This document discusses probabilistic models for inference using Hidden Markov Models (HMM) and Bayesian networks. It provides references on HMM, Bayesian probability, and temporal models. It explains that probabilistic models are needed to handle uncertain knowledge and probabilistic reasoning, unlike logic-based models. The document outlines contents on learning and inference in HMM and Bayesian networks. It discusses uncertainty, Bayesian probability, generative models, inferences in Bayesian networks, and using temporal models like HMM. Mathematical representations of inference in HMM are also presented.

1. INFERENCE IN HMM AND BAYESIAN NETWORKS
DR MINAKSHI PRADEEP ATRE
PVG’S COET, PUNE

2. REFERENCES
 Journal paper, titled. “An Introduction of Hidden Markov Model and Bayesian Network by
Zoubain Ghahramani (2001)
 https://www.cs.ubc.ca/~murphyk/Bayes/bnintro.html
 https://www.youtube.com/watch?v=kqSzLo9fenk (Bayes Theorem and HMM)
 https://www.youtube.com/watch?v=YlL0YARYK-o (HMM)
 https://www.youtube.com/watch?v=EqUfuT3CC8s (Markov models)
 https://www.youtube.com/watch?v=5araDjcBHMQ (maths for Markov Model & time series)
 https://aimacode.github.io/aima-exercises/
 https://aimacode.github.io/aima-exercises/bayesian-learning-exercises/
 15-381: Artificial Intelligence (ppt)

3. WHY WE NEED PROBABILISTIC MODELS?
Probabilist
ic Models
Inferences
Unable to handle:
1) uncertain knowledge
and 2) probabilistic
reasoning
FOL

4. ONTOLOGY
 a set of concepts and categories in a subject area or
domain
 that shows their properties and the relations between
them

5. WHY WE NEED PROBABILISTIC MODELS?
 The FOL is used for inference
 There were 2 methods:
 Automated inference
 Forward checking
 Backward checking
 Resolution refutation
Propositional logic
has a very limited
ontology, making
only the
commitment that
the world consists
of facts
First order logic
overcomes the
limitation of this
ontology and has
different inference
methods
Objects
Propertie
s
Relations
Unable to handle:
uncertain
knowledge and
probabilistic
reasoning

6. CONTENTS : PROBABILISTIC MODELS FOR INFERENCE
 Learning and inference in hidden
Markov Model (HMM) in context with
Bayesian network
 Uncertainty and methods,
 Bayesian Probability and Belief network,
 probabilistic Reasoning,
 Generative models: Bayesian networks,
 inferences in Bayesian networks,
 Temporal models: Hidden Markov
models
Represents knowledge
and data as a fixed set
of random variables
with a joint probability
distribution

7. UNCERTAINTY AND METHODS

8. BAYESIAN PROBABILITY
 Solved examples on Bayesian Probability

9. SUM OF BASIC PROBABILITY FORMULAE

10. LIMITATIONS OF BAYESIAN
 What’s wrong with Bayesian networks
 Bayesian networks are very useful for modeling joint distributions
 • But they have their limitations:
 - Cannot account for temporal / sequence models
 - DAG’s (no self or any other loops)

11. CONCLUSION ( BAYESIAN LEARNING METHODS)
 Bayesian learning methods are firmly based on probability theory
and exploit advanced methods developed in statistics.
 Naïve Bayes is a simple generative model that works fairly well in
practice.
 A Bayesian network allows specifying a limited set of dependencies
using a directed graph.
 Inference algorithms allow determining the probability of values
for query variables given values for evidence variables.

12. WHAT ARE BELIEF NETWORKS?
 Def = efficient reasoning with probability is so new that there is one main approach—belief networks
 Conditional independence information is a vital and robust way to structure information about an
uncertain domain
 Belief networks are a natural way to represent conditional independence information
 The links between nodes represent the qualitative aspects of the domain, and the conditional probability
tables represent the quantitative aspects
 A belief network is a complete representation for the joint probability distribution for the domain, but is
often exponentially smaller in size
 Inference in belief networks means computing the probability distribution of a set of query variables,
given a set of evidence variables.

