The Predictor neither has a time machine nor some gift for backward causation. It is assumed, however, that the Predictor is rather reliable, though not necessarily infallible. Should the Player choose the opaque box only or both boxes?

A natural starting-point is to specify pay-off matrix or utility function for the Player. Letting `1′ and `2′ be shorthand for “the opaque box only” and “both boxes”, respectively, the pay-off matrix can be written as below.

At first sight, one may think that the next step is to specify a probability distribution and compute the expected utility conditional on the Player’s actual choice. This requires some careful thought, however, because in standard decision theory there are no probabilities associated with one’s own actions. From the Player’s perspective, a decision theoretic treatment assumes that the actual choice is simply a variable under the Player’s control, not a random variable that is correlated with previous events. But it is part of the problem formulation that the Predictor is reliable, though perhaps not infallible. Therefore the Player’s actual choice must be strongly correlated in some way with the Predictor’s prediction.

One possible conclusion at this point is that the problem formulation is simply contradictory or impossible. But let’s be open-minded enough to accept the problem formulation. How, then, can one incorporate the stipulated correlation between the predicted choice and the actual choice into an analysis of the Player’s optimal decision strategy? Because exotic scenarios like time travel and backward causation are not considered, the most natural approach is to postulate a common cause between the Predictor’s prediction and the Player’s actual choice. The Predictor might, for instance, be a psychologist or a close friend with extensive knowledge about the Player’s personality traits. This suggests a probability distribution that conforms to the following Bayesian network:

Bayesian network for the Newcomb problem.

The arrows in this graph define the statistical dependencies between variables. For a sufficiently pragmatic understanding of causation, they also represent causal relationships between variables. Mathematically, the Bayesian network expresses the claim that the joint probability distribution factorizes as

.

Because the content of the opaque box is deterministically determined by the prediction, the variable may be eliminated from the probability space to give

.

In order to specify the probabilities, let’s for simplicity suppose that the Player’s choice is the outcome of a two-step process. First, the opaque box only () is chosen with probability and both boxes () with probability . Of special interest are the cases (a Player trustful of Predictor’s abilities) and (a greedy distrustful Player). Secondly, this choice may be overturned due to the Player’s impulsivity with probability . Let’s further say that the Predictor’s prediction is the result of a similar two-step process. The Predictor has knowledge about the common cause and the outcome of the first step of the Player’s decision procedure. But the knowledge of the common cause is slightly corrupted by noise. Thus, the prediction is set by the common cause with probability , and the opposite of the common cause with probability .

Common cause (C)	Predicted choice (P)	Actual choice (A)	Reward (R)	Probability
1	1	1	$1,000
1	1	2	$1,001
1	2	1	$0
1	2	2	$1
2	1	1	$1,000
2	1	2	$1,001
2	2	1	$0
2	2	2	$1

The Predictor’s prediction accuracy is seen to be . According to the problem formulation, this accuracy is supposed to be high, which constrains the permissible parameter values to . Expectation values for different model parameters are in the table below.

0	0.99	0.999	990.001	990.000	991.000
0.5	0.99	0.999	500.500	989.020	11.980
1	0.99	0.999	10.999	10.000	11.000
0	0.99	0.9	990.100	990.000	991.000
0.5	0.99	0.9	500.500	892.000	109.000
1	0.99	0.9	10.900	10.000	11.000

It is not entirely clear to me which quantity is actually the most relevant as a measure of the success of the Player’s strategy. My tentative conclusion is that it is most reasonable to think of the model parameters and as representing quantities that the Player can choose, and the outcomes or as a random variable beyond the Player’s direct control. This may seem a bit odd, since represents the Player’s choice. However, this is mostly a feature of the representation rather than a substantive point. A Player that deterministically chooses between the opaque box only or both boxes is represented using (always choose the opaque box) or (always choose both boxes). In addition, there is a possibility for a randomized strategy, in which the Player determines the probabilities rather than the outcome. Thus, for a deterministic Player the model is equivalent to considering to be under the Player’s direct control. For a Player utilizing a randomized strategy, it is anyway necessary to introduce a probability distribution over .

