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chapter 16. structured probabilistic models for deep learning the model until it is just barely possible to train or use. we often use models whose marginal distributions cannot be computed, and are satisﬁed simply to draw approximate samples from these models. we often train models with an intractable objective function that we cannot even approximate in a reasonable amount of time, but we are still able to approximately train the model if we can [UNK] obtain an estimate of the gradient of such a function. the deep learning approach is often to ﬁgure out what the minimum amount of information we absolutely need is, and then to ﬁgure out how to get a reasonable approximation of that information as quickly as possible. 16. 7. 1 example : the restricted boltzmann machine the restricted boltzmann machine ( rbm ) (, ) or smolensky 1986 harmonium is the quintessential example of how graphical models are used for deep learning. the rbm is not itself a deep model. instead, it has a single layer of latent variables that may be used to learn a representation for the input. in chapter, we will 20 see how rbms can be used to build many deeper models. here, we show how the rbm exempliﬁ	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	602	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
single layer of latent variables that may be used to learn a representation for the input. in chapter, we will 20 see how rbms can be used to build many deeper models. here, we show how the rbm exempliﬁes many of the practices used in a wide variety of deep graphical models : its units are organized into large groups called layers, the connectivity between layers is described by a matrix, the connectivity is relatively dense, the model is designed to allow [UNK] gibbs sampling, and the emphasis of the model design is on freeing the training algorithm to learn latent variables whose semantics were not speciﬁed by the designer. later, in section, we will revisit the rbm 20. 2 in more detail. the canonical rbm is an energy - based model with binary visible and hidden units. its energy function is e, ( v h b ) = −v c − h v − w h, ( 16. 10 ) where b, c, and w are unconstrained, real - valued, learnable parameters. we can see that the model is divided into two groups of units : v and h, and the interaction between them is described by a matrix w. the model is depicted graphically in ﬁgure	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	602	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
##ined, real - valued, learnable parameters. we can see that the model is divided into two groups of units : v and h, and the interaction between them is described by a matrix w. the model is depicted graphically in ﬁgure. as this ﬁgure makes clear, an important aspect of this model is 16. 14 that there are no direct interactions between any two visible units or between any two hidden units ( hence the “ restricted, ” a general boltzmann machine may have arbitrary connections ). the restrictions on the rbm structure yield the nice properties p ( ) = π h v \| ip ( hi \| v ) ( 16. 11 ) 587	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	602	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
chapter 16. structured probabilistic models for deep learning h1h1 h2h2 h3h3 v1 v1 v2 v2 v3 v3 h4h4 figure 16. 14 : an rbm drawn as a markov network. and p ( ) = π v h \| ip ( vi \| h ). ( 16. 12 ) the individual conditionals are simple to compute as well. for the binary rbm we obtain : p ( hi = 1 ) = \| v σ vw :, i + bi, ( 16. 13 ) p ( hi = 0 ) = 1 \| v −σ vw :, i + bi. ( 16. 14 ) together these properties allow for [UNK] block gibbs sampling, which alter - nates between sampling all of h simultaneously and sampling all of v simultane - ously. samples generated by gibbs sampling from an rbm model are shown in ﬁgure. 16. 15 since the energy function itself is just a linear function of the parameters, it is easy to take its derivatives. for example, ∂ ∂wi, j e, ( v h ) = −vihj. ( 16. 15 ) these two properties — [UNK] gibbs sampling and [UNK] derivatives — make	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	603	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
function of the parameters, it is easy to take its derivatives. for example, ∂ ∂wi, j e, ( v h ) = −vihj. ( 16. 15 ) these two properties — [UNK] gibbs sampling and [UNK] derivatives — make training convenient. in chapter, we will see that undirected models may be 18 trained by computing such derivatives applied to samples from the model. training the model induces a representation h of the data v. we can often use eh h [UNK] ( \| v ) [ ] h as a set of features to describe. v overall, the rbm demonstrates the typical deep learning approach to graph - ical models : representation learning accomplished via layers of latent variables, combined with [UNK] interactions between layers parametrized by matrices. the language of graphical models provides an elegant, ﬂexible and clear language for describing probabilistic models. in the chapters ahead, we use this language, among other perspectives, to describe a wide variety of deep probabilistic models. 588	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	603	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
