Dobbiaco Lectures 2010 (4)Solved and Unsolved Problems in

Transcription

1 Dobbiaco Lectures 2010 (4) Solved and Unsolved Problems in Biology Bud Mishra Room 1002, 715 Broadway, Courant Institute, NYU, New York, USA Dobbiaco

2 Outline 1 2 3

3 PART IX: Information

4 Main theses Outline There are seven main propositions in the text. These are:... 1 The world is everything that is the case. 2 What is the case (a fact) is the existence of states of affairs. 3 A logical picture of facts is a thought. 4 A thought is a proposition with sense. 5 A proposition is a truth-function of elementary propositions. 6 The general form of a proposition is the general form of a truth function, which is: p, ξ, ξ 7 Where (or of what) one cannot speak, one must pass over in silence. Ludwig Wittgenstein, Tractatus Logico-Philosophicus, 1921.

5 Outline 1 2 3

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7 Bayesian Interpretation Probability P(e) our certainty about whether event e is true or false in the real world. (Given whatever information we have available.) Degree of Belief. More rigorously, we should write Conditional probability P(e L) Represents a degree of belief with respect to L The background information upon which our belief is based.

8 Probability as a Dynamic Entity We update the degree of belief as more data arrives: using Bayes Theorem: P(e D) = P(D e)p(e). P(D) Posterior is proportional to the prior in a manner that depends on the data P(D e)/p(d). Prior Probability: P(e) is one s belief in the event e before any data is observed. Posterior Probability: P(e D) is one s updated belief in e given the observed data. Likelihood: P(D e) Probability of the data under the assumption e

9 Dynamics Outline Note: P(e D 1, D 2 ) = P(D 2 D 1, e)p(e D 1 ) P(D 2 D 1 ) = P(D 2 D 1, e)p(d 1 e)p(e) P(D 2 D 1 ) Further, note: The effects of prior diminish as the number of data points increase. The Law of Large Number: With large number of data points, Bayesian and frequentist viewpoints become indistinguishable.

10 Parameter Estimation Functional form for a model M 1 Model depends on some parameters Θ 2 What is the best estimation of Θ? Typically the parameters Θ are a set of real-valued numbers Both prior P(Θ) and posterior P(Θ D) are defining probability density functions.

11 MAP Method: Maximum A Posteriori Find the set of parameters Θ 1 Maximizing the posterior P(Θ D) or minimizing a score log P(Θ D) E (Θ) = log P(Θ D) = log P(D Θ) log P(Θ) + log P(D) 2 If prior P(Θ) is uniform over the entire parameter space (i.e., uninformative) min arg Θ E L (Θ) = log P(D Θ). Maximum Likelihood Solution

12 Robustness Outline Find a distribution (pdf) for parameters Θ 1 Maximizing the expectation over parameters E = P(Θ, D) log P(Θ D) = P(Θ) P(Θ D) log P(Θ D) Empirical Bayes Methods: P(Θ) is a compressed (lossy) representation of P(D).

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14 Information theory Information theory is based on probability theory (and statistics). Basic concepts: Entropy (the information in a random variable) and Mutual Information (the amount of information in common between two random variables). The most common unit of information is the bit (based log 2). Other units include the nat, and the hartley.

15 Entropy Outline The entropy H of a discrete random variable X is a measure of the amount uncertainty associated with the value X. Suppose one transmits 1000 bits (0s and 1s). If these bits are known ahead of transmission (to be a certain value with absolute probability), logic dictates that no information has been transmitted. If, however, each is equally and independently likely to be 0 or 1, 1000 bits (in the information theoretic sense) have been transmitted.

16 Entropy Outline Between these two extremes, information can be quantified as follows. If X is the set of all messages x that X could be, and p(x) is the probability of X given x, then the entropy of X is defined as H(x) = E X [I(x)] = x X p(x) log p(x). Here, I(x) is the self-information, which is the entropy contribution of an individual message, and E X is the expected value.

17 An important property of entropy is that it is maximized when all the messages in the message space are equiprobable p(x) = 1/n, i.e., most unpredictable, in which case H(X) = log n. The binary entropy function (for a random variable with two outcomes {0, 1} or {H, T }: H b (p, q) = p log p q log q, p + q = 1.

18 Joint entropy Outline The joint entropy of two discrete random variables X and Y is merely the entropy of their pairing: X, Y. Thus, if X and Y are independent, then their joint entropy is the sum of their individual entropies. H(X, Y) = E X,Y [ log p(x, y)] = x,y log p(x, y). For example, if (X,Y) represents the position of a chess piece Ñ X the row and Y the column, then the joint entropy of the row of the piece and the column of the piece will be the entropy of the position of the piece.

