https://xkcd.com/1236/
Graphical models
Plate notation
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Conditional independence
Quiz 2 remark
Everything comes
from the sum rule and the product
models provide a rich set of modeling language and inference procedures.
[Bishop 8.1 and 8.2]
What is graphical models /
Bayesian network
rule … but graphical
Motivating scenario
– computer vision
Motivation – encoding
(in)dependence
Motivation – encoding
(in)dependence
Motivation – encoding
(in)dependence
Motivation – encoding
(in)dependence
Probabilistic graphical model –
Nodes: random variables
Edges: probabilistic
Graph: captures ¡°structures¡± in
relationships
joint distribution
terminologies
Inference (Ch 8.4) – compute the posterior distributions of one or more
subsets of other nodes, given values of a subset of nodes (in a known graphical model with known
parameters)
Learning – estimate the parameters
(conditional probabilities) for a given graphical model, from a dataset of observed values.
Probabilistic graphical model –
Nodes: random variables
Edges: probabilistic relationships
Graph: captures ¡°structures¡± in the joint distribution
overarching goals
LHS of (8.2) is symmetric w.r.t a, b, c,
¡ú Fig 8.2 implicitly chose an ordering
but RHS is not.
¡ú possible to have different ordering, different decomposition, different graphical models
networks for any distribution
This decomposition holds for any distribution over (x1, … xK).
Graph is ¡°fully connected¡± — there is a link (in either direction) between any pair
joint distributions ¡ú Graph
… it is the absence of links in the graph that conveys interesting information about the properties of the class of distributions that the graph represents.
Graph ¡ú joint distributions
General Bayes nets
general, the joint distribution is given
General Bayes nets
general, the joint distribution is given
Polynomial regression: more variables
Generative
Ancestral sampling:
allows us to draw samples from
¡ñ Work though each node in order
¡ñ Draw lower-number nodes (parents)
¡ñ Draw xk with values of its parents fixed
¡ñ To obtain samples from marginals, retain
disregard the rest
a (joint) distribution.
the variables in question and
Generative
a typical application
¡ñ Higher-numbered nodes: terminal nodes, observed
¡ñ Lower-numbered nodes: latent variables, unobserved
¡ú to allow a complex distribution over the observed to factor into simpler conditional distributions
BN graphs represent causal processes.
Polynomial regression example isn¡¯t a generative model (but can be made so with more complexity)
Latent variables need not have physical interpretation (can be introduced representational/computational reasons)
Number of para
meters for
discrete variables
Read: 8.1.3 in Bishop
Graphical models
What is graphical models /
Plate notation
Conditional independence
Bayesian network
Three-node graphs
Goal: infer
conditional
independence of a and b
Larger goal: spot general subgraph patterns so as to reason about conditional independence in general.
Head-Tail (HT) pattern
Head-Head (HH) pattern
bit more subtle
Head-Head (HH) pattern
Head-Head (HH) pattern
Head-Head (HH) pattern
pattern and
¡°explaining away¡±
Conditional
¡ú known as
independence for
¡°D-separation¡±
D-separation:
blocked paths
D-separation
D-separation:
Graphical model as filters for probability distributions
Graphical models: Bayes
What is graphical models /
Plate notation
Conditional independence
Bayesian network
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