Why the Best Predictive What Do We Mean by . . . Models Are Often - PowerPoint PPT Presentation

Predictive vs. . . . Remaining Problem: . . . Need for Formalization What Do We Mean by . . . Why the Best Predictive What Do We Mean by . . . Models Are Often Different Main Result: . . . Proof for Predictive Case from the Best Explanatory Proof for Explanatory . . . Discussion Models: A Theoretical Home Page Explanation Title Page ◭◭ ◮◮ Songsak Sriboonchitta 1 , Luc Longpr´ e 3 ◭ ◮ Vladik Kreinovich 3 , and Thongchai Dumrongpokaphan 2 Page 1 of 28 1 Faculty of Economics, 2 Dept. of Mathematics, Chiang Mai University, Go Back Thailand, songsakecon@gmail.com, tcd43@hotmail.com 3 University of Texas at El Paso, El Paso, Texas 79968, USA, Full Screen longpre@utep.edu, vladik@utep.edu Close Quit

Predictive vs. . . . Remaining Problem: . . . 1. Predictive vs. Explanatory Models: Traditional Need for Formalization Confusion What Do We Mean by . . . • Many researchers implicitly assume that predictive and What Do We Mean by . . . explanatory powers are strongly correlated. Main Result: . . . Proof for Predictive Case • They assumed that a statistical model that leads to Proof for Explanatory . . . accurate predictions also provides a good explanation. Discussion • They also assume that models providing a good expla- Home Page nation lead to accurate predictions. Title Page • In practice, models that lead to good predictions do ◭◭ ◮◮ not always explain the observed phenomena. ◭ ◮ • Vice versa, models that explain do not always lead to Page 2 of 28 most accurate predictions. Go Back Full Screen Close Quit

Predictive vs. . . . Remaining Problem: . . . 2. Predictive vs. Explanatory Models: Example Need for Formalization • Newton’s equations provide a very clear explanation of What Do We Mean by . . . why and how celestial bodies move. What Do We Mean by . . . Main Result: . . . • In principle, we can predict the trajectories of celestial Proof for Predictive Case bodies by integrating the corresponding equations. Proof for Explanatory . . . • This would, however, require a lot of computation time Discussion on modern computers. Home Page • On the other hand, people successfully predicted the Title Page observed positions of planets way before Newton. ◭◭ ◮◮ • For that, they use epicycles , i.e., in effect, trigonomet- ◭ ◮ ric series. Page 3 of 28 • Such series are still used in celestial mechanics to pre- dict the positions of celestial bodies. Go Back Full Screen • They are very good for predictions, but they are abso- lutely useless in explanations. Close Quit

Predictive vs. . . . Remaining Problem: . . . 3. Remaining Problem: Why? Need for Formalization • The empirical fact that the best predictive models are What Do We Mean by . . . often different from the best explanatory models. What Do We Mean by . . . Main Result: . . . • But from the theoretical viewpoint, this empirical fact Proof for Predictive Case still remains a puzzle. Proof for Explanatory . . . • In this talk, we provide a theoretical explanation for Discussion this empirical phenomenon. Home Page Title Page ◭◭ ◮◮ ◭ ◮ Page 4 of 28 Go Back Full Screen Close Quit

Predictive vs. . . . Remaining Problem: . . . 4. Need for Formalization Need for Formalization • In order to provide a theoretical explanation for the What Do We Mean by . . . difference, we need to first formally describe: What Do We Mean by . . . Main Result: . . . – what it means for a model to be the best predictive Proof for Predictive Case model, and Proof for Explanatory . . . – what it means for a model to be the best explana- Discussion tory model. Home Page • The “explanatory” part is intuitively understandable. Title Page • We have some equations or formulas that explain all ◭◭ ◮◮ the observed data. ◭ ◮ • This means that all the observed data satisfy these Page 5 of 28 equations. Go Back Full Screen Close Quit

Predictive vs. . . . Remaining Problem: . . . 5. Need for Formalization (cont-d) Need for Formalization • Of course, these equations must be checkable – else: What Do We Mean by . . . What Do We Mean by . . . – if they are formulated purely in terms of complex Main Result: . . . abstract mathematics, Proof for Predictive Case – so that no one knows how to check whether ob- Proof for Explanatory . . . served data satisfy these equations or formulas, Discussion – then how can we know that the data satisfies them? Home Page • Thus, when we say that we have an explanatory model, Title Page what we are saying is that we have an algorithm that: ◭◭ ◮◮ – given the data, ◭ ◮ – checks whether the data is consistent with the cor- Page 6 of 28 responding equations or formulas. Go Back • From this pragmatic viewpoint, by an explanatory model, we simply means a program. Full Screen • Of course, this program must be non-trivial. Close Quit

