22,183 research outputs found
Estimating Ratios of Normalizing Constants Using Linked Importance Sampling
Ratios of normalizing constants for two distributions are needed in both
Bayesian statistics, where they are used to compare models, and in statistical
physics, where they correspond to differences in free energy. Two approaches
have long been used to estimate ratios of normalizing constants. The `simple
importance sampling' (SIS) or `free energy perturbation' method uses a sample
drawn from just one of the two distributions. The `bridge sampling' or
`acceptance ratio' estimate can be viewed as the ratio of two SIS estimates
involving a bridge distribution. For both methods, difficult problems must be
handled by introducing a sequence of intermediate distributions linking the two
distributions of interest, with the final ratio of normalizing constants being
estimated by the product of estimates of ratios for adjacent distributions in
this sequence. Recently, work by Jarzynski, and independently by Neal, has
shown how one can view such a product of estimates, each based on simple
importance sampling using a single point, as an SIS estimate on an extended
state space. This `Annealed Importance Sampling' (AIS) method produces an
exactly unbiased estimate for the ratio of normalizing constants even when the
Markov transitions used do not reach equilibrium. In this paper, I show how a
corresponding `Linked Importance Sampling' (LIS) method can be constructed in
which the estimates for individual ratios are similar to bridge sampling
estimates. I show empirically that for some problems, LIS estimates are much
more accurate than AIS estimates found using the same computation time,
although for other problems the two methods have similar performance. Linked
sampling methods similar to LIS are useful for other purposes as well
Puzzles of Anthropic Reasoning Resolved Using Full Non-indexical Conditioning
I consider the puzzles arising from four interrelated problems involving
`anthropic' reasoning, and in particular the `Self-Sampling Assumption' (SSA) -
that one should reason as if one were randomly chosen from the set of all
observers in a suitable reference class. The problem of Freak Observers might
appear to force acceptance of SSA if any empirical evidence is to be credited.
The Sleeping Beauty problem arguably shows that one should also accept the
`Self-Indication Assumption' (SIA) - that one should take one's own existence
as evidence that the number of observers is more likely to be large than small.
But this assumption produces apparently absurd results in the Presumptuous
Philosopher problem. Without SIA, however, a definitive refutation of the
counterintuitive Doomsday Argument seems difficult. I show that these problems
are satisfyingly resolved by applying the principle that one should always
condition on all evidence - not just on the fact that you are an intelligent
observer, or that you are human, but on the fact that you are a human with a
specific set of memories. This `Full Non-indexical Conditioning' (FNC) approach
usually produces the same results as assuming both SSA and SIA, with a
sufficiently broad reference class, while avoiding their ad hoc aspects. I
argue that the results of FNC are correct using the device of hypothetical
``companion'' observers, whose existence clarifies what principles of reasoning
are valid. I conclude by discussing how one can use FNC to infer how densely we
should expect intelligent species to occur, and by examining recent anthropic
arguments in inflationary and string theory cosmology
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