On the above grounds, I take the unconditional expectation value to be the relevant quantity. Some simple algebra shows that, in general,

.

Consequently, , meaning that greater impulsivity is beneficial (detrimental) if the Player’s strategy amounts to (). Because of the constraint on from the assumption that Predictor is reliable, cannot be varied too much and it is more interesting to study the dependence on . One has . Again, the assumption that Predictor is reliable means that and . It follows that greater bias towards choosing the opaque box only is beneficial.

Conclusion (tentative)

Newcomb’s problem should not be analyzed as a standard decision theoretic problem. Rather, it requires the introduction of a probability distribution that describes how the Player’s choice is correlated, via a common cause, to the Predictor’s prediction. The relevant quantity for assessing the Player’s strategy is the unconditional expectation value and the model parameters and specify which strategy is employed. In all relevant parameter ranges, is the optimal strategy, meaning that the optimal strategy is to deterministically choose the opaque box only. should be as small as is permissible given the assumptions stipulated in the problem formulation.

A. S. Hsu, N. Chater and P. M. B. Vitanyi. The probabilistic analysis of language acquisition: Theoretical, computational, and experimental analysis. arXiv:1006.3271 (2010)

W. R. L. Anderegg et al. Expert credibility in climate change. PNAS, doi: 10.1073/pnas.1003187107

Abstract: Although preliminary estimates from published literature and expert surveys suggest striking agreement among climate scientists on the tenets of anthropogenic climate change (ACC), the American public expresses substantial doubt about both the anthropogenic cause and the level of scientific agreement underpinning ACC. A broad analysis of the climate scientist community itself, the distribution of credibility of dissenting researchers relative to agreeing researchers, and the level of agreement among top climate experts has not been conducted and would inform future ACC discussions. Here, we use an extensive dataset of 1,372 climate researchers and their publication and citation data to show that (i) 97–98% of the climate researchers most actively publishing in the field support the tenets of ACC outlined by the Intergovernmental Panel on Climate Change, and (ii) the relative climate expertise and scientific prominence of the researchers unconvinced of ACC are substantially below that of the convinced researchers.

In the theory of relativity, the relation “event A happened before event B” is not a total ordering relation. When events have space-like separation, the relation “before” is observer-dependent. By restricting attention to a partial ordering relation, that only applies to pairs of events with time-like separation, an observer-independent relation is obtained.

The theory of relativity is concerned with continuous, smooth manifolds. Computation is usually conceived of in terms of a discretized state space. To connect some aspects of the two, the new eprints defines a partial ordering relation on computational events (e.g. state transitions in a Turing machine) that represents a “before” relation. The resulting visualizations of the partial ordering relations as graphs are intriguing. Whether or not one believes this sort of mathematical structure will have useful applications in physics, the eprint is worth a read just for its visualizations.

At first sight, it might appear that the posterior probability of a hypothesis conditional on observed evidence is the perfect measure for a Bayesian. However, a hypothesis sometimes has a high posterior probability simply by virtue of a high prior probability, without necessarily being support/confirmed/justified by virtue of the observed evidence. The second obvious candidate is the likelihood of the evidence given the hypothesis, but this too fails to capture what Bayesians mean by support/confirmation/justification. Instead, several more complicated measures have been proposed by different authors. Here’s a sample (from Atkinson and Crupi et al., 2007):

• Carnap (1950)
  

• Keynes (1921)
  

• Kemeney and Oppenheim (1952)
  

• Good (1950)
  

• Carnap (1950)
  

• Christensen (1999)
  

• Nozick (1981)
  

• Mortimer (1988)
  

• Finch (1960)
  

• Rips (1960)
  

The above measures are all normalized so that the hypothesis is considered supported, confirmed or justified by the evidence when the measure is positive and not so when the measure is negative. From the above embarrassment of riches, it’s at least clear that there is agreement that support/confirmation/justification is a relation between a hypothesis and available evidence, via a Bayesian probability. Shogenji reasons from a new perspective and arrives at the following measure of justification:

This measure has the convenient property that, for any two hypotheses that are independent both a priori and a posteriori (i.e. and are both satisfied), the condition implies . Other relations, where the equalities are replaced by inequalities, also hold. Up to monotonic transformations, Shogenji’s measure is furthermore unique in having this property.