chapter 16. structured probabilistic models for deep learning figure 16. 15 : samples from a trained rbm, and its weights. image reproduced with permission from ( ). lisa 2008 ( left ) samples from a model trained on mnist, drawn using gibbs sampling. each column is a separate gibbs sampling process. each row represents the output of another 1, 000 steps of gibbs sampling. successive samples are highly correlated with one another. the corresponding weight vectors. compare ( right ) this to the samples and weights of a linear factor model, shown in ﬁgure. the samples 13. 2 here are much better because the rbm priorp ( h ) is not constrained to be factorial. the rbm can learn which features should appear together when sampling. on the other hand, the rbm posterior is factorial, while the sparse coding posterior is not, p ( ) h v \| p ( ) h v \| so the sparse coding model may be better for feature extraction. other models are able to have both a non - factorial and a non - factorial. p ( ) h p ( ) h v \| 589	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	604	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
chapter 17 monte carlo methods randomized algorithms fall into two rough categories : las vegas algorithms and monte carlo algorithms. las vegas algorithms always return precisely the correct answer ( or report that they failed ). these algorithms consume a random amount of resources, usually memory or time. in contrast, monte carlo algorithms return answers with a random amount of error. the amount of error can typically be reduced by expending more resources ( usually running time and memory ). for any ﬁxed computational budget, a monte carlo algorithm can provide an approximate answer. many problems in machine learning are so [UNK] that we can never expect to obtain precise answers to them. this excludes precise deterministic algorithms and las vegas algorithms. instead, we must use deterministic approximate algorithms or monte carlo approximations. both approaches are ubiquitous in machine learning. in this chapter, we focus on monte carlo methods. 17. 1 sampling and monte carlo methods many important technologies used to accomplish machine learning goals are based on drawing samples from some probability distribution and using these samples to form a monte carlo estimate of some desired quantity. 17. 1. 1 why sampling? there are many reasons that we may wish to draw samples from a probability distribution. sampling provides a ﬂexible way to approximate many sums and 590	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	605	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
chapter 17. monte carlo methods integrals at reduced cost. sometimes we use this to provide a signiﬁcant speedup to a costly but tractable sum, as in the case when we subsample the full training cost with minibatches. in other cases, our learning algorithm requires us to approximate an intractable sum or integral, such as the gradient of the log partition function of an undirected model. in many other cases, sampling is actually our goal, in the sense that we want to train a model that can sample from the training distribution. 17. 1. 2 basics of monte carlo sampling when a sum or an integral cannot be computed exactly ( for example the sum has an exponential number of terms and no exact simpliﬁcation is known ) it is often possible to approximate it using monte carlo sampling. the idea is to view the sum or integral as if it was an expectation under some distribution and to approximate the expectation by a corresponding average. let s = x p f e ( ) x ( ) = x p [ ( ) ] f x ( 17. 1 ) or s = p f d e ( ) x ( ) x x = p [ ( ) ] f x ( 17. 2 ) be the sum or integral to	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	606	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
x ( ) = x p [ ( ) ] f x ( 17. 1 ) or s = p f d e ( ) x ( ) x x = p [ ( ) ] f x ( 17. 2 ) be the sum or integral to estimate, rewritten as an expectation, with the constraint that p is a probability distribution ( for the sum ) or a probability density ( for the integral ) over random variable. x we can approximate s by drawing n samples x ( 1 ),..., x ( ) n from p and then forming the empirical average [UNK] = 1 n n i = 1 f ( x ( ) i ). ( 17. 3 ) this approximation is justiﬁed by a few [UNK] properties. the ﬁrst trivial observation is that the estimator [UNK] is unbiased, since e [ [UNK] ] = 1 n n i = 1 e [ ( f x ( ) i ) ] = 1 n n i = 1 s s. = ( 17. 4 ) but in addition, the law of large numbers states that if the samples x ( ) i are i. i. d., then the average converges almost surely to the expected value : lim [UNK] = s, ( 17. 5 ) 591	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	606	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
chapter 17. monte carlo methods provided that the variance of the individual terms, var [ f ( x ( ) i ) ], is bounded. to see this more clearly, consider the variance of [UNK] as n increases. the variance var [ [UNK] ] decreases and converges to 0, so long as var [ ( f x ( ) i ) ] < ∞ : var [ [UNK] ] = 1 n2 n i = 1 var [ ( ) ] f x ( 17. 6 ) = var [ ( ) ] f x n. ( 17. 7 ) this convenient result also tells us how to estimate the uncertainty in a monte carlo average or equivalently the amount of expected error of the monte carlo approximation. we compute both the empirical average of the f ( x ( ) i ) and their empirical variance, 1 and then divide the estimated variance by the number of samples n to obtain an estimator of var [ [UNK] ]. the central limit theorem tells us that the distribution of the average, [UNK], converges to a normal distribution with mean s and variance var [ ( ) ] f x n. this allows us to estimate conﬁdence intervals around the estimate [UNK], using the cumulative distribution of the normal density. however, all this relies on our ability to easily	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	607	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