13.  Belief networks can reason causally, diagnostically, in mixed mode, or intercausally.
 No other uncertain reasoning mechanism can handle all these modes
 The complexity of belief network inference depends on the network structure
 In polytrees (singly connected networks), the computation time is linear in the size of the network
 There are various inference techniques for general belief networks, all of which have exponential complexity in
the worst case.
 In real domains, the local structure tends to make things more feasible, but care is needed to construct a
tractable network with more than a hundred nodes
 It is also possible to use approximation techniques, including stochastic simulation, to get an estimate of the
true probabilities with less computation
 Various alternative systems for reasoning with uncertainty have been suggested. All the truth-functional
systems have serious problems with mixed or intercausal reasoning

14.  In the context of using Bayes' rule, conditional independence relationships among variables can simplify
the computation of query results and greatly reduce the number of conditional probabilities that need
to be specified.
 We use a data structure called a belief network' to represent the dependence between variables and to
give a concise specification of the joint probability distribution.
 A belief network is a graph in which the following holds:
 1. A set of random variables makes up the nodes of the network.
 2. A set of directed links or arrows connects pairs of nodes. The intuitive meaning of an arrow from node X to
node Y is that X has a direct influence on Y.
 3. Each node has a conditional probability table that quantifies the effects that the parents have on the node. The
parents of a node are all those nodes that have arrows pointing to it.
 4. The graph has no directed cycles (hence is a directed, acyclic graph, or DAG).

15. A TYPICAL BELIEF NETWORK
 Consider the following situation.
 You have a new burglar alarm installed at home.
 It is fairly reliable at detecting a burglary, but also responds
on occasion to minor earthquakes. (This example is due to
Judea Pearl, a resident of Los Angeles; hence the acute
interest in earthquakes.)
 You also have two neighbors, John and Mary, who have
promised to call you at work when they hear the alarm.
 John always calls when he hears the alarm, but sometimes
confuses the telephone ringing with the alarm and calls
then, too.
 Mary, on the other hand, likes rather loud music and
sometimes misses the alarm altogether.
 Given the evidence of who has or has not called, we would
like to estimate the probability of a burglary.
 This simple domain is described by the belief network in
Figure

16. DISCUSSION (PAGE 456 RUSSELL & NORVIG)
 The topology of the network can be thought of as an abstract knowledge
base that holds in a wide variety of different settings, because it represents
the general structure of the causal processes in the domain rather than any
details of the population of individuals.
 In the case of the burglary network, the topology shows that burglary and
earthquakes directly affect the probability of the alarm going off, but whether
or not John and Mary call depends only on the alarm—the network thus
represents our assumption that they do not perceive any burglaries directly,
and they do not feel the minor earthquakes.

17. BELIEF NETWORKS : LEARNING IN BELIEF NETWORKS
 There are four kinds of belief networks, depending upon whether the structure of the network is known or
unknown, and whether the variables in the network are observable or hidden.
 known structure, fully observable -- In this case the only learnable part is the conditional probability tables.
These can be estimated directly using the statistics of the sample data set.
 unknown structure, fully observable -- Here the problem is to reconstruct the network topology. The
problem can be thought of as a search through structure space, and fitting data to each structure reduces to
the fixed-structure problem, so the MAP or ML probability value can be used as a heuristic in hill-climbing or
SA search.
 known structure, hidden variables -- This is analagous to neural network learning.
 unknown structure, hidden variables -- When some variables are unobservable, it becomes difficult to apply
prior techniques for recovering structure, but they require averaging over all possible values of the unknown
variables. No good general algorithms are known for handling this case.

18. HMM: TEMPORAL MODELS
Inference
has 3
factors
filtering
prediction
smoothing

19. HMM: TEMPORAL MODELS
 Inference in temporal model :
 filtering
 prediction
 smoothing

20. INFERENCE: MATHEMATICAL REPRESENTATION

21. MARKOV CHAIN

22. MARKOV CHAIN & EXAMPLE

23. WHAT’S NEXT : UNIT 4 LEARNING

24. LEARNING ALGORITHMS
1
Bayesian (naïve Bayes)
2
Decision tree
3
Neural Networks

25. THANK YOU