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Graphical Models

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Graphical Models

Tamara L Berg

CSE 595 Words & Pictures

HW3 online tonight

Start thinking about project ideas

Project proposals in class Oct 30

Come to office hours Oct 23-25 to discuss ideas

Projects should be completed in groups of 3

Topics are open to your choice, but need to involve some text and some visual analysis.

- Implement an existing paper
- Do something new
- Do something related to your research

- Flickr
- API provided for search. Example code to retrieve images and associated textual meta-data available here: http://graphics.cs.cmu.edu/projects/im2gps/

- SBU Captioned Photo Dataset
- 1 million user captioned photographs. Data links available here: http://dsl1.cewit.stonybrook.edu/~vicente/sbucaptions/

- ImageClef
- 20,000 images with associated segmentations, tags and descriptions available here: http://www.imageclef.org/

- Scrape data from your favorite web source (e.g. Wikipedia, pinterest, facebook)
- Rip data from a movie/tv show and find associated text scripts
- Sign language
- 6000 frames continuous signing sequence, for 296 frames, the position of the left and right, upper arm, lower arm and hand were manually segmented http://www.robots.ox.ac.uk/~vgg/data/sign_language/index.html

Discriminative version – build a classifier to discriminate between monkeys and non-monkeys.

P(monkey|image)

Generative version - build a model of the joint distribution.

P(image,monkey)

Can use Bayes rule to compute p(monkey|image) if we know p(image,monkey)

Can use Bayes rule to compute p(monkey|image) if we know p(image,monkey)

Generative

Discriminative

1. Quick introduction to graphical models

2. Examples:

- Naïve Bayes, pLSA, LDA

Slide from Dan Klein

Random variables

Let

be a realization of

Random variables

Let

be a realization of

A random variable is some aspect of the world about which we (may) have uncertainty.

Random variables can be:

Binary (e.g. {true,false}, {spam/ham}),

Take on a discrete set of values

(e.g. {Spring, Summer, Fall, Winter}),

Or be continuous (e.g. [0 1]).

Random variables

Let

be a realization of

Joint Probability Distribution:

Also written

Gives a real value for all possible assignments.

Also written

Joint Probability Distribution:

Given a joint distribution, we can reason about unobserved variables given observations (evidence):

Stuff you care about

Stuff you already know

Also written

Joint Probability Distribution:

One way to represent the joint probability distribution for discrete is as an n-dimensional table, each cell containing the probability for a setting of X. This would have entries if each ranges over values.

Also written

Joint Probability Distribution:

One way to represent the joint probability distribution for discrete is as an n-dimensional table, each cell containing the probability for a setting of X. This would have entries if each ranges over values.

Graphical Models!

Also written

Joint Probability Distribution:

Graphical models represent joint probability distributions more economically, using a set of “local” relationships among variables.

Graphical models offer several useful properties:

1. They provide a simple way to visualize the structure of a probabilistic model and can be used to design and motivate new models.

2. Insights into the properties of the model, including conditional independence properties, can be obtained by inspection of the graph.

3. Complex computations, required to perform inference and learning in sophisticated models, can be expressed in terms of graphical manipulations, in which underlying mathematical expressions are carried along implicitly.

from Chris Bishop

- Undirected (also called Markov Random Fields) - links express constraints between variables.
- Directed (also called Bayesian Networks) - have a notion of causality -- one can regard an arc from A to B as indicating that A "causes" B.

- Undirected (also called Markov Random Fields) - links express constraints between variables.
- Directed (also called Bayesian Networks) - have a notion of causality -- one can regard an arc from A to B as indicating that A "causes" B.