normal distribution with mean s and variance var [ ( ) ] f x n. this allows us to estimate conﬁdence intervals around the estimate [UNK], using the cumulative distribution of the normal density. however, all this relies on our ability to easily sample from the base distribution p ( x ), but doing so is not always possible. when it is not feasible to sample from p, an alternative is to use importance sampling, presented in section. a 17. 2 more general approach is to form a sequence of estimators that converge towards the distribution of interest. that is the approach of monte carlo markov chains ( section ). 17. 3 17. 2 importance sampling an important step in the decomposition of the integrand ( or summand ) used by the monte carlo method in equation is deciding which part of the integrand should 17. 2 play the role the probability p ( x ) and which part of the integrand should play the role of the quantity f ( x ) whose expected value ( under that probability distribution ) is to be estimated. there is no unique decomposition because p ( x ) f ( x ) can always be rewritten as p f q ( ) x ( ) = x ( ) x p f ( ) x ( )	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	607	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
under that probability distribution ) is to be estimated. there is no unique decomposition because p ( x ) f ( x ) can always be rewritten as p f q ( ) x ( ) = x ( ) x p f ( ) x ( ) x q ( ) x, ( 17. 8 ) where we now sample from q and average pf q. in many cases, we wish to compute an expectation for a given p and an f, and the fact that the problem is speciﬁed 1the unbiased estimator of the variance is often preferred, in which the sum of squared [UNK] is divided by instead of. n −1 n 592	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	607	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
chapter 17. monte carlo methods from the start as an expectation suggests that this p and f would be a natural choice of decomposition. however, the original speciﬁcation of the problem may not be the the optimal choice in terms of the number of samples required to obtain a given level of accuracy. fortunately, the form of the optimal choice q∗can be derived easily. the optimal q∗corresponds to what is called optimal importance sampling. because of the identity shown in equation, any monte carlo estimator 17. 8 [UNK] = 1 n n i, = 1 x ( ) i [UNK] f ( x ( ) i ) ( 17. 9 ) can be transformed into an importance sampling estimator [UNK] = 1 n n i, = 1 x ( ) i [UNK] p ( x ( ) i ) ( f x ( ) i ) q ( x ( ) i ). ( 17. 10 ) we see readily that the expected value of the estimator does not depend on : q eq [ [UNK] ] = eq [ [UNK] ] = s. ( 17. 11 ) however, the variance of an importance sampling estimator can be greatly sensitive to the choice of. the variance is given by q var [ [UNK] ] = var [ p	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	608	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
[UNK] ] = eq [ [UNK] ] = s. ( 17. 11 ) however, the variance of an importance sampling estimator can be greatly sensitive to the choice of. the variance is given by q var [ [UNK] ] = var [ p f ( ) x ( ) x q ( ) x ] / n. ( 17. 12 ) the minimum variance occurs when is q q∗ ( ) = x p f ( ) x \| ( ) x \| z, ( 17. 13 ) where z is the normalization constant, chosen so that q∗ ( x ) sums or integrates to 1 as appropriate. better importance sampling distributions put more weight where the integrand is larger. in fact, when f ( x ) does not change sign, var [ [UNK] ] = 0, meaning that when the optimal distribution is used. a single sample is [UNK] of course, this is only because the computation of q∗has essentially solved the original problem, so it is usually not practical to use this approach of drawing a single sample from the optimal distribution. any choice of sampling distribution q is valid ( in the sense of yielding the correct expected value ) and q∗is the optimal one ( in the sense of yielding minimum variance ). sampling from q∗is usually in	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	608	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
single sample from the optimal distribution. any choice of sampling distribution q is valid ( in the sense of yielding the correct expected value ) and q∗is the optimal one ( in the sense of yielding minimum variance ). sampling from q∗is usually infeasible, but other choices of q can be feasible while still reducing the variance somewhat. 593	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	608	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
chapter 17. monte carlo methods another approach is to use biased importance sampling, which has the advantage of not requiring normalized p or q. in the case of discrete variables, the biased importance sampling estimator is given by [UNK] = n i = 1 p ( x ( ) i ) q ( x ( ) i ) f ( x ( ) i ) n i = 1 p ( x ( ) i ) q ( x ( ) i ) ( 17. 14 ) = n i = 1 p ( x ( ) i ) [UNK] ( x ( ) i ) f ( x ( ) i ) n i = 1 p ( x ( ) i ) [UNK] ( x ( ) i ) ( 17. 15 ) = n i = 1 [UNK] ( x ( ) i ) [UNK] ( x ( ) i ) f ( x ( ) i ) n i = 1 [UNK] ( x ( ) i ) [UNK] ( x ( ) i ), ( 17. 16 ) where [UNK] and [UNK] are the unnormalized forms of p and q and the x ( ) i are the samples from q. this estimator is biased because e [ [UNK] bis ] = s, except asymptotically when n →∞and the denominator of equation converges to	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	609	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
