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By Michael K. Murray, John W. Rice (auth.)

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Example text

The point of relativity theory is that each observer creates an a priori different co-ordinate system for the set of all events in the cosmos. The principle that the laws of physics should not give preference to any particular parametrisations has given great impetus to the formulation of the concepts of calculus in terms of the underlying abstract sets of states or events. In statistics as well, we shall pursue the goal of formulating statistical theory in terms of the underlying set of probability distributions, rather than in terms of convenient parametrisations of it.

1) for every point p in the set P. Note that whereas f is a function on the set P, the domain of 1 is some subset of R n. 1) is called the co-ordinate expression of f, and functions on a set are commonly defined by giving their co-ordinate expressions in some co-ordinate system. For example, one is often invited to consider the function x 2 + y2 on the plane. 4 by determining the co-ordinates of p, and then taking the sums of the squares of the resulting pair of numbers (x(p), y(p)). This last step, which is the one of greatest interest, is a purely arithmetic operation which applies to any pair of numbers.

Differentiating at t = 0 gives the equation EII(iI'(O)) = 0 Hence those random variables I in Ro which represent tangent vectors to 1/ within the space of probability measures 'P are the o. Conversely if EII(J) 0 ones satisfying the condition EII(J) then the variation I/(t) = 1/ + tl, that is, the family defined by = = dl/(t) = exp(tf)dl/ and dividing by its total mass gives a family of probability measures through 1/ whose velocity is represented by f. CALCULUS ON MANIFOLDS 42 On the other hand P is an affine space in its own right, regarded as finite positive measures up to scale.

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