Anthropic probabilities

In calculating the conditional probabilities for predicting some of our observations given others, there can be no objection of principle to including a description of 'us' as part of the conditions:

Drawing inferences using such probabilities is called anthropic reasoning. The motivation is the idea that probabilities for certain features of the Universe might be sensitive to this inclusion.

The utility of anthropic reasoning depends on how sensitive probabilities like Eq. (18.6) are to the inclusion of 'us'. To make this concrete, consider the probabilities for a hypothetical cosmological parameter we will call A. We will assume that H and imply that A is constant over the visible Universe, but only supply probabilities for the various constant values it might take through Eq. (18.4). We seek to compare p (AH, with p (A|'us', H, In principle, both are calculable from Eqs. (18.4) and (18.5). Figure 18.1 shows three possible ways in which they might be related.

• p (A|H, is peaked around one value, as in Fig. 18.1(a). The parameter A is determined either by H or or by both.4 Anthropic reasoning is not necessary; the parameter is already determined by fundamental physics.

• p (A|H, is distributed and p (A|'us',H, is also distributed, as in Fig. 18.1(b). Anthropic reasoning is inconclusive. One might as well measure the value of A and use this as a condition for making further predictions,5 i.e. work with probabilities of the form p (a|A, H,

• p (AH, is distributed but p (A|'us',H, is peaked, as in Fig. 18.1(c). Anthropic reasoning helps to explain the value of A.

The important point to emphasize is that a theoretical hypothesis for H and is needed to carry out anthropic reasoning. Put differently, a theoretical context is needed to decide whether a parameter like A can vary, and to find out how it varies, before using anthropic reasoning to restrict its range. The Hamiltonian and quantum state provide this context. In Section 18.5, we will consider the situation where the state is imperfectly known.

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