p and q and the x ( ) i are the samples from q. this estimator is biased because e [ [UNK] bis ] = s, except asymptotically when n →∞and the denominator of equation converges to 1. hence this estimator 17. 14 is called asymptotically unbiased. although a good choice of q can greatly improve the [UNK] of monte carlo estimation, a poor choice of q can make the [UNK] much worse. going back to equation, we see that if there are samples of 17. 12 q for which p f ( ) x \| ( ) x \| q ( ) x is large, then the variance of the estimator can get very large. this may happen when q ( x ) is tiny while neither p ( x ) nor f ( x ) are small enough to cancel it. the q distribution is usually chosen to be a very simple distribution so that it is easy to sample from. when x is high - dimensional, this simplicity in q causes it to match p or p f \| \| poorly. when q ( x ( ) i ) p ( x ( ) i ) \| f ( x ( ) i ) \|, importance sampling collects useless samples ( summing tiny numbers or	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	609	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
in q causes it to match p or p f \| \| poorly. when q ( x ( ) i ) p ( x ( ) i ) \| f ( x ( ) i ) \|, importance sampling collects useless samples ( summing tiny numbers or zeros ). on the other hand, when q ( x ( ) i ) p ( x ( ) i ) \| f ( x ( ) i ) \|, which will happen more rarely, the ratio can be huge. because these latter events are rare, they may not show up in a typical sample, yielding typical underestimation of s, compensated rarely by gross overestimation. such very large or very small numbers are typical when x is high dimensional, because in high dimension the dynamic range of joint probabilities can be very large. in spite of this danger, importance sampling and its variants have been found very useful in many machine learning algorithms, including deep learning algorithms. for example, see the use of importance sampling to accelerate training in neural language models with a large vocabulary ( section ) or other neural nets 12. 4. 3. 3 with a large number of outputs. see also how importance sampling has been used to estimate a partition function ( the normalization constant of a probability 594	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	609	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
chapter 17. monte carlo methods distribution ) in section, and to estimate the log - likelihood in deep directed 18. 7 models such as the variational autoencoder, in section. importance sampling 20. 10. 3 may also be used to improve the estimate of the gradient of the cost function used to train model parameters with stochastic gradient descent, particularly for models such as classiﬁers where most of the total value of the cost function comes from a small number of misclassiﬁed examples. sampling more [UNK] examples more frequently can reduce the variance of the gradient in such cases (, ). hinton 2006 17. 3 markov chain monte carlo methods in many cases, we wish to use a monte carlo technique but there is no tractable method for drawing exact samples from the distribution pmodel ( x ) or from a good ( low variance ) importance sampling distribution q ( x ). in the context of deep learning, this most often happens when pmodel ( x ) is represented by an undirected model. in these cases, we introduce a mathematical tool called a markov chain to approximately sample from pmodel ( x ). the family of algorithms that use markov chains to perform monte carlo estimates is called markov chain monte	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	610	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
undirected model. in these cases, we introduce a mathematical tool called a markov chain to approximately sample from pmodel ( x ). the family of algorithms that use markov chains to perform monte carlo estimates is called markov chain monte carlo methods ( mcmc ). markov chain monte carlo methods for machine learning are described at greater length in koller and friedman 2009 ( ). the most standard, generic guarantees for mcmc techniques are only applicable when the model does not assign zero probability to any state. therefore, it is most convenient to present these techniques as sampling from an energy - based model ( ebm ) p ( x ) [UNK] − exp ( e ( ) ) x as described in section. in the ebm formulation, every 16. 2. 4 state is guaranteed to have non - zero probability. mcmc methods are in fact more broadly applicable and can be used with many probability distributions that contain zero probability states. however, the theoretical guarantees concerning the behavior of mcmc methods must be proven on a case - by - case basis for [UNK] families of such distributions. in the context of deep learning, it is most common to rely on the most general theoretical guarantees that naturally apply to all energy - based models. to understand why drawing samples from	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	610	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
