We show that the natural scaling of measurement for a particular problem defines the most likely probability distribution of observations taken from that measurement scale. Our approach extends the method of maximum entropy to use measurement scale as a type of information constraint. We argue that a very common measurement scale is linear at small magnitudes grading into logarithmic at large magnitudes, leading to observations that often follow Student's probability distribution which has a Gaussian shape for small fluctuations from the mean and a power law shape for large fluctuations from the mean. An inverse scaling often arises in which measures naturally grade from logarithmic to linear as one moves from small to large magnitudes, leading to observations that often follow a gamma probability distribution. A gamma distribution has a power law shape for small magnitudes and an exponential shape for large magnitudes. The two measurement scales are natural inverses connected by the Laplace integral transform. This inversion connects the two major scaling patterns commonly found in nature. We also show that superstatistics is a special case of an integral transform, and thus can be understood as a particular way in which to change the scale of measurement. Incorporating information about measurement scale into maximum entropy provides a general approach to the relations between measurement, information and probability.

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Most modern financial markets use a continuous double auction mechanism to store and match orders and facilitate trading. In this chapter we use a microscopic dynamical statistical model for the continuous double auction under the assumption of IID random order flow. The analysis is based on simulation, dimensional analysis, and theoretical tools based on mean-field approximations. The model makes testable predictions for all the basic properties of markets, including price volatility, the depth of stored supply and demand, the bid-ask spread, the price impact function, and the time and probability of filling orders. These predictions are based on properties of order flow such as share volume of market and limit orders, cancellations, typical order size, and tick size. Because these quantities can all be measured directly in real data sets there are no free parameters. We show that the order size, which can be cast as a nondimensional granularity parameter, is in most cases a more significant determinant of market behavior than tick size. We also provide an explanation for the observed highly concave nature of the price im