complex model: K=3 Component Mixing coefficient

Determining parameters ¹, §, and ¼ using maximum log likelihood Solution: use standard, iterative, numeric optimization methods or the expectation maximization algorithm (Chapter 9). Log of a sum; no closed form maximum.

The Exponential Family (1)
where ´ is the natural parameter and so g(´) can be interpreted as a normalization coefficient.

The Exponential Family (2.1)
The Bernoulli Distribution Comparing with the general form we see that and so Logistic sigmoid

The Bernoulli distribution can hence be written as where

The Multinomial Distribution where, , and NOTE: The ´ k parameters are not independent since the corresponding ¹k must satisfy

Let . This leads to and Here the ´ k parameters are independent. Note that Softmax

The Multinomial distribution can then be written as where

The Exponential Family (4)
The Gaussian Distribution where

ML for the Exponential Family (1)
From the definition of g(´) we get Thus

ML for the Exponential Family (2)
Give a data set, , the likelihood function is given by Thus we have Sufficient statistic

Prior corresponds to º pseudo-observations with value Â.
Conjugate priors For any member of the exponential family, there exists a prior Combining with the likelihood function, we get Prior corresponds to º pseudo-observations with value Â.

Noninformative Priors (1)
With little or no information available a-priori, we might choose a non-informative prior. ¸ discrete, K-nomial : ¸2[a,b] real and bounded: ¸ real and unbounded: improper! A constant prior may no longer be constant after a change of variable; consider p(¸) constant and ¸=´2:

Translation invariant priors. Consider For a corresponding prior over ¹, we have for any A and B. Thus p(¹) = p(¹ { c) and p(¹) must be constant.

Example: The mean of a Gaussian, ¹ ; the conjugate prior is also a Gaussian, As , this will become constant over ¹ .

Scale invariant priors. Consider and make the change of variable For a corresponding prior over ¾, we have for any A and B. Thus p(¾) / 1/¾ and so this prior is improper too. Note that this corresponds to p(ln ¾) being constant.

Example: For the variance of a Gaussian, ¾2, we have If ¸ = 1/¾2 and p(¾) / 1/¾ , then p(¸) / 1/ ¸. We know that the conjugate distribution for ¸ is the Gamma distribution, A noninformative prior is obtained when a0 = 0 and b0 = 0.

Nonparametric Methods (1)
Parametric distribution models are restricted to specific forms, which may not always be suitable; for example, consider modelling a multimodal distribution with a single, unimodal model. Nonparametric approaches make few assumptions about the overall shape of the distribution being modelled.

Histogram methods partition the data space into distinct bins with widths ¢i and count the number of observations, ni, in each bin. Often, the same width is used for all bins, ¢i = ¢. ¢ acts as a smoothing parameter. In a D-dimensional space, using M bins in each dimension will require MD bins!

Assume observations drawn from a density p(x) and consider a small region R containing x such that The probability that K out of N observations lie inside R is Bin(KjN,P ) and if N is large If the volume of R, V, is sufficiently small, p(x) is approximately constant over R and Thus V small, yet K>0, therefore N large?

Kernel Density Estimation: fix V, estimate K from the data. Let R be a hypercube centred on x and define the kernel function (Parzen window) It follows that and hence

To avoid discontinuities in p(x), use a smooth kernel, e.g. a Gaussian Any kernel such that will work. h acts as a smoother.

Nearest Neighbour Density Estimation: fix K, estimate V from the data. Consider a hypersphere centred on x and let it grow to a volume, V ?, that includes K of the given N data points. Then K acts as a smoother.

Nonparametric models (not histograms) requires storing and computing with the entire data set. Parametric models, once fitted, are much more efficient in terms of storage and computation.

K-Nearest-Neighbours for Classification (1)
Given a data set with Nk data points from class Ck and , we have and correspondingly Since , Bayes’ theorem gives

K acts as a smother For , the error rate of the 1-nearest-neighbour classifier is never more than twice the optimal error (obtained from the true conditional class distributions).

Chapter 3: Linear models for regression

Linear Basis Function Models (1)
Example: Polynomial Curve Fitting

Generally where Áj(x) are known as basis functions. Typically, Á0(x) = 1, so that w0 acts as a bias. In the simplest case, we use linear basis functions : Ád(x) = xd.

Polynomial basis functions: These are global; a small change in x affect all basis functions.

Gaussian basis functions: These are local; a small change in x only affect nearby basis functions. ¹j and s control location and scale (width).

Sigmoidal basis functions: where Also these are local; a small change in x only affect nearby basis functions. ¹j and s control location and scale (slope).

Maximum Likelihood and Least Squares (1)
Assume observations from a deterministic function with added Gaussian noise: which is the same as saying, Given observed inputs, , and targets, , we obtain the likelihood function where

Taking the logarithm, we get where is the sum-of-squares error.

