Resolution of the Miller-Popper paradox

Abstract

A longstanding paradox was first reported by David Miller in 1975 and highlighted by Karl Popper in 1979. Miller showed that the ranking of predictions from two theories, in terms of closeness to observation, appears to be reversed when the problem is transformed into a different mathematical space. He concluded that “… no false theory can … be closer to the truth than is another theory”. This flies in the face of normal scientific practice and is thus paradoxical; it is named here the “Miller-Popper paradox”. This paper proposes a resolution of the paradox, through consideration of the inevitable errors and uncertainties in both observations and predictions. It is proved that, for linear transformations and Gaussian error distributions, the transformation between spaces creates no change in quantitative measures of “closeness-to-observation” when these measures are based in probability theory. The extension of this result to nonlinear transformations and to non-Gaussian error distributions is also discussed. These results demonstrate that concepts used in comparison of predictions with observations—concepts of “closeness”, “consistency”, “agreement”, “falsification”, etc.—all imply some knowledge of the uncertainty characteristics of both predictions and observations.

Appendix: Nonlinear transformations

When the two spaces are related in a nonlinear way, (3.1) becomes

$${\mathbf{x}} = A\left( {\mathbf{p}} \right)\;\;\;\;\;{\text{and}}\;\;\;\;\;{\mathbf{p}} = B\left( {\mathbf{x}} \right),$$

(A.1)

where $A$ and $B$ are nonlinear matrix functions. We still assume, for this problem, that ${\mathbf{x}}$ and ${\mathbf{p}}$ have the same dimension.

Small departures from ${\mathbf{x}}$ and ${\mathbf{p}}$ are related by

$$\delta {\mathbf{x}} \approx {\mathbf{A}}\left( {\mathbf{p}} \right)\delta {\mathbf{p}}\;\;\;{\text{and}}\;\;\;\delta {\mathbf{p}} \approx {\mathbf{B}}\left( {\mathbf{x}} \right)\delta {\mathbf{x}},$$

(A.2)

where ${\mathbf{A}}\left( {\mathbf{p}} \right) = \nabla_{{\mathbf{p}}} A\left( {\mathbf{p}} \right)$, ${\mathbf{B}}\left( {\mathbf{x}} \right) = \nabla_{{\mathbf{x}}} B\left( {\mathbf{x}} \right)$ and ${\mathbf{B}}\left( {\mathbf{x}} \right) = {\mathbf{A}}\left( {\mathbf{p}} \right)^{ - 1}$. The relations in (A.2) become exact as $\delta {\mathbf{x}} \to 0$ and $\delta {\mathbf{p}} \to 0$.

Therefore (3.2) becomes

$${{\varvec{\upvarepsilon}}}_{p} \approx {\mathbf{B}}\left( {\mathbf{x}} \right){{\varvec{\upvarepsilon}}}_{x} ,$$

(A.3)

and (3.3) becomes

$${\mathbf{C}}_{p} \left( {\mathbf{p}} \right) \approx {\mathbf{B}}\left( {\mathbf{x}} \right){\mathbf{C}}_{x} \left( {\mathbf{x}} \right){\mathbf{B}}\left( {\mathbf{x}} \right)^{T} .$$

(A.4)

Then (3.10) becomes

$$J_{p}^{n} \left( {\mathbf{x}} \right) \approx J_{x}^{n} \left( {\mathbf{x}} \right),$$

(A.5)

and this also becomes exact in the linear limit and in the limit of small errors.

Consider the two examples of nonlinear transformations given by Miller (2006).

Example 1

Transformation from the length and breadth of a rectangle to its area and perimeter.

In this case, (2.1) (ii) becomes ${\text{p}} = {\text{xy}}$ and ${\text{q}} = 2\left( {{\text{x}} + {\text{y}}} \right)$.

Differentiation gives ${\mathbf{B}} = {\mathbf{A}}^{ - 1} = \left[ {\begin{array}{*{20}c} {\text{y}} & {\text{x}} \\ 2 & 2 \\ \end{array} } \right]$.