It’s interesting to apply this measure to a modified version of Linda the feminist bank teller. In its original form, it concerns the Bayesian posterior probability and is taken to illustrates incorrect but intuitive probabilistic reasoning. From Wikipedia:

Linda is 31 years old, single, outspoken, and very bright. She majored in philosophy. As a student, she was deeply concerned with issues of discrimination and social justice, and also participated in anti-nuclear demonstrations.

Which is more probable?

1. Linda is a bank teller.
2. Linda is a bank teller and is active in the feminist movement.

Clearly, , with denoting the presented information about Linda, the hypothesis that she is a bank teller, and the hypothesis that she is active in the feminist movement. But what if the question is changed to concern degree of support/confirmation/justification instead of posterior probability (as discussed by Crupi et al., 2007b)? One then have an interesting test case for competing measures! Let

be the degree of justification for the hypothesis that Linda is a bank teller. Then

or, equivalently,

Assuming for simplicity that the hypotheses and are independent both a priori and on the available evidence,

or, equivalently,

The degree of justification in the conjunction of both hypotheses is seen to be an interpolation of the degrees of justification for the individual hypotheses. Since it is intuitively felt that the information about Linda confirms the hypothesis fairly strongly while being at best irrelevant for , it is natural that some of this degree of confirmation is retained for the conjunction and that . This illustrates the unique way in which Shogenji’s measure deals with the problem of irrelevant conjunction.

Shogenji’s article and Atkinson’s commentary are recommended for further discussion.

References

D. Atkinson. Confirmation and justification. A commentary on Shogenji’s measure. Synthese (online first), DOI: 10.1007/s11229-009-9696-4

V. Crupi, K. Tentori, and M. Gonzalez. On Bayesian Measures of Evidential Support: Theoretical and Empirical Issues. Philosophy of Science 74:229 (2007) [Available here.]

V. Crupi, B. Fitelson and K. Tentori. Probability, Confirmation, and the Conjunction Fallacy. Thinking & Reasoning 14:182 (2007b) [Also available at the Phil-Sci Archive and here.]

T. Shogenji. The degree of epistemic justification and the conjunction fallacy. Synthese (online first), DOI: 10.1007/s11229-009-9699-1

• The fact that the No Free Lunch theorem does not hold for a continuous search domain may have some consequences for the project pursued by Dembski and Marks. (That project, by the way, seems to have inspired at least two blogs devoted to criticizing it, DIEBLOG and Bounded Science. Some interesting points are raised there.) Their idea of studying search algorithms/heuristics for a target, then metaheuristics to select such heuristics, then meta-meta….-heuristics, and so on quickly generates uncountably infinite search domains for the meta-…-heuristics. Even if the search domain containing the target is finite, the set of all probabilistic search heuristics is already uncountably infinite, and it doesn’t get better in the space of metaheuristics. Dembski and Marks don’t define their targets in terms of a fitness function, and therefore aren’t directly affected by failure of the No Free Lunch theorem for uncountably infinite search domains. However, they have left it unspecified how the target is to be selected. A very natural way to fill in those details would be in fact let the target by defined in terms of a fitness function; so indirectly I think their project is undermined.
• In their new article, Bernoulli’s Principle of Insufficient Reason and Conservation of Information in Computer Search, they restrict their endorsal of the Principle of Insufficient Reason to finite sample spaces. This is a bit interesting: (i) It allows them to avoid the most damning criticism against that principle and shows some awareness of the serious problems with it, though it isn’t exactly unproblematic for finite spaces either. Furthermore, the conditions under which the Principle of Insufficient Reason are supposed to apply are actually never satisfied in practice, as was noted by some commenters at one of the author’s blog. (ii) It undermines their previous motivation for using uniform probability measures over search heuristics, metaheuristics, etc. (iii) It pretty much commits them to a Bayesian view of what probabilities are. Nothing wrong with that—some of my best friends are Bayesians—but the first author has previously argued against such views, arguing that prior probabilities are hard to justify, at least in the context of “design inferences” (but it is hard to see why design inferences would be in any way special in this regard):
  