19 Conditional Entropy or Equivocation The conditional entropy or conditional uncertainty of X given random variable Y (also called the equivocation of X about Y ) is the average conditional entropy over Y : H(X Y) = E Y [H(X y)] = y Y p(y) x X p(x y) log p(x y) = x,y p(x, y) log p(x, y) p(y) A basic property of this form of conditional entropy is that: H(X Y) = H(X, Y) H(Y).

20 Mutual Information (Transinformation) Mutual information measures the amount of information that can be obtained about one random variable by observing another. The mutual information of X relative to Y is given by: I(X; Y) = E X,Y [SI(x, y)] = x,y p(x, y) log p(x, y) p(x)p(y). where SI (Specific mutual Information) is the pointwise mutual information.

21 A basic property of the mutual information is that I(X; Y) = H(X) H(X Y) = H(X)+H(Y) H(X, Y) = I(Y; X). That is, knowing Y, we can save an average of I(X; Y) bits in encoding X compared to not knowing Y. Note that mutual information is symmetric. It is important in communication where it can be used to maximize the amount of information shared between sent and received signals.

22 Kullback-Leibler Divergence (Information Gain) The Kullback-Leibler divergence (or information divergence, information gain, or relative entropy) is a way of comparing two distributions: a true probability distribution p(x), and an arbitrary probability distribution q(x). D KL (p(x) q(x)) = x X p(x) log p(x) q(x) = x X[ p(x) log q(x)] [ p(x) log p(x)]

23 If we compress data in a manner that assumes q(x) is the distribution underlying some data, when, in reality, p(x) is the correct distribution, the Kullback-Leibler divergence is the number of average additional bits per datum necessary for compression. Although it is sometimes used as a distance metric, it is not a true metric since it is not symmetric and does not satisfy the triangle inequality (making it a semi-quasimetric).

24 Mutual information can be expressed as the average Kullback-Leibler divergence (information gain) of the posterior probability distribution of X given the value of Y to the prior distribution on X: I(X; Y) = E p(y) [D KL (p(x Y = y) p(x)] = D KL (p(x, Y) p(x)p(y)). In other words, mutual information I(X, Y) is a measure of how much, on the average, the probability distribution on X will change if we are given the value of Y. This is often recalculated as the divergence from the product of the marginal distributions to the actual joint distribution. Mutual information is closely related to the log-likelihood ratio test in the context of contingency tables and the multinomial distribution and to Pearson s χ 2 test.

25 Source theory Any process that generates successive messages can be considered a source of information. A memoryless source is one in which each message is an independent identically-distributed random variable, whereas the properties of ergodicity and stationarity impose more general constraints. All such sources are stochastic.

26 Information Rate Rate Information rate is the average entropy per symbol. For memoryless sources, this is merely the entropy of each symbol, while, in the case of a stationary stochastic process, it is r = lim n H(X n X n 1, X n 2...) In general (e.g., nonstationary), it is defined as 1 r = lim n n H(X n, X n 1, X n 2...) In information theory, one may thus speak of the rate or entropy of a language.

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28 Rate Distortion Theory R(D) = Minimum achievable rate under a given constraint on the expected distortion. X = random variable; T = alphabet for a compressed representation. If x X is represented by t T, there is a distortion d(x, t) R(D) = d(x, t) = x,t min I(T,X). {p(t x): d(x,t) D} p(x, t)d(x, t) = x,t p(x)p(t x)d(x, t)

29 Introduce a Lagrange multiplier parameter β and Solve the following variational problem L min [p(t x)] = I(T;X) + β d(x, t) p(x)p(t x). We need L p(t x) = 0. Since L = x p(x) t p(t x) log p(t x) p(t) +β x p(x) t p(t x)d(x, t), we have [ p(x) log p(t x) ] + βd(x, t) = 0. p(t) p(t x) e βd(x,t). p(t)

30 Summary Outline In summary, p(t x) = p(t) Z(x,β) e βd(x,t) p(t) = x p(x)p(t x). Z(x,β) = t p(t) exp[ βd(x, t)] is a Partition Function. The Lagrange parameter in this case is positive; It is determined by the upper bound on distortion: R D = β.

31 Redescription Some hidden object may be observed via two views X and Y (two random variables.) Create a common descriptor T Example X = words, Y = topics. R(D) = L min I(T;X) p(t x):i(t:y) D = I(T : X) βi(t;y)

32 Proceeding as before, we have p(t x) = p(t) = x p(t) Z(x,β) e βd KL[p(y x) p(y t)] p(x)p(t x) p(y t) = p(y x) = 1 p(t) p(x, y)p(t x) x p(x, y) p(x) Information Bottleneck = T.