Directed Graph, G = (X,E)

Nodes

Edges

- Each node is associated with a random variable

Directed Graph, G = (X,E)

Nodes

Edges

- Each node is associated with a random variable

- Definition of joint probability in a graphical model:

- where are the parents of

Joint Probability:

0

0

0

0

1

1

1

1

0

0

0

0

1

0

1

1

1

1

0

1

0

0

1

1

Independence:

Conditional Independence:

Or,

By Chain Rule (using the usual arithmetic ordering)

Joint Probability:

By Chain Rule (using the usual arithmetic ordering)

Joint distribution from the example graph:

By Chain Rule (using the usual arithmetic ordering)

Joint distribution from the example graph:

Missing variables in the local conditional probability functions correspond to missing edges in the underlying graph.

Removing an edge into node i eliminates an argument from the conditional probability factor

- Graphs can have observed (shaded) and unobserved nodes. If nodes are always unobserved they are called hidden or latent variables
- Probabilistic inference in graphical models is the problem of computing a conditional probability distribution over the values of some of the nodes (the “hidden” or “unobserved” nodes), given the values of other nodes (the “evidence” or “observed” nodes).

Conditional Probabilities:

Marginalization:

- Exact algorithms
- Elimination algorithm
- Sum-product algorithm
- Junction tree algorithm

- Sampling algorithms
- Importance sampling
- Markov chain Monte Carlo

- Variational algorithms
- Mean field methods
- Sum-product algorithm and variations
- Semidefinite relaxations

1. Quick introduction to graphical models

2. Examples:

- Naïve Bayes, pLSA, LDA

C – Class

F - Features

We only specify (parameters):

prior over class labels

how each feature depends on the class

C – Class

F - Features

We only specify (parameters):

prior over class labels

how each feature depends on the class

C – Class

F - Features

We only specify (parameters):

prior over class labels

how each feature depends on the class

Slide from Dan Klein

Slide from Dan Klein

Slide from Dan Klein

Percentage of documents in training set labeled as spam/ham

Slide from Dan Klein

In the documents labeled as spam, occurrence percentage of each word

(e.g. # times “the” occurred/# total words).

Slide from Dan Klein

In the documents labeled as ham, occurrence percentage of each word

(e.g. # times “the” occurred/# total words).

Slide from Dan Klein

The class that maximizes:

- In practice
- Multiplying lots of small probabilities can result in floating point underflow

- In practice
- Multiplying lots of small probabilities can result in floating point underflow
- Since log(xy) = log(x) + log(y), we can sum log probabilities instead of multiplying probabilities.

- In practice
- Multiplying lots of small probabilities can result in floating point underflow
- Since log(xy) = log(x) + log(y), we can sum log probabilities instead of multiplying probabilities.
- Since log is a monotonic function, the class with the highest score does not change.

- In practice
- Multiplying lots of small probabilities can result in floating point underflow
- Since log(xy) = log(x) + log(y), we can sum log probabilities instead of multiplying probabilities.
- Since log is a monotonic function, the class with the highest score does not change.
- So, what we usually compute in practice is:

Download images from the web via a search query (e.g. penguin).

Re-rank images using a naïve Bayes model trained on text surrounding the images and meta-data features (image alt tag, image title tag, image filename).

Top ranked images used to train an SVM classifier to further improve ranking.

Steps:

- Detect and describe image patches
- Assign patch descriptors to a set of predetermined clusters (a vocabulary) with a vector quantization algorithm
- Construct a bag of keypoints, which counts the number of patches assigned to each cluster
- Apply a multi-class classifier (naïve Bayes), treating the bag of keypoints as the feature vector, and thus determine which category or categories to assign to the image.

C – Class

F - Features

We only specify (parameters):

prior over class labels

how each feature depends on the class

Problem: Categorize images as one of 7 object classes using Naïve Bayes classifier:

- Classes: object categories (face, car, bicycle, etc)
- Features – Images represented as a histogram where bins are the cluster centers or visual word vocabulary. Features are vocabulary counts.
treated as uniform.

learned from training data – images labeled with category.

