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KTH/CSC October 13, 2014Erik Aurell, KTH & Aalto U1 The max-entropy fallacy Erik Aurell International Mini-workshop on Collective Dynamics in Information Systems 2014 Beijing, October 13, 2014 Kavli Institute for Theoretical Physics China (KITPC) Loosely based on G. Del Ferraro & E.A. J. Phys. Soc. Japan (2014) C. Feinauer, M. Skwark, A. Pagnani & E.A. PLoS Comp Biol 10 e (2014)

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KTH/CSC October 13, 2014Erik Aurell, KTH & Aalto U2 What entropy? By entropy I will mean the Shannon entropy of a probability distribution: Maximizing S[p] subject to the constraint gives What maximization? The idea that other probability distributions than equilibrium statistical mechanics can be derived by maximizing entropy given suitable constraints. What max-entropy?

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KTH/CSC October 13, 2014Erik Aurell, KTH & Aalto U3 Two reasons to give this talk Max-entropy: E.T. Jaynes proposed in 1957 that both equilibrium and non-equilibrium statistical mechanics be based upon this criterion. Max-entropy inference: in the last decade considerable attention has been given to learning pairwise interaction models from data, motivated by max-entropy arguments. This research is highly interesting, but does it support max-entropy? [...] the probability distribution over microscopic states which has maximum entropy, subject to whatever is known, provides the most unbiased representation of our knowledge of the system. E.T Jaynes, “Information Theory and Statistical Mechanics II”, Physical Review (1957)

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KTH/CSC October 13, 2014Erik Aurell, KTH & Aalto U4 Why oppose max-entropy? Are probabilities in Physics objective or subjective? (2) does it give the right answers in principle? ” [...]one must recognize the fact that probability theory has developed in two very different directions as regards fundamental notions.” ”[..] the ’objective’ school of thought..” ”[..] the ’subjective’ school of thought... ” ”[...] the probability of an event is merely a formal expression of our expectation that the event did or will occur, based on whatever information is available” E.T Jaynes, “Information Theory and Statistical Mechanics I”, Physical Review (1957) (4) is max-entropy inference a scientific methodology e.g. in the sense of Popper? (3) is it necessary to explain the recent successes in inference? (1) is it a practical method to study non-equilibrium processes (say, on graphs)?

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KTH/CSC October 13, 2014Erik Aurell, KTH & Aalto U5 (1) is it practical? [master equation] Auxiliary maximum entropy distribution True distribution We consider continuous-time dynamics on graphs. Dynamics could be driven out of equilibrium, or relaxing towards equilibrium. Observables in the sense of max-entropy

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KTH/CSC Graph of dynamics Overlayed possible terms Factor graph of auxiliary model v October 13, 2014Erik Aurell, KTH & Aalto U6 This is a dimensional reduction Dynamics of the observables according to the master equation Dynamics of the observables according to the auxiliary distr. If the auxiliary distribution is a good model both ways of computing the dynamics must agree. In this way the changes of the β’s can be computed and the master equation reduced to a (complicated) finite-dimensional ODE. The averages have to be computed by the cavity method (or something else).

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KTH/CSC October 13, 2014Erik Aurell, KTH & Aalto U7 The approach not new. But not (much) tested on single graphs Simplest non-trivial case: the ID Ising spin chain Simplest max-entropy theory built on observing magnetization and energy Roy J Glauber, “Time-dependent statistics of the Ising model”, Journal of mathematical physics, 4:294, (1963) Obeys detailed balance Simple ferromagnetic Hamiltonian Essentially solved 51 years ago Already this is not totally trivial to do…because of the averages…

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KTH/CSC October 13, 2014Erik Aurell, KTH & Aalto U8 …plus, in every time-step, an implicit three-variable equation change from master equation Solving equations by Newton …works reasonably well… computed by cavity Energy vs time Difference to the Glauber theory, in energy vs time

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KTH/CSC October 13, 2014Erik Aurell, KTH & Aalto U9 Joint spin-field distribution… …and works better, though not perfectly. The longer range in the auxiliary distribution the more complicated the cavity calculation and equation. A. C. C Coolen, S. N. Laughton, and D. Sherrington. Physical Review B 53: 8184, (1996) In principle similar, but needs three cavity fields and solving an eight-dimensional implicit equation at every step… Difference to the Glauber theory

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KTH/CSC October 13, 2014Erik Aurell, KTH & Aalto U10 Internal consistency check… Consider again the two ways of computing the changes of observables They work also if O l is not in the theory. But then they do not have to agree, and the discrepancy between the two sides is an internal consistency check. Simplest tests are for longer-range pairwise correlations. Magnetization-energy theory Joint spin-field distribution theory

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KTH/CSC October 13, 2014Erik Aurell, KTH & Aalto U11 Answer to: (1) is it practical? δQ forward: ε+Lθ δQ backward: ε (2) does max-entropy give right answers outside equilibrium? No. It is complicated to implement, even in a simple 1D model of a dynamics relaxing towards equilibrium. One can consider successive approximations with longer “interactions”, but the complexity grows very quickly. Which brings us to the next question:

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KTH/CSC October 13, 2014Erik Aurell, KTH & Aalto U12 Traditionally there were no exact (relevant) results [Gibbs-Boltzmann distribution] [putative non-equilibrium distribution] If max-entropy is relevant for non-equilibrium then the probability distributions should, as Gibbs-Boltzmann distribution, be exponential. SSEP Now known for years this is the case, but the functional V is very non-trivial. B. Derrida, J. Stat. Mech. (2007) P07023E. Akkermans et al, EPL (2013) Recently extended to multi- dimensional systems, for the related question of fluctuations of the current.

