The bread and butter of classical statistics is maximum likelihood estimation: setting up an equation that measures the distance between a model’s predictions and the actual data you observe, and finding what parameters of the model make that distance as small as possible. (When errors are normally distributed, this reduces to the familiar ‘least squares’ model fitting. When the error structure is more ‘interesting’, the distance-minimizing problem is correspondingly harder.)
In this age of fast computers, it is rarely necessary to accept shoehorning your data into a 1950s-era model that could be fit to the data with pencil and paper. One of our specialities at Excelsior is doing exact maximum likelihood fitting of custom models. Read about a simple real-world application we worked on the in the winter of 2016-2017: finding the pressure and volume of an unknown quantity of gas by expanding that gas into a series of chambers of known volume and observing the pressure drop. A more complicated example, estimating the dimensions of irradiated fuel particles that could not be directly observed, appeared in the April 2017 Journal of Nuclear Materials.
Other maximization problems
Many other optimization problems that don’t have a random component can also be solved by writing out an equation and finding its maximum or minimum using calculus.
For a simple example, read about how the apparent brightness of Venus varies through the year: in 2017, Venus was brightest on February 17th, not on January 12th when it appeared farthest away from the sun in the sky, and not on March 25th when it is closest to earth.
Contact us if you need a model fit to your data using maximum likelihood, or if you have an optimization problem you need help solving. Or continue on and read about how we can help with resource allocation problems, finding optimal strategies of games, and discrete optimization problems.