1. Quick introduction to graphical models

2. Examples:

- Naïve Bayes, pLSA, LDA

Joint Probability:

Marginalizing over topics determines the conditional

probability:

Need to:

Determine the topic vectors common to all documents.

Determine the mixture components specific to each document.

Goal: a model that gives high probability to the words that appear in the corpus.

Maximum likelihood estimation of the parameters is obtained by maximizing the objective function:

Documents – Images

Words – visual words (vector quantized SIFT descriptors)

Topics – object categories

Images are modeled as a mixture of topics (objects).

They investigate three areas:

- (i) topic discovery, where categories are discovered by pLSA clustering on all available images.
- (ii) classification of unseen images, where topics corresponding to object categories are learnt on one set of images, and then used to determine the object categories present in another set.
- (iii) object detection, where you want to determine the location and approximate segmentation of object(s) in each image.

Most likely words for 4 learnt topics (face, motorbike, airplane, car)

Confusion table for unseen test images against pLSA trained on images containing four object categories, but no background images.

Confusion table for unseen test images against pLSA trained on images containing four object categories, and background images. Performance is not quite as good.

1. Quick introduction to graphical models

2. Examples:

- Naïve Bayes, pLSA, LDA

Per-document topic proportions

Per-word topic assignment

Observed word

pLSA

Per-document topic proportions

Per-word topic assignment

Observed word

Dirichlet parameter

Per-document topic proportions

Per-word topic assignment

Observed word

topics

Dirichlet parameter

Per-document topic proportions

Per-word topic assignment

Observed word

Joint distribution:

Topic discovery from a text corpus. Highly ranked words for 4 topics.

LDA in

Animals on the Web

Tamara L Berg, David Forsyth

Animals on the Web Outline:

- Harvest pictures of animals from the web using Google Text Search.
- Select visual exemplars using text based information LDA
- Use vision and text cues to extend to similar images.

Text Model

Latent Dirichlet Allocation (LDA) on the words in collected web pages to discover 10 latent topics for each category.

Each topic defines a distribution over words. Select the 50 most likely words for each topic.

Example Frog Topics:

1.) frog frogs water tree toad leopard green southern music king irish eggs folk princess river ball range eyes game species legs golden bullfrog session head spring book deep spotted de am free mouse information round poison yellow upon collection nature paper pond re lived center talk buy arrow common prince

2.) frog information january links common red transparent music king water hop tree pictures pond green people available book call press toad funny pottery toads section eggs bullet photo nature march movies commercial november re clear eyed survey link news boston list frogs bull sites butterfly court legs type dot blue

Rank images according to whether they have these likely words near the image in the associated page.

Select up to 30 images per topic as exemplars.

1.) frog frogs water tree toad leopard green southern music king irish eggs folk princess river ball range eyes game species legs golden bullfrog session head ...

2.) frog information january links common red transparent music king water hop tree pictures pond green people available book call press ...

An unsupervised approach to learn and recognize natural scene categories. A scene is represented by a collection of local regions. Each region is represented as part of a “theme” (e.g. rock, grass etc) learned from data.

1.) Choose a category label (e.g. mountain scene).

2.) Given the mountain class, draw a probability vector that will determine what intermediate theme(s) (grass rock etc) to select while generating each patch of the scene.

3.) For creating each patch in the image, first determine a particular theme out of the mixture of possible themes, and then draw a codeword given this theme. For example, if a “rock” theme is selected, this will in turn privilege some codewords that occur more frequently in rocks (e.g. slanted lines).

4.) Repeat the process of drawing both the theme and codeword many times, eventually forming an entire bag of patches that would construct a scene of mountains.

Left - distribution of the 40 intermediate themes.

Right - distribution of codewords as well as the appearance of 10 codewords selected from the top 20 most likely codewords for this category model.

Left - distribution of the 40 intermediate themes.

Right - distribution of codewords as well as the appearance of 10 codewords selected from the top 20 most likely codewords for this category model.

correct

incorrect

correct

incorrect