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KTH/CSC October 13, 2014Erik Aurell, KTH & Aalto U13 Answer to: (2) does max-entropy give right answers outside equilibrium? No, because there is no way that a complicated long-range effective interaction potential can be deduced from maximizing entropy and a limited number of simple constraints. For the experts: both systems relaxing to equilibrium such as the Ising spin chain and the SSEP (and other such solved models) are covered by the macroscopic fluctuation theory of Jona-Lasinio and co-workers. But only SSEP-like systems lead to long-range effective interactions. For the relaxing Ising spin chain the max-entropy approach should hence probably eventually work, though remain computationally cumbersome.

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KTH/CSC October 13, 2014Erik Aurell, KTH & Aalto U14 (3) is it necessary to explain the recent successes in inference? The main success is contact prediction in proteins. Folding proteins in silico is hard, and not a solved problem – unless you have an already solved structure as template. Predicting which amino acids are in contact in a structure can be done from co-variation in similar proteins.

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KTH/CSC X1X1 X6X6 X4X4 X3X3 X2X2 X5X5 X7X7 Relation btw positional correlation and structure known since 20 ys August 27, 2014 Erik Aurell, KTH & Aalto15 Neher (1994) Göbel, Sander, Schneider, Valencia (1994) Lapedes et al 2001 Weigt et al PNAS 2009 Burger & van Nimwegen 2010 Balakrishnan et al 2011 Morcos et al PNAS 2011 Hopf et al Cell 2012 Jones et al Bioinformatics 2012 Ekeberg et al Phys Rev E 2013 Skwark et al Bioinformatics 2013 Kamisetty et al PNAS 2013

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KTH/CSC The recent success is to learn a Potts model from data February 26, 2014Erik Aurell, KTH & Aalto16 ” The prediction method applies a maximum entropy approach to infer evolutionary covariation […]” T. Hopf et al, Cell 149: (2012) ”The maximum-entropy approach to potentially solving the problem of protein structure prediction from residue covariation patterns […]” D. Marks et al, Nat Biotechnol. 30: (2012) ” To disentangle direct and indirect couplings, we aim at inferring a statistical model P(A 1,...,A L ) for entire protein sequences (A 1,...,A L ) […] aim at the most general, least-constrained model […] achieved by applying the maximum-entropy principle ” F. Morcos et al, PNAS 108:E1293–E1301 (2011)

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KTH/CSC August 27, 2014Erik Aurell, KTH & Aalto17 Actually we have all the data It is a choice to reduce multiple sequence alignments to nucleotide frequencies and correlations for data analysis. But we start from all the data. The conceptual basis of max-entropy is therefore not there. Furthermore, the best available methods to learn these Potts models use all the data. M Ekeberg et al, Phys Rev. E (2013) M Ekeberg et al, J Comp Phys (2014)

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KTH/CSC February 26, 2014Erik Aurell, KTH & Aalto18 Learning better models... Multiple sequence alignments generally have stretches of gaps. Not generated with high probability from a Potts model. Marcin Skwark and Christoph Feinauer, AISTATS (2014) C. Feinauer, M. Skwark, A. Pagnani & E.A. PLoS Comp Biol 10 e (2014) This (and previous) slide show the – real, but admittedly not very large – improvement in contact prediction by learning two models (gplmDCA and plmDCA20) which do take into account gap stretches.

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KTH/CSC October 13, 2014Erik Aurell, KTH & Aalto U19 Answer to: (3) is max-entropy necessary to explain the recent successes in inference? No. Why that is or should be so? Nobody knows! Perhaps an important problem for evolutionary theory? And perhaps has other uses? The successes are better explained by the distribution of amino acids in homologous proteins, as a result of all evolution of life, is actually in an exponential family, and rather close to a Potts model. We are back to the objective / subjective interpretations of probability, from the start the most contentious issue surrounding max-entropy.

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KTH/CSC October 13, 2014Erik Aurell, KTH & Aalto U20 (4) is max-entropy inference a scientific methodology? According to Popper science is based on falsifiability. This same basic idea has been stated by many others, before and after. We are trying to prove ourselves wrong as quickly as possible, because only in that way can we find progress R.P. Feynman, as on famousquotes.org. You cannot falsify anything by a single experiment or single data set with no theory of prediction beforehand to falsify. Also according to Popper, scientific knowledge is built as a collective enterprise of scientists. Therefore, Jaynes’ conditional... […] subject to whatever is known [..]...implicitly includes all human knowledge up to that time – which is not a simple constraint. A similar philosophical objection can be made against Rissanen’s Minimum Description Length principle.

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KTH/CSC October 13, 2014Erik Aurell, KTH & Aalto U21 Thanks to Gino Del Ferraro Alexander Mozeika Marcin Skwark Christoph Feinauer Andrea Pagnani Magnus Ekeberg Angelo Vulpiani

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