Bayesian Belief Networks BBNs

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Bayesian Belief Networks (BBNs)

• Microsoft Belief Networks (MSBNx)
• Examples are from
• http//www.dcs.qmw.ac.uk/norman/BBNs/Definition_o
  f_BBNs__graphs_and_probability_tables.htm

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• This ppt demonstrates the Microsoft Belief
  Networks software which can be found at
  http//research.microsoft.com/adapt/MSBNx/

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Bayesian Belief Networks

• A BBN is a directed, acyclic graph together with
  an associated set of probability tables. The
  graph consists of nodes and arcs.
• The nodes represent random variables which can be
  discrete or continuous. For example, a node might
  represent the variable 'Train strike' which is
  discrete, having the two possible values 'true'
  and 'false'.
• The arcs can be thought as causal relationships
  between variables, but in general,
• an arc from X to Y means
• X has direct influence on our belief in Y.

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• The key feature of BBNs is that they enable us to
  model and reason about uncertainty.
• In our example, a train strike does not imply
  that Norman will definitely be late (he might
  leave early and drive), but there is an increased
  probability that he will be late.
• In the BBN we model this by filling in a
  conditional probability table for each node.

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Conditional Probabilities in BBN

• This is actually the conditional probability of
  the variable 'Norman late' given the variable
  'train strike'.

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Microsoft MSBNx

• Add nodes (right click on the background)
• Once the BBN is fixed, it does not let you add
  new nodes!
• Add arcs (choose two nodes with Ctrl and choose
  Add Dependency Arc)
• Add conditional probabilities (right click on one
  of the nodes, and choose assess, and modify the
  given distribution (which is set at 0.5 as
  default)
• See the current probabilities (right click on one
  of the nodes, and choose barchart)
• Entering hard evidence (right click on the node,
  and choose evidence, and choose yes (or
  no))

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• Having entered the probabilities we can now use
  Bayesian probability to do various types of
  analysis.
• For example, we might want to calculate the
  probability that Norman is late
• p(Norman late) p(Norman late,train strike)
  p(Norman late,?train strike)
• p(Norman late train
  strike)p( train strike)
• p(Norman late ?
  train strike)p(? train strike)
• (0.8 0.1) (0.1
  0.9) 0.17
• This is the unconditional probability of Norman
  being late.
• Similarly, the unconditional probability that
  Martin is late is 0.51.

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You see the network like this (with current
probabilities) if you click on BarChart over a
node.
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Entering Hard Evidence - 1

• However, the most important use of BBNs is in
  revising probabilities in the light of actual
  observations of events.
• Suppose, for example, that we know there is a
  train strike. In this case we can enter the
  evidence that 'train strike' true.
• The conditional probability tables already tell
  us the revised probabilities for Norman being
  late (0.8) and Martin being late (0.6).

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Note This direction is easy, what about belief
update in the other direction?
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Entering Hard Evidence - 2

• Suppose now that we do not know if there is a
  train strike but we do know that Norman is late.
  Then we can enter the evidence that 'Norman late'
  true and we can use this observation to
  determine
• a) the (revised) probability that there is a
  train strike and
• b) the (revised) probability that Martin will be
  late.
• To calculate a) we use Bayes theorem
• p(trainstrikenormanlate)
• p(normanlatetrainstrike)
  p(trainstrike)/ p(normanlate)
• 0.80.1/0.17 0.47
• Notice that we use p(normanlate) as we know
  it, the fact that we observed it this time does
  not change its probability.

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Enterin Hard Evidence - 2

• Now we can use this revised probability to
  calculate probability of Martin being late
• p(martinlate) p(martinlate trainstrike)p(
  trainstrike)
• p(martinlate ?
  trainstrike)p(? trainstrike)
• (0.6 0.47) (0.5
  0.53) 0.547
• from the revised probabilities
  from the CPT

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• Lets look at a more complex network with 4
  random variables
• In this case we have to construct a new
  conditional probability table for node B ('Martin
  late') to reflect the fact that it is conditional
  on two parent nodes (A and D).
• We also have to provide a probability table for
  the new root node D ('Martin oversleeps').
• Martin oversleeps True 0.4 False 0.6

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a more complicated model, showing PRIOR
probabilities
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Diverging connection
Entering hard evidence we saw that Norman is
late. Note that our belief in TrainStrike and
Martin being late are both increased.
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Diverging Connections Blocking Propagation

• Suppose now that we have hard evidence about
  TrainStrike, that is we know for certain whether
  or not there is a train strike.
• In this case any evidence about Martinlate does
  not change in any way our belief about
  Normanlate this is because the certainty of
  TrainStrike blocks the evidence from being
  transmitted (it becomes irrelevant once we know
  TrainStrike for certain).
• Because the independence of Normanlate and
  Martinlate is conditional on the certainty of
  TrainStrike, we say formally that Normanlate and
  Martinlate are conditionally independent (given
  TrainStrike).

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Converging connection
Entering some evidence (hard or soft) about
MartinLate is propagated to TrainStrike and
Oversleep (and also NormanLate).
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Converging connection
However, if we have no info about MartinLate,
Oversleep and TrainStrike are independent no
evidence is transmitted between them.
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Converging Connections - Overview
Clearly any evidence about B or C is transmitted
to A. On the other hand, if anything is known
about A (even so-called soft evidence ) then the
parents of A become dependent. For example,
suppose that Martin usually hangs up his coat in
the hall as soon as he gets in to work. Then if
we observe that Martin's coat is not hung up
after 9.00am our belief that he is late increases
(note that we do not know for certain that he is
late - we have soft as opposed to hard evidence -
because on some days Martin does not wear a
coat). Even this 'soft' evidence about Martin
being late increases our belief in both BMartin
oversleeping and CTrain delays.
It follows that in a converging connection,
evidence can only be transmitted between the
parents B and C when the converging node A has
received some evidence (which can be soft or
hard).
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Propagation Serial connections
What about the other direction? (we have some
evidence about C)?
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• You are responsible in understanding the basics
• what the BBN graph representation mean
  (conditional independence)
• how to form the simplest graph using conditional
  independence assumptions
• how a single hard evidence updates the other
  probabilities (past several slides)
• how inference is done (chapter14b-BBN.pdf)
• exhaustive
• modifications (factoring)
• stochastic