on a case - by - case basis for [UNK] families of such distributions. in the context of deep learning, it is most common to rely on the most general theoretical guarantees that naturally apply to all energy - based models. to understand why drawing samples from an energy - based model is [UNK], consider an ebm over just two variables, deﬁning a distribution a b. in order p (, ) to sample a, we must draw a from p ( a b \| ), and in order to sample b, we must draw it from p ( b a \| ). it seems to be an intractable chicken - and - egg problem. directed models avoid this because their graph is directed and acyclic. to perform ancestral sampling one simply samples each of the variables in topological order, conditioning on each variable ’ s parents, which are guaranteed to have already been sampled ( section ). ancestral sampling deﬁnes an [UNK], single - pass method 16. 3 595	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	610	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
chapter 17. monte carlo methods of obtaining a sample. in an ebm, we can avoid this chicken and egg problem by sampling using a markov chain. the core idea of a markov chain is to have a state x that begins as an arbitrary value. over time, we randomly update x repeatedly. eventually x becomes ( very nearly ) a fair sample from p ( x ). formally, a markov chain is deﬁned by a random state x and a transition distribution t ( x \| x ) specifying the probability that a random update will go to state xif it starts in state x. running the markov chain means repeatedly updating the state x to a value x sampled from t ( x \| x ). to gain some theoretical understanding of how mcmc methods work, it is useful to reparametrize the problem. first, we restrict our attention to the case where the random variable x has countably many states. we can then represent the state as just a positive integer x. [UNK] integer values of x map back to [UNK] states in the original problem. x consider what happens when we run inﬁnitely many markov chains in parallel. all of the states of the [UNK] markov chains are drawn from some distribution q ( )	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	611	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
values of x map back to [UNK] states in the original problem. x consider what happens when we run inﬁnitely many markov chains in parallel. all of the states of the [UNK] markov chains are drawn from some distribution q ( ) t ( x ), where t indicates the number of time steps that have elapsed. at the beginning, q ( 0 ) is some distribution that we used to arbitrarily initialize x for each markov chain. later, q ( ) t is inﬂuenced by all of the markov chain steps that have run so far. our goal is for q ( ) t ( ) x to converge to. p x ( ) because we have reparametrized the problem in terms of positive integer x, we can describe the probability distribution using a vector, with q v q i v ( = x ) = i. ( 17. 17 ) consider what happens when we update a single markov chain ’ s state x to a new state x. the probability of a single state landing in state xis given by q ( + 1 ) t ( x ) = x q ( ) t ( ) ( x t x \| x. ) ( 17. 18 ) using our integer parametrization,	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	611	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
the probability of a single state landing in state xis given by q ( + 1 ) t ( x ) = x q ( ) t ( ) ( x t x \| x. ) ( 17. 18 ) using our integer parametrization, we can represent the [UNK] of the transition operator using a matrix. we deﬁne so that t a a ai, j = ( t x = = ) i \| x j. ( 17. 19 ) using this deﬁnition, we can now rewrite equation. rather than writing it in 17. 18 terms of q and t to understand how a single state is updated, we may now use v and a to describe how the entire distribution over all the [UNK] markov chains ( running in parallel ) shifts as we apply an update : v ( ) t = av ( 1 ) t−. ( 17. 20 ) 596	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	611	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
chapter 17. monte carlo methods applying the markov chain update repeatedly corresponds to multiplying by the matrix a repeatedly. in other words, we can think of the process as exponentiating the matrix : a v ( ) t = atv ( 0 ). ( 17. 21 ) the matrix a has special structure because each of its columns represents a probability distribution. such matrices are called stochastic matrices. if there is a non - zero probability of transitioning from any state x to any other state xfor some power t, then the perron - frobenius theorem (, ; perron 1907 frobenius 1908, ) guarantees that the largest eigenvalue is real and equal to. over time, we can 1 see that all of the eigenvalues are exponentiated : v ( ) t = v λ v diag ( ) −1t v ( 0 ) = ( ) v diag λ tv −1v ( 0 ). ( 17. 22 ) this process causes all of the eigenvalues that are not equal to to decay to zero. 1 under some additional mild conditions, a is guaranteed to have only one eigenvector with eigenvalue. the process thus converges to a 1 stationary distribution,	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	612	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
the eigenvalues that are not equal to to decay to zero. 1 under some additional mild conditions, a is guaranteed to have only one eigenvector with eigenvalue. the process thus converges to a 1 stationary distribution, sometimes also called the. at convergence, equilibrium distribution v = = av v, ( 17. 23 ) and this same condition holds for every additional step. this is an eigenvector equation. to be a stationary point, v must be an eigenvector with corresponding eigenvalue. this condition guarantees that once we have reached the stationary 1 distribution, repeated applications of the transition sampling procedure do not change the over the states of all the various markov chains ( although distribution transition operator does change each individual state, of course ). if we have chosen t correctly, then the stationary distribution q will be equal to the distribution p we wish to sample from. we will describe how to choose t shortly, in section. 17. 4 most properties of markov chains with countable states can be generalized to continuous variables. in this situation, some authors call the markov chain a harris chain but we use the term markov chain to describe both conditions. in general, a markov chain with transition operator t will	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	612	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