  “As we’ve already seen, for the Bayesian approach to work requires prior probabilities. Yet prior probabilities are often impossible to justify. Unlike the example of the urn and two coins discussed earlier, in which drawing a ball from an urn neatly determines the prior probabilities regarding which coin will be tossed, for most design inferences, especially the interesting ones like whether there is design in biological systems, we have no handle on the prior probability of a design hypothesis, or that prior probability is fiercely disputed (theists, for instance, might regard the prior probability as high whereas atheists would regard it as low).” (Dembski, 2004)

  (iv) The endorsement of the more general the Principle of Maximum Entropy puts them in danger of committing themselves to a Bayesian methodology as Bayesian conditionalization is closely linked to this principle. A further generalization, to the Principle of Minimal Discrimination Information (or the Principle of Minimum Relative Entropy) contains both conditionalization and Maximum Entropy as special cases. Again, nothing wrong with this, except that the first author has in the past rejected a Bayesian methodology.

References

William A. Dembski and Robert J. Marks II. Bernoulli’s Principle of Insufficient Reason and Conservation of Information in Computer Search. Proceedings of the 2009 IEEE International Conference on Systems, Man, and Cybernetics. San Antonio, TX, USA – 2009 [Available at the second author's homepage, here]

William A. Dembski. The Design Revolution: Answering the Toughest Questions About Intelligent Design. InterVarsity Press (2004) [Chapter 33, the source of the above quote, is available here]

A number of commentaries and extensions of the original analysis have been published (as of today, a Cited Reference Search at Web of Science yields more than 700 hits for Wolpert and Macready’s 1997 paper). One of the most interesting and original is an article by Griffiths and Orponen, which specifically studies binary-valued functions () and a particular performance measure—the maximum value in the sequence of function values for a fixed, arbitrary number of points . The NFL theorems only hold for a rather trivial class of probability distributions over , but the specialization to maximum-value performance allows NFL-like results to be proved for certain non-trivial distributions as well. Griffiths and Orponen’s work deserves to be more well-known and it would be interesting to see it extended to functions that are not binary-valued.

But what prompted me to write this note was another development altogether. Only elementary mathematics is required to state, prove and reason about the NFL theorems when the search domain and set of values are both finite sets. This was the case considered by Wolpert and Macready. Allowing infinite sets requires care, but the real mathematical difficulties arise in the case of an infinitely large search domain . These cases are explored in a new paper by Auger and Teytaud. Their analysis gets rather technical, but concludes that for countably infinite search domains, the NFL theorem holds in a weaker form than for finite domains. For continuous search domains, the NFL theorem do not hold. The reason for this is that there is no way for the function values and of any two points and in a continuous domain to be uncorrelated (independent). Uncountably infinite search domains are, in a sense, too big too allow the assumption that there are absolutely no correlations. The authors remark:

The deep reason for this fact is that in “bigger” space (and continuous spaces are “very” big), random fields (and distributions of fitness functions are non-trivial random fields in the continuous case) necessarily have correlations [11].

Auger and Teytaud are also able to construct optimal algorithms, which turn out to be related to a class of algorithms called Estimation of Distribution Algorithms.

References

E. J. Griffiths and P. Orponen. Optimization, block designs and no free lunch theorems. Inform. Proc. Lett. 94:55 (2005)

A. Auger and O. Teytaud. Continuous Lunches Are Free Plus the Design of Optimal Optimization Algorithms. Algorithmica (Online First, DOI: 10.1007/s00453-008-9244-5) [Also available as eprint inria-00369788]