33 Blahut-Arimoto Algorithm Start with the basic formulation for RDT; Can be changed mutatis mutandis for IB. Input: p(x), T, and β Output: p(t x) Step 1. Randomly initialize p(t) Step 2. loop until p(t x) converges (to a fixed point) Step 3. Step 4. Step 5. endloop p(t x) := p(t) Z(x,β) e βd(x,t) p(t) := x p(x)p(t x) Convex Programming: Optimization of a convex function over a convex set Global optimum exists!

34 PART X: Models

35 Outline 1 2 3

36 Information Bottleneck The computational principle uses Information Bottleneck Theory (a special case of RDT) It creates a Kripke model (with states and state transitions)

37 Data Outline D i = The random variable; Samples are the rows in submatrix of D i

38 States Outline States Cluster D i =; effectively identifying the state variable X i such that the mutual information between D i and X i is minimized min I(D i ; X i ) subj DISTORTION

39 Transitions Outline State-Transitions Preserve ontologies... Relate the conditional distribution P(O X i ) with P(O X i+1 ) (1 i < k) & P(O X i 1 ) (1 < i k) Measure distortions using Kullback-Liebler Distances.

40 Algorithm Outline Cluster the data over time as well as gene sets... Cluster and cluster-edges optimize the mutual information terms I(D i ; X i ) β + D KL (O X i O X i+1 ) β D KL (O X i O X i 1 ) Measure distortions using Kullback-Liebler Distances.

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43 Cancer Outline What causes cancer? 1 Cancer as a disease of the genomeé 2 Cancer as a somatic evolutionary processé 3 Cancer as a price of symbiosis (mitochondrial)é 4 Cancer as a response to multi-cellularityé 5 Cancer as a price of repair/regeneration (stem cells)é 6 Cancer as a consequence of energy consumption (Anaerobic Glycolysis)É Warburg Effect 7 Cancer as a response to external stressé 8 Cancer as a response to the micro-environment (hyperand hypo-methylation)

44 Processes involved in Cancer Coordination of processes in cancer progression 1 Inflammation 2 Autophagy and mitophagy 3 Apoptosis 4 Hypoxia 5 Anaerobic glycolysis 6 Fibrosis 7 Signaling

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46 PART XI: Laws

47 Outline 1 2 3

48 Character of Physical Laws For many philosophers of science, physics is the science and the way in which it has developed forms a model that all other sciences are implicitly expected to follow. But biology is not physics, and there will never be a biological equivalent of the physicists dream of a theory of everything, apart from the modern synthesis of natural selection and population genetics. Making sense of life: explaining biological development with models, metaphors and machines: Reviewed by M. Cobb

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50 Feynman Outline I want to discuss now the art of guessing nature s laws. It is an art. How is it done? One way you might suggest is to look at history to see ow the other guys did it. So we look at history. We must start with Newton. He had a situation where he had incomplete knowledge, and he was able to guess the laws by putting together ideas which were relatively close to experiment... Feynman: The Character of Physical Law

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52 Hypotheses non fingo Tycho Brahe s Data. Kepler s Laws: (Especially the Third Law) Galilean Principles: Law of Inertia, Galilean Identity (Inertial Mass = Gravitational Mass)

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55 Idler Wheels and Gears The next guy who did something great was maxwell, who obtained the laws of electricity and magnetism. What he did was this. He put together all the laws of electricity, due to Faraday and other people who came before him, and he looked at them and realized that they were mathematically inconsistent. In order to straighten it out he had to add one term to an equation. He did this by inventing for himself a model of idler wheels and geras and so on in space... So the logic may be wrong, but the answer right... Feynman: The Character of Physical Law

56 Einstein Outline In the case of [Einstein s] relativity the discovery was completely different. There was an accumulation of paradoxes; the known laws gave inconsistent results. This was a new kind of thinking, a thinking in terms of discussing the possible symmetries of laws... Also it was difficult to accept that ordinary ideas of time and space, which seemed so instinctive, could be wrong. Feynman: The Character of Physical Law

57 Thoughts.. Outline Newton Leibnitz Euler Bolzano Weirstrasse Frege Cantor Russell Wittgenstein

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59 Feynman Outline What is it about nature that lets this happen, that it is possible to guess from one part what the rest is going to do? That is an unscientific question: I do not know how to answer it, and therefore I am going to give an unscientific answer... I think it is because nature has a simplicity and therefore a great beauty. Feynman: The Character of Physical Law

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61 [End of Lecture #4]