Then, if ${\mathbf{C}}_{x} = a^{2} \left[ {\begin{array}{*{20}c} 1 & 0 \\ 0 & 1 \\ \end{array} } \right]$, ${\mathbf{C}}_{p} = a^{2} \left[ {\begin{array}{*{20}c} {{\text{y}}^{2} + 4} & {{\text{xy}} + 4} \\ {{\text{xy}} + 4} & {{\text{x}}^{2} + 4} \\ \end{array} } \right].$

Therefore, for a circular ${\mathbf{C}}_{x}$, ${\mathbf{C}}_{p}$. has an elliptical shape for small uncertainties and becomes a distorted ellipse (oval) for large uncertainties. For the example given by Miller (a rectangle of 34 × 23), ${\mathbf{C}}_{p}$. is highly elliptical, and it also becomes singular when ${\text{x}} = {\text{y}}$. Moreover, the elliptical nature of ${\mathbf{C}}_{p}$ is such that the ellipse for the prediction that is closer to observion in ${\mathbf{x}}$-space remains closer in ${\mathbf{p}}$-space (similar to Fig. 2).

Example 2

Transformation from Cartesian to polar coordinates.

In this case (2.1) becomes

• (i) ${\text{x}} = {\text{p }}\cos {\text{q}}$ and y $= {\text{p }}\sin {\text{q}}$,

• (ii) ${\text{p}} = \sqrt {{\text{x}}^{2} + {\text{y}}^{2} }$ and ${\text{q}} = \tan^{ - 1} ({\text{y}}/{\text{x}})$.

Differentiation gives ${\mathbf{A}} = \left[ {\begin{array}{*{20}c} {\cos {\text{q}}} & { - {\text{p }}\sin {\text{q}}} \\ {\sin {\text{q}}} & {{\text{p }}\cos {\text{q}}} \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {{\text{x}}/{\text{p}}} & { - {\text{y}}} \\ {{\text{y}}/{\text{p}}} & {\text{x}} \\ \end{array} } \right],$

and ${\mathbf{B}} = {\mathbf{A}}^{ - 1} = \frac{1}{{\text{p}}}\left[ {\begin{array}{*{20}c} {\text{x}} & {\text{y}} \\ { - {\text{y}}/{\text{p}}} & {{\text{x}}/{\text{p}}} \\ \end{array} } \right]$.

Then, if ${\mathbf{C}}_{x} = a^{2} \left[ {\begin{array}{*{20}c} 1 & 0 \\ 0 & 1 \\ \end{array} } \right]$, ${\mathbf{C}}_{p} = a^{2} \left[ {\begin{array}{*{20}c} 1 & 0 \\ 0 & {1/{\text{p}}} \\ \end{array} } \right]$.

Therefore, for a circular ${\mathbf{C}}_{x}$, ${\mathbf{C}}_{p}$ has an elliptical shape for small uncertainties and becomes a distorted ellipse (oval) for large uncertainties.

Note that, for a given value of $({\text{y}}/{\text{x}})$, ${\text{q}} = \tan^{ - 1} ({\text{y}}/{\text{x}})$ can take values, ${\text{q}} + {\text{n}}{{\varvec{\uppi}}}$, where ${\text{n}}$ is any integer. This makes the transformation from ${\mathbf{x}}$-space to ${\mathbf{p}}$-space multi-valued. However, this ambiguity can easily be resolved when the computing the “closeness” between two points, by considering only values of ${\text{q}}$ with the least difference between them.

The transformation collapses at ${\text{p}} = 0$. Also, when ${\text{p}}$ is small (smaller than the uncertainty in ${\text{x}}$ and y), the ellipse of uncertainty in ${\mathbf{p}}$-space distorts such that it no longer surrounds the point ${\mathbf{p}}$. This issue suggests that some transformations, whilst mathematically possible, are not physically realistic.