Causal Decision Theory and Conditioning: a Primer

Standard Savage decision theory as well as Richard Jeffrey’s alternative, address a normative problem for an odd doxastic condition.  an agent fully believes:

·      a set of all of the available, mutually exclusive actions;

·      a set of exhaustive and mutually exclusive possible states of the world;

·      a set of consequences—outcomes—of each possible state of the world/action pair.

and the agent:

·      Has coherent degrees of belief in the possible states of the world;

·      Has utilities (or in Jeffrey’s version, desirabiities) for the outcomes.

The normative question is which action the agent ought to take. The answer offered is the action, or one of them, that maximizes the expected utility, where the expectation is with respect to the degrees of belief in the states of the world.

From an ideal Bayesian perspective, what is essential is the distinction between actions and outcomes and their costs or values.  The ideal Bayesian knows which actions have the maximal expected utility. The states of the world are gratuitous.  Followers of Savage, or Jeffrey’s in effect assume the agent only obtains the expected utilities by calculating them using the specified states of the world and probabilities of outcomes, given the various possible states of the world and actions.

What is odd is that no epistemological problem is considered about how an agent knows, or could know, or rationally assess, the possible states of the world and their probabilities, the possible actions, or the probabilities of outcomes effected by alternative actions in the several possible states of the world.  a thorough subjectivist such as Jeffrey would answer these questions: all that is relevant are the agent’s degrees of belief about actions, states of the world,and outcomes and their desirabilities.  Epistemology reduces to observing, Bayesian updating, and rather trivial computation. Be that as it may, or may not, causal decision theory considers two kinds of complications.

1.     The agent believes that the action chosen will influence the state of the world.

2.     The agent believes that the state of the world will influence the action chosen;

This is already a conceptual expansion for the agent, to include causal relations and probabilities of actions.

In case 1, how should the agent take account of the belief that the choice of action will be influenced by the state of the world?  For simplicity, first assume the outcome is a deterministic function of the action, a, and the state, s, of the world, and the utility is U(o(a, s) where o is some function actions and states.

Proposal 1:  Calculate the expected utility for each action as the sum over states of the world of the utility of each action in that state of the world multiplied by the probability of that state of the world given the action:

(1) Exp(U(a))  = Σs U(o(a,s)) Prob(s | a)

In Savage theory the last factor on the right hand side of (1) and (2) is just Prob(s)

One “partition question” concerns whether the action that maximizes utility is the same depending on how the set of states is “partitioned.” Let S be a variable that ranges over some finite set of values, s1,…,sn.  a coarsening of S is a set S1 = {{s1 v..v sk}, {sk +1 v …v sm},….{sm v…v sn}}, etc. a refinement is the inverse.

Coarsening can change the probability of an outcome on an action. Let S = {s1, s2, s3} and suppose S’ is a coarsening of S to {(s1 v s2), s3}. For all outcomes o and actions a, let o and a be independent conditional on s1 and likewise on s2 and s3, but S not be independent of A.  Then for any outcome in O:

P(O | a, (s1 v s2)) = P(O, | (a,s1 v a,s2) = P (O, a, (s1 v s2)) / (P(a,s1 v a,s2))  =

P((O, a, s1) v  P(O, a, s2)) / (P(a,s1 v a,s2)) =

P(O, a, s1) + P(O, a, s2) / ((P(a,s1) +P( a,s2)) =

[P(O | a, s1) P(a, s1) + P(O | a, s2)] / ((P(a,s1) +P( a,s2)) =

[P(O | s1) P(a, s1) + P(O | s2) P(a, s2)] / ((P(a,s1) +P( a,s2)) =

(P(a) [P(O | s1) P(s1 | a) + P(O | s2) P(s2 | a)]) / (P(a)( (P(s1 | a) +P(s2) | a)) =

[P(O | s1) P(s1 | a) + P(O | s2) P(s2 | a)] / (P(s1 | a) + P(s2) | a))

The probability distribution of 0 given the state s1 v s2 in S’ varies as the conditional probabilities of s1 and, respectively, of s2 vary with the value of A they are conditioned on, and O and A are not independent in S’ but they are independent—by assumption—in S.  

For case 2, the results and the argument are similar.  The general point is an old one, Yule’s (on the mixture of records).

The partitioning problem does not apply to Savage’s theory—it makes no difference how the range of possible state values are cut up into new coarsened variables.  

So decision theory when the actions influence the states or the states influence the actions is up in the air—the right decision depends on the right way to characterize the states.  Various writers, Lewis, Skyrms, Woodruff and others, have proposed vague or ad hoc or infeasible solutions. Lewis proposed to chose the most specific “causally relevant” partition, which I take to mean the finest partition for which there is a difference  in elements of the partition in the probabilities of outcomes conditional on actions. Skyrms objects that this is often unknowable, and proposes an intricate set of alternative conditions, which Woodruff generalizes. The general strategy is to embed the problem in a logical theory of conditonals, and entwine it with accounts of “chance”and relations of chance and degrees of belief, e.g., the principal principle. The general point is hard to extract.

When states influence actions Meek and Glymour propose that there are two theories. One simply calculates the expected values of the outcomes on various actions as with Jeffrey’s decision theory, the other assumes that a decisive act is done with freedom of the will, represented as an exogenous variable, that breaks the influence of the state on the act.  

Appealing as the second story may be to our convictions about our own acts as we do them, or deliberate on what to do, it is of no avail when the actions influence the states, not vice-versa. For that case, one either knows the total effect of an action on the outcome, or one doesn’t, and if one doesn't, there is nothing for it except to know what the states are that make a difference.  One would think serious philosophy would have focused then on means to acquire such knowledge. One would be wrong.