with countable states can be generalized to continuous variables. in this situation, some authors call the markov chain a harris chain but we use the term markov chain to describe both conditions. in general, a markov chain with transition operator t will converge, under mild conditions, to a ﬁxed point described by the equation q ( x ) = [UNK] ( x \| x ), ( 17. 24 ) which in the discrete case is just rewriting equation. when 17. 23 x is discrete, the expectation corresponds to a sum, and when x is continuous, the expectation corresponds to an integral. 597	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	612	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
chapter 17. monte carlo methods regardless of whether the state is continuous or discrete, all markov chain methods consist of repeatedly applying stochastic updates until eventually the state begins to yield samples from the equilibrium distribution. running the markov chain until it reaches its equilibrium distribution is called “ burning in ” the markov chain. after the chain has reached equilibrium, a sequence of inﬁnitely many samples may be drawn from from the equilibrium distribution. they are identically distributed but any two successive samples will be highly correlated with each other. a ﬁnite sequence of samples may thus not be very representative of the equilibrium distribution. one way to mitigate this problem is to return only every n successive samples, so that our estimate of the statistics of the equilibrium distribution is not as biased by the correlation between an mcmc sample and the next several samples. markov chains are thus expensive to use because of the time required to burn in to the equilibrium distribution and the time required to transition from one sample to another reasonably decorrelated sample after reaching equilibrium. if one desires truly independent samples, one can run multiple markov chains in parallel. this approach uses extra parallel computation to eliminate latency. the strategy of using only a single markov chain to generate all samples and the strategy	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	613	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
sample after reaching equilibrium. if one desires truly independent samples, one can run multiple markov chains in parallel. this approach uses extra parallel computation to eliminate latency. the strategy of using only a single markov chain to generate all samples and the strategy of using one markov chain for each desired sample are two extremes ; deep learning practitioners usually use a number of chains that is similar to the number of examples in a minibatch and then draw as many samples as are needed from this ﬁxed set of markov chains. a commonly used number of markov chains is 100. another [UNK] is that we do not know in advance how many steps the markov chain must run before reaching its equilibrium distribution. this length of time is called the mixing time. it is also very [UNK] to test whether a markov chain has reached equilibrium. we do not have a precise enough theory for guiding us in answering this question. theory tells us that the chain will converge, but not much more. if we analyze the markov chain from the point of view of a matrix a acting on a vector of probabilities v, then we know that the chain mixes when at has [UNK] lost all of the eigenvalues from a besides the unique eigenvalue of. 1 this means	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	613	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
point of view of a matrix a acting on a vector of probabilities v, then we know that the chain mixes when at has [UNK] lost all of the eigenvalues from a besides the unique eigenvalue of. 1 this means that the magnitude of the second largest eigenvalue will determine the mixing time. however, in practice, we cannot actually represent our markov chain in terms of a matrix. the number of states that our probabilistic model can visit is exponentially large in the number of variables, so it is infeasible to represent v, a, or the eigenvalues of a. due to these and other obstacles, we usually do not know whether a markov chain has mixed. instead, we simply run the markov chain for an amount of time that we roughly estimate to be [UNK], and use heuristic methods to determine whether the chain has mixed. these heuristic methods include manually inspecting samples or measuring correlations between 598	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	613	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