Predictive vs. . . . Remaining Problem: . . . 6. Need for Formalization (cont-d) Need for Formalization • It is not enough for the data to be simply consistent What Do We Mean by . . . with the data. What Do We Mean by . . . Main Result: . . . • Explanatory means that we must explain all this data; Proof for Predictive Case for example: Proof for Explanatory . . . – if we simply state that, in general, the trade volume Discussion grows when the GDP grows, Home Page – all the data may be consistent with this rule. Title Page • However, this consistency is not enough: for a model ◭◭ ◮◮ to be truly explanatory. ◭ ◮ • It needs to explain why in some cases, the growth in Page 7 of 28 trade is small and in other cases, it is huge. Go Back • In other words, it must explain the exact growth rate. Full Screen • Of course, this is economics, not fundamental physics. Close Quit

Predictive vs. . . . Remaining Problem: . . . 7. Need for Formalization (cont-d) Need for Formalization • We cannot explain all the numbers based on first prin- What Do We Mean by . . . ciples only. What Do We Mean by . . . Main Result: . . . • We have to take into account some quantities that af- Proof for Predictive Case fect our processes. Proof for Explanatory . . . • But for the model to be truly explanatory we must be Discussion sure that, Home Page – once the values of these additional quantities are Title Page fixed, ◭◭ ◮◮ – there should be only one sequence of numbers that ◭ ◮ satisfies the corresponding equations or formulas, Page 8 of 28 – namely, the sequence that we observe (ignoring noise, of course). Go Back • This is not that different from physics. Full Screen Close Quit

Predictive vs. . . . Remaining Problem: . . . 8. Need for Formalization (cont-d) Need for Formalization • For example, Newton’s laws of gravitation allow many What Do We Mean by . . . possible orbits of celestial bodies. What Do We Mean by . . . Main Result: . . . • However, once you fix the masses and initial conditions, Proof for Predictive Case Newton’s laws uniquely determine the orbits. Proof for Explanatory . . . • In algorithmic terms, if: Discussion – to the original program for checking whether the Home Page data satisfies the given equations and/or formulas, Title Page – we add checking the values of additional quantities, ◭◭ ◮◮ – then the observed data is the only possible sequence ◭ ◮ of observations that is consistent with this program. Page 9 of 28 • Once we know such a program that uniquely deter- mines all the data, we can, in principle, find this data. Go Back Full Screen • We can try all possible combinations of possible data values until we satisfy all the corresponding conditions. Close Quit

Predictive vs. . . . Remaining Problem: . . . 9. Need for Formalization (cont-d) Need for Formalization • How can we describe this in precise terms? What Do We Mean by . . . What Do We Mean by . . . • All the observations can be stored in the computer, Main Result: . . . and in the computer, everything is stored as 0s and 1s. Proof for Predictive Case • From this viewpoint, the whole set of observed data is Proof for Explanatory . . . simply a finite sequence x of 0s and 1s. Discussion Home Page • The length n of this sequence is known. • There are 2 n sequences of length n . Title Page ◭◭ ◮◮ • There are finitely many such sequences, so we must potentially check them all. ◭ ◮ Page 10 of 28 • Thus, we find the desired sequence x – the only one that satisfies all the required conditions. Go Back • Of course, for large n , the time 2 n can be unrealistically Full Screen astronomically large. Close Quit

Predictive vs. . . . Remaining Problem: . . . 10. Need for Formalization (cont-d) Need for Formalization • So, we are talking about potential possibility to com- What Do We Mean by . . . pute – not practical computations. What Do We Mean by . . . Main Result: . . . • One does not solve Newton’s equations by trying all Proof for Predictive Case possible trajectories. Proof for Explanatory . . . • But it is OK, since our goal here is: Discussion – not to provide a practical solution to the problem, Home Page – but rather to provide a formal definition of an ex- Title Page planatory model. ◭◭ ◮◮ • For the purpose of this definition, we can associate each ◭ ◮ explanatory model: Page 11 of 28 – not only with the original checking program, Go Back – but also with the related exhaustive-search pro- gram p that generates the data. Full Screen • The exhaustive search part is easy to program. Close Quit