Computing the gradient and setting it to zero yields Solving for w, we get where The Moore-Penrose pseudo-inverse,

Geometry of Least Squares
Consider S is spanned by . wML minimizes the distance between t and its orthogonal projection on S, i.e. y. N-dimensional M-dimensional

Sequential Learning Data items considered one at a time (a.k.a. online learning); use stochastic (sequential) gradient descent: This is known as the least-mean-squares (LMS) algorithm. Issue: how to choose ´?

Regularized Least Squares (1)
Consider the error function: With the sum-of-squares error function and a quadratic regularizer, we get which is minimized by Data term + Regularization term ¸ is called the regularization coefficient.

With a more general regularizer, we have Lasso Quadratic

Lasso tends to generate sparser solutions than a quadratic regularizer.

Multiple Outputs (1) Analogously to the single output case we have: Given observed inputs, , and targets, , we obtain the log likelihood function

Multiple Outputs (2) Maximizing with respect to W, we obtain If we consider a single target variable, tk, we see that where , which is identical with the single output case.

The Bias-Variance Decomposition (1)
Recall the expected squared loss, where The second term of E[L] corresponds to the noise inherent in the random variable t. What about the first term?

Suppose we were given multiple data sets, each of size N. Any particular data set, D, will give a particular function y(x;D). We then have

Taking the expectation over D yields

Thus we can write where

Example: 25 data sets from the sinusoidal, varying the degree of regularization, ¸.

The Bias-Variance Trade-off
From these plots, we note that an over-regularized model (large ¸) will have a high bias, while an under-regularized model (small ¸) will have a high variance.

Bayesian Linear Regression (1)
Define a conjugate prior over w Combining this with the likelihood function and using results for marginal and conditional Gaussian distributions, gives the posterior where

A common choice for the prior is for which Next we consider an example …

0 data points observed Prior Data Space

1 data point observed Likelihood Posterior Data Space

2 data points observed Likelihood Posterior Data Space

20 data points observed Likelihood Posterior Data Space

Predictive Distribution (1)
Predict t for new values of x by integrating over w: where

Example: Sinusoidal data, 9 Gaussian basis functions, 1 data point

Example: Sinusoidal data, 9 Gaussian basis functions, 2 data points

Equivalent Kernel (1) The predictive mean can be written This is a weighted sum of the training data target values, tn. Equivalent kernel or smoother matrix.

Equivalent Kernel (2) Weight of tn depends on distance between x and xn; nearby xn carry more weight.

Equivalent Kernel (3) Non-local basis functions have local equivalent kernels: Polynomial Sigmoidal

Equivalent Kernel (4) The kernel as a covariance function: consider We can avoid the use of basis functions and define the kernel function directly, leading to Gaussian Processes (Chapter 6).

Equivalent Kernel (5) for all values of x; however, the equivalent kernel may be negative for some values of x. Like all kernel functions, the equivalent kernel can be expressed as an inner product: where .

Bayesian Model Comparison (1)
How do we choose the ‘right’ model? Assume we want to compare models Mi, i=1, …,L, using data D; this requires computing Bayes Factor: ratio of evidence for two models Posterior Prior Model evidence or marginal likelihood

Having computed p(MijD), we can compute the predictive (mixture) distribution A simpler approximation, known as model selection, is to use the model with the highest evidence.

For a model with parameters w, we get the model evidence by marginalizing over w Note that

For a given model with a single parameter, w, con-sider the approximation where the posterior is assumed to be sharply peaked.

Taking logarithms, we obtain With M parameters, all assumed to have the same ratio , we get Negative Negative and linear in M.

Matching data and model complexity

The Evidence Approximation (1)
The fully Bayesian predictive distribution is given by but this integral is intractable. Approximate with where is the mode of , which is assumed to be sharply peaked; a.k.a. empirical Bayes, type II or gene-ralized maximum likelihood, or evidence approximation.

From Bayes’ theorem we have and if we assume p(®,¯) to be flat we see that General results for Gaussian integrals give

Example: sinusoidal data, M th degree polynomial,

Maximizing the Evidence Function (1)
To maximise w.r.t. ® and ¯, we define the eigenvector equation Thus has eigenvalues ¸i + ®.

Maximizing the Evidence Function (2)
We can now differentiate w.r.t. ® and ¯, and set the results to zero, to get where N.B. ° depends on both ® and ¯.

Effective Number of Parameters (3)
w1 is not well determined by the likelihood w2 is well determined by the likelihood ° is the number of well determined parameters Likelihood Prior

Example: sinusoidal data, 9 Gaussian basis functions, ¯ = 11.1.

Example: sinusoidal data, 9 Gaussian basis functions, ¯ = 11.1. Test set error

Example: sinusoidal data, 9 Gaussian basis functions, ¯ = 11.1.

In the limit , ° = M and we can consider using the easy-to-compute approximation

Limitations of Fixed Basis Functions
M basis function along each dimension of a D-dimensional input space requires MD basis functions: the curse of dimensionality. In later chapters, we shall see how we can get away with fewer basis functions, by choosing these using the training data.