chapter 17. monte carlo methods successive samples. 17. 4 gibbs sampling so far we have described how to draw samples from a distribution q ( x ) by repeatedly updating x x ← [UNK] ( x \| x ). however, we have not described how to ensure that q ( x ) is a useful distribution. two basic approaches are considered in this book. the ﬁrst one is to derive t from a given learned pmodel, described below with the case of sampling from ebms. the second one is to directly parametrize t and learn it, so that its stationary distribution implicitly deﬁnes the pmodel of interest. examples of this second approach are discussed in sections and. 20. 12 20. 13 in the context of deep learning, we commonly use markov chains to draw samples from an energy - based model deﬁning a distribution pmodel ( x ). in this case, we want the q ( x ) for the markov chain to be pmodel ( x ). to obtain the desired q ( ) x, we must choose an appropriate t ( x \| x ). a conceptually simple and [UNK] approach to building a markov chain that samples from pmodel ( x ) is to use gibbs sampling,	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	614	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
. to obtain the desired q ( ) x, we must choose an appropriate t ( x \| x ). a conceptually simple and [UNK] approach to building a markov chain that samples from pmodel ( x ) is to use gibbs sampling, in which sampling from t ( x \| x ) is accomplished by selecting one variable xi and sampling it from pmodel conditioned on its neighbors in the undirected graph g deﬁning the structure of the energy - based model. it is also possible to sample several variables at the same time so long as they are conditionally independent given all of their neighbors. as shown in the rbm example in section, all of the hidden units of an 16. 7. 1 rbm may be sampled simultaneously because they are conditionally independent from each other given all of the visible units. likewise, all of the visible units may be sampled simultaneously because they are conditionally independent from each other given all of the hidden units. gibbs sampling approaches that update many variables simultaneously in this way are called block gibbs sampling. alternate approaches to designing markov chains to sample from pmodel are possible. for example, the metropolis - hastings algorithm is widely used in other disciplines. in the context of the deep learning approach to undirected modeling,	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	614	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
called block gibbs sampling. alternate approaches to designing markov chains to sample from pmodel are possible. for example, the metropolis - hastings algorithm is widely used in other disciplines. in the context of the deep learning approach to undirected modeling, it is rare to use any approach other than gibbs sampling. improved sampling techniques are one possible research frontier. 17. 5 the challenge of mixing between separated modes the primary [UNK] involved with mcmc methods is that they have a tendency to mix poorly. ideally, successive samples from a markov chain designed to sample 599	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	614	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
chapter 17. monte carlo methods from p ( x ) would be completely independent from each other and would visit many [UNK] regions in x space proportional to their probability. instead, especially in high dimensional cases, mcmc samples become very correlated. we refer to such behavior as slow mixing or even failure to mix. mcmc methods with slow mixing can be seen as inadvertently performing something resembling noisy gradient descent on the energy function, or equivalently noisy hill climbing on the probability, with respect to the state of the chain ( the random variables being sampled ). the chain tends to take small steps ( in the space of the state of the markov chain ), from a conﬁguration x ( 1 ) t− to a conﬁguration x ( ) t, with the energy e ( x ( ) t ) generally lower or approximately equal to the energy e ( x ( 1 ) t− ), with a preference for moves that yield lower energy conﬁgurations. when starting from a rather improbable conﬁguration ( higher energy than the typical ones from p ( x ) ), the chain tends to gradually reduce the energy of the state and only occasionally move to another mode. once the chain has found a region of low energy ( for example, if	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	615	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
##guration ( higher energy than the typical ones from p ( x ) ), the chain tends to gradually reduce the energy of the state and only occasionally move to another mode. once the chain has found a region of low energy ( for example, if the variables are pixels in an image, a region of low energy might be a connected manifold of images of the same object ), which we call a mode, the chain will tend to walk around that mode ( following a kind of random walk ). once in a while it will step out of that mode and generally return to it or ( if it ﬁnds an escape route ) move towards another mode. the problem is that successful escape routes are rare for many interesting distributions, so the markov chain will continue to sample the same mode longer than it should. this is very clear when we consider the gibbs sampling algorithm ( section ). 17. 4 in this context, consider the probability of going from one mode to a nearby mode within a given number of steps. what will determine that probability is the shape of the “ energy barrier ” between these modes. transitions between two modes that are separated by a high energy barrier ( a region of low probability ) are exponentially less likely ( in terms of the height of the energy barrier )	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	615	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
