Pattern 4 (Examining several consequences in succession) The verication of a new consequence enhances our condence in the conjecture, unless the new
4.3 Extracting a state of belief
4.3 Extracting a state of belief
Observation 4.1.1 is powerful in extracting states of belief because of a claim that independence assertions are induced by the \causal structure" underlying the domain of interest. Let me rst dene the notion of a causal structure and then state the claim.
Denition 4.3.1
A causal structure over primitive propositions N is a binary rela-tion DC N N, where iDC j reads as i directly causes j. We say that i causes j (or, j is an eect of i) precisely when1. i directly causes j, or
2. i causes k and k directly causes j.
In a causal structure, a proposition cannot cause itself.
It is common to represent a causal structure using a directed acyclic graph. For example, the causal structure
Rain directly causes Slippery Road Rain directly causes Wet Grass Sprinkler On directly causes Wet Grass
is depicted graphically in Figure 2. We simply create a node for each primitive proposition, and then add an arc from nodeito nodejprecisely whenidirectly causes j. By denition of a causal structure, the resulting directed graph is guaranteed to be acyclic.
Below is a key claim that underlies the way independence assertions are induced in many practical applications.
Claim 4.3.2
The expert who identies a causal structure nds a proposition independent from its non{eects given its direct causes.Slippery Road
Sprinkler On
Wet Grass Rain
Figure 2: A graphical representation of a causal structure.
The intuition here is that information about the non{eects of a proposition is rele-vant to the proposition only because it conveys information about its direct causes.
Therefore, the information becomes useless once the state of direct causes is known.
For example, in Figure 2, information about Slippery Road is relevant to Wet Grass only because it conveys information about Rain. However, once the state of Rain is known, information about Slippery Road is no longer relevant to Wet Grass.
In practical applications, it is common to ask a domain expert to identify a causal structure, and then use Claim 4.3.2 to induce constraints on the state of belief held by the expert. The soundness of this practice hinges on the correctness of Claim 4.3.2, which must be relative to some formal denition of causation. However, due to the lack of a universally accepted denition of causation, we are in no position to provide a satisfactory proof of Claim 4.3.2. Nevertheless, the role played by Claim 4.3.2 in practical applications is not minor.
Given the discussion of Section 4.1, and Claim 4.3.2, an expert who identies a causal structure is in fact supplying us with constraints on her state of belief. We have seen in Section 4.1 how these constraints reduce the exponential number of supports required to specify a state of belief. This section focuses on the supports needed to complete the specication of a state of belief.
One completes the specication of a state of belief by quantifying a causal struc-ture. Quantication is the process in which an expert assesses her support for each
4.3. EXTRACTINGA STATE OF BELIEF 43 primitive proposition given a particular state of its direct causes. For example, the node Slippery Road in Figure 2 is quantied by providing four conditional supports:
1. The conditional support for Slippery Road given Rain.
2. The conditional support for Slippery Road given :Rain.
3. The conditional support for:Slippery Road given Rain.
4. The conditional support for:Slippery Road given :Rain.
These supports constitute the quantication of node Slippery Road. The quantica-tion of node Wet Grass, however, consists of eight condiquantica-tional supports because there are four possible states of the direct causes of Wet Grass.
A quantied causal structure is called a causal network and is usually depicted graphically using two components:
A direct acyclic graph, which species a causal structure.
A set of tables, which contain node quantications. Each node has a table associated with it. The table has two columns corresponding to the states of the node. It has one row for each state of the node's direct causes. Each entry in the table is a support for some state of the node conditioned on a particular state of its direct causes. The entries of each row must sum up to the full support.
Figure 3 depicts a causal network with respect to the support structure h[0;1];+;=i. Figure 4 depicts another causal network but with respect to the support structure
hf0;1;...;1g;min;?i. The causal networks in Figures 3 and 4 share the same causal structure, but they dier in the way they are quantied. We shall look at more methods of quantication in the following chapters.
The formal denition of a causal network is given below.
Not Slippery Slippery
.8 .2
.05 .95
No Rain Rain
Dry .99 .85 .98 .01
.01 .15
.99 .02 Wet Rain and On Rain and Off No Rain and On No Rain and Off
Probabilistic Causal Network
On .5 .5
Off Rain No Rain
0 1
Rain Sprinkler On
Wet Grass Slippery Road
Figure 3: A probabilistic causal network. The top left entry of the bottom left table reads, The probability ofRaingivenSlippery Road is .8.
Notation
idenotes the parents of node i. L(N) denotes a propositional language constructed from primitive propositions N.Denition 4.3.3
A causal network with respect to a support structure hS;;i is a triple hN;G;CSi, whereN is a set of primitive propositions.
G is a directed acyclic graph over N.
CS is a partial function L(N)L(N)!S such that
{
CSi(i) is dened, and{
CSi(i)CSi(:i) =1
,for every node i.
CS is called a conditional support function.
4.3. EXTRACTINGA STATE OF BELIEF 45
On Off Rain No Rain
Not Slippery Slippery
No Rain Rain
Dry Wet Rain and On Rain and Off No Rain and On No Rain and Off
0 17
44 0
0 63
20 0
63 0
57 0
0 0 0
Spohnian Causal Network
infinity
Rain Sprinkler On
Wet Grass Slippery Road
Figure 4: A Spohnian causal network. The top left entry of the bottom left table reads, The impossibility ofRaingiven Slippery Roadis 0.
A causal network consists of two sets of constraints on a state of belief. The rst set is about independence assertions, while the second is about conditional supports.
Since the goal of constructing a causal network is to specify a state of belief, it is most important to know whether a given state of belief satises the constraints imposed by a causal network.
Notation
i/ denotes the non{descendents of node i.Denition 4.3.4
A state of belief over propositions N satises a causal networkhN;G;CSi precisely when
IN(i;i;i/ni) and i(i) =CSi(i) for every node i.
If we view a causal network as a set of constraints, then the denition of a causal network does not always guarantee the consistency of these constraints. Although we can nd a state of belief that satises the independences asserted by a causal structure, it is not always possible to nd a state of belief that satises the conditional supports quantifying the causal structure. However, causal networks with respect to
a distributive and bijective support structure are always consistent.
Theorem 4.3.5
An abstract causal network that is induced with respect to a distribu-tive and bijecdistribu-tive support structure is satised by exactly one state of belief.Probabilistic and Sphonian causal networks are induced with respect to distributive and bijective support structures.