Chapter 8: graphical models

Bayesian Networks Directed Acyclic Graph (DAG)

Bayesian Networks General Factorization

Bayesian Curve Fitting (1)
Polynomial

Plate

Input variables and explicit hyperparameters

Bayesian Curve Fitting —Learning
Condition on data

Bayesian Curve Fitting —Prediction
Predictive distribution: where

Generative Models Causal process for generating images

Discrete Variables (1) General joint distribution: K 2 { 1 parameters Independent joint distribution: 2(K { 1) parameters

Discrete Variables (2) General joint distribution over M variables: KM { 1 parameters M -node Markov chain: K { 1 + (M { 1) K(K { 1) parameters

Discrete Variables: Bayesian Parameters (1)

Discrete Variables: Bayesian Parameters (2)
Shared prior

Parameterized Conditional Distributions
If are discrete, K-state variables, in general has O(K M) parameters. The parameterized form requires only M + 1 parameters

Linear-Gaussian Models
Directed Graph Vector-valued Gaussian Nodes Each node is Gaussian, the mean is a linear function of the parents.

Conditional Independence
a is independent of b given c Equivalently Notation

Conditional Independence: Example 1

Note: this is the opposite of Example 1, with c unobserved.

Note: this is the opposite of Example 1, with c observed.

“Am I out of fuel?” B = Battery (0=flat, 1=fully charged)
and hence B = Battery (0=flat, 1=fully charged) F = Fuel Tank (0=empty, 1=full) G = Fuel Gauge Reading (0=empty, 1=full)

“Am I out of fuel?” Probability of an empty tank increased by observing G = 0.

“Am I out of fuel?” Probability of an empty tank reduced by observing B = 0. This referred to as “explaining away”.

D-separation A, B, and C are non-intersecting subsets of nodes in a directed graph. A path from A to B is blocked if it contains a node such that either the arrows on the path meet either head-to-tail or tail-to-tail at the node, and the node is in the set C, or the arrows meet head-to-head at the node, and neither the node, nor any of its descendants, are in the set C. If all paths from A to B are blocked, A is said to be d-separated from B by C. If A is d-separated from B by C, the joint distribution over all variables in the graph satisfies

D-separation: Example

D-separation: I.I.D. Data

Directed Graphs as Distribution Filters

The Markov Blanket Factors independent of xi cancel between numerator and denominator.

Cliques and Maximal Cliques

Joint Distribution where is the potential over clique C and is the normalization coefficient; note: M K-state variables  KM terms in Z. Energies and the Boltzmann distribution

Illustration: Image De-Noising (1)
Original Image Noisy Image

Noisy Image Restored Image (ICM)

Restored Image (ICM) Restored Image (Graph cuts)

Converting Directed to Undirected Graphs (1)

Converting Directed to Undirected Graphs (2)
Additional links

Directed vs. Undirected Graphs (1)

Directed vs. Undirected Graphs (2)

Inference in Graphical Models

Inference on a Chain

Inference on a Chain To compute local marginals:
Compute and store all forward messages, Compute and store all backward messages, Compute Z at any node xm Compute for all variables required.

Trees Undirected Tree Directed Tree Polytree

Factor Graphs

Factor Graphs from Directed Graphs

Factor Graphs from Undirected Graphs

The Sum-Product Algorithm (1)
Objective: to obtain an efficient, exact inference algorithm for finding marginals; in situations where several marginals are required, to allow computations to be shared efficiently. Key idea: Distributive Law

Initialization

To compute local marginals: Pick an arbitrary node as root Compute and propagate messages from the leaf nodes to the root, storing received messages at every node. Compute and propagate messages from the root to the leaf nodes, storing received messages at every node. Compute the product of received messages at each node for which the marginal is required, and normalize if necessary.

Sum-Product: Example (1)

The Max-Sum Algorithm (1)
Objective: an efficient algorithm for finding the value xmax that maximises p(x); the value of p(xmax). In general, maximum marginals  joint maximum.

Maximizing over a chain (max-product)

Generalizes to tree-structured factor graph maximizing as close to the leaf nodes as possible

Max-Product  Max-Sum For numerical reasons, use Again, use distributive law

Initialization (leaf nodes) Recursion

Termination (root node) Back-track, for all nodes i with l factor nodes to the root (l=0)

Example: Markov chain

The Junction Tree Algorithm
Exact inference on general graphs. Works by turning the initial graph into a junction tree and then running a sum-product-like algorithm. Intractable on graphs with large cliques.

Loopy Belief Propagation
Sum-Product on general graphs. Initial unit messages passed across all links, after which messages are passed around until convergence (not guaranteed!). Approximate but tractable for large graphs. Sometime works well, sometimes not at all.

Pattern Recognition and Machine Learning

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