that probability is the shape of the “ energy barrier ” between these modes. transitions between two modes that are separated by a high energy barrier ( a region of low probability ) are exponentially less likely ( in terms of the height of the energy barrier ). this is illustrated in ﬁgure. the problem arises when there are multiple modes with 17. 1 high probability that are separated by regions of low probability, especially when each gibbs sampling step must update only a small subset of variables whose values are largely determined by the other variables. as a simple example, consider an energy - based model over two variables a and b, which are both binary with a sign, taking on values −1 1 and. if e ( a b, ) = −wab for some large positive number w, then the model expresses a strong belief that a and b have the same sign. consider updating b using a gibbs sampling step with a = 1. the conditional distribution over b is given by p ( b = 1 \| a = 1 ) = σ ( w ). if w is large, the sigmoid saturates, and the probability of also assigning b to be 1 is close to 1. likewise, if a = −1, the probability of assigning b to	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	615	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
= σ ( w ). if w is large, the sigmoid saturates, and the probability of also assigning b to be 1 is close to 1. likewise, if a = −1, the probability of assigning b to be −1 is close to 1. according to pmodel ( a b, ), both signs of both variables are equally likely. 600	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	615	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
chapter 17. monte carlo methods figure 17. 1 : paths followed by gibbs sampling for three distributions, with the markov chain initialized at the mode in both cases. ( left ) a multivariate normal distribution with two independent variables. gibbs sampling mixes well because the variables are independent. a multivariate normal distribution with highly correlated variables. ( center ) the correlation between variables makes it [UNK] for the markov chain to mix. because the update for each variable must be conditioned on the other variable, the correlation reduces the rate at which the markov chain can move away from the starting point. ( right ) a mixture of gaussians with widely separated modes that are not axis - aligned. gibbs sampling mixes very slowly because it is [UNK] to change modes while altering only one variable at a time. according to pmodel ( a b \| ), both variables should have the same sign. this means that gibbs sampling will only very rarely ﬂip the signs of these variables. in more practical scenarios, the challenge is even greater because we care not only about making transitions between two modes but more generally between all the many modes that a real model might contain. if several such transitions are [UNK] because of the [UNK] of mixing between modes, then it becomes very expensive to obtain a	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	616	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
greater because we care not only about making transitions between two modes but more generally between all the many modes that a real model might contain. if several such transitions are [UNK] because of the [UNK] of mixing between modes, then it becomes very expensive to obtain a reliable set of samples covering most of the modes, and convergence of the chain to its stationary distribution is very slow. sometimes this problem can be resolved by ﬁnding groups of highly dependent units and updating all of them simultaneously in a block. unfortunately, when the dependencies are complicated, it can be computationally intractable to draw a sample from the group. after all, the problem that the markov chain was originally introduced to solve is this problem of sampling from a large group of variables. in the context of models with latent variables, which deﬁne a joint distribution pmodel ( x h, ), we often draw samples of x by alternating between sampling from pmodel ( x h \| ) and sampling from pmodel ( h x \| ). from the point of view of mixing 601	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	616	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
chapter 17. monte carlo methods figure 17. 2 : an illustration of the slow mixing problem in deep probabilistic models. each panel should be read left to right, top to bottom. ( left ) consecutive samples from gibbs sampling applied to a deep boltzmann machine trained on the mnist dataset. consecutive samples are similar to each other. because the gibbs sampling is performed in a deep graphical model, this similarity is based more on semantic rather than raw visual features, but it is still [UNK] for the gibbs chain to transition from one mode of the distribution to another, for example by changing the digit identity. consecutive ( right ) ancestral samples from a generative adversarial network. because ancestral sampling generates each sample independently from the others, there is no mixing problem. rapidly, we would like pmodel ( h x \| ) to have very high entropy. however, from the point of view of learning a useful representation of h, we would like h to encode enough information about x to reconstruct it well, which implies that h and x should have very high mutual information. these two goals are at odds with each other. we often learn generative models that very precisely encode x into h but are not able to mix very well. this	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	617	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)