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##tive learning algorithms, but they are still deservedly recognized for their important role in deep learning history. deep belief networks are generative models with several layers of latent variables. the latent variables are typically binary, while the visible units may be binary or real. there are no intralayer connections. usually, every unit in each layer is connected to every unit in each neighboring layer, though it is possible to construct more sparsely connected dbns. the connections between the top two layers are undirected. the connections between all other layers are directed, with the arrows pointed toward the layer that is closest to the data. see ﬁgure b for an example. 20. 1 a dbn with l hidden layers contains l weight matrices : w ( 1 ),..., w ( ) l. it also contains l + 1 bias vectors : b ( 0 ),..., b ( ) l, with b ( 0 ) providing the biases for the visible layer. the probability distribution represented by the dbn is given by p ( h ( ) l, h ( 1 ) l− ) exp [UNK] b ( ) l h ( ) l + b ( 1 ) l− h ( 1 ) l− + h ( 1 ) l− w	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	675	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
##n is given by p ( h ( ) l, h ( 1 ) l− ) exp [UNK] b ( ) l h ( ) l + b ( 1 ) l− h ( 1 ) l− + h ( 1 ) l− w ( ) l h ( ) l, ( 20. 17 ) p h ( ( ) k i = 1 \| h ( + 1 ) k ) = σ b ( ) k i + w ( + 1 ) k :, i h ( + 1 ) k [UNK] ∈ − i, k 1,..., l 2, ( 20. 18 ) p v ( i = 1 \| h ( 1 ) ) = σ b ( 0 ) i + w ( 1 ) :, i h ( 1 ) [UNK]. ( 20. 19 ) in the case of real - valued visible units, substitute v [UNK] v b ; ( 0 ) + w ( 1 ) h ( 1 ), β −1 ( 20. 20 ) 660	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	675	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
chapter 20. deep generative models with β diagonal for tractability. generalizations to other exponential family visible units are straightforward, at least in theory. a dbn with only one hidden layer is just an rbm. to generate a sample from a dbn, we ﬁrst run several steps of gibbs sampling on the top two hidden layers. this stage is essentially drawing a sample from the rbm deﬁned by the top two hidden layers. we can then use a single pass of ancestral sampling through the rest of the model to draw a sample from the visible units. deep belief networks incur many of the problems associated with both directed models and undirected models. inference in a deep belief network is intractable due to the explaining away [UNK] within each directed layer, and due to the interaction between the two hidden layers that have undirected connections. evaluating or maximizing the standard evidence lower bound on the log - likelihood is also intractable, because the evidence lower bound takes the expectation of cliques whose size is equal to the network width. evaluating or maximizing the log - likelihood requires not just confronting the problem of intractable inference to marginalize out the latent variables, but also the problem of an intractable partition function within the undirected model of the	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	676	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
to the network width. evaluating or maximizing the log - likelihood requires not just confronting the problem of intractable inference to marginalize out the latent variables, but also the problem of an intractable partition function within the undirected model of the top two layers. to train a deep belief network, one begins by training an rbm to maximize [UNK] log p ( v ) using contrastive divergence or stochastic maximum likelihood. the parameters of the rbm then deﬁne the parameters of the ﬁrst layer of the dbn. next, a second rbm is trained to approximately maximize [UNK] ( 1 ) [UNK] ( 1 ) ( h ( 1 ) \| v ) log p ( 2 ) ( h ( 1 ) ) ( 20. 21 ) where p ( 1 ) is the probability distribution represented by the ﬁrst rbm and p ( 2 ) is the probability distribution represented by the second rbm. in other words, the second rbm is trained to model the distribution deﬁned by sampling the hidden units of the ﬁrst rbm, when the ﬁrst rbm is driven by the data. this procedure can be repeated indeﬁnitely, to add as many layers to the dbn as desired, with each new rbm modeling	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	676	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
units of the ﬁrst rbm, when the ﬁrst rbm is driven by the data. this procedure can be repeated indeﬁnitely, to add as many layers to the dbn as desired, with each new rbm modeling the samples of the previous one. each rbm deﬁnes another layer of the dbn. this procedure can be justiﬁed as increasing a variational lower bound on the log - likelihood of the data under the dbn ( hinton et al., ). 2006 in most applications, no [UNK] is made to jointly train the dbn after the greedy layer - wise procedure is complete. however, it is possible to perform generative ﬁne - tuning using the wake - sleep algorithm. 661	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	676	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
chapter 20. deep generative models the trained dbn may be used directly as a generative model, but most of the interest in dbns arose from their ability to improve classiﬁcation models. we can take the weights from the dbn and use them to deﬁne an mlp : h ( 1 ) = σ b ( 1 ) + v w ( 1 ). ( 20. 22 ) h ( ) l = σ b ( ) l i + h ( 1 ) l− w ( ) l [UNK] l 2,..., m, ( 20. 23 ) after initializing this mlp with the weights and biases learned via generative training of the dbn, we may train the mlp to perform a classiﬁcation task. this additional training of the mlp is an example of discriminative ﬁne - tuning. this speciﬁc choice of mlp is somewhat arbitrary, compared to many of the inference equations in chapter that are derived from ﬁrst principles. this mlp 19 is a heuristic choice that seems to work well in practice and is used consistently in the literature. many approximate inference techniques are motivated by their ability to ﬁnd a maximally variational lower bound on the log - likelihood	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	677	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
. this mlp 19 is a heuristic choice that seems to work well in practice and is used consistently in the literature. many approximate inference techniques are motivated by their ability to ﬁnd a maximally variational lower bound on the log - likelihood tight under some set of constraints. one can construct a variational lower bound on the log - likelihood using the hidden unit expectations deﬁned by the dbn ’ s mlp, but this is true of probability distribution over the hidden units, and there is no any reason to believe that this mlp provides a particularly tight bound. in particular, the mlp ignores many important interactions in the dbn graphical model. the mlp propagates information upward from the visible units to the deepest hidden units, but does not propagate any information downward or sideways. the dbn graphical model has explaining away interactions between all of the hidden units within the same layer as well as top - down interactions between layers. while the log - likelihood of a dbn is intractable, it may be approximated with ais ( salakhutdinov and murray 2008, ). this permits evaluating its quality as a generative model. the term “ deep belief network ” is commonly used incorrectly to refer to any kind of deep neural	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	677	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
it may be approximated with ais ( salakhutdinov and murray 2008, ). this permits evaluating its quality as a generative model. the term “ deep belief network ” is commonly used incorrectly to refer to any kind of deep neural network, even networks without latent variable semantics. the term “ deep belief network ” should refer speciﬁcally to models with undirected connections in the deepest layer and directed connections pointing downward between all other pairs of consecutive layers. the term “ deep belief network ” may also cause some confusion because the term “ belief network ” is sometimes used to refer to purely directed models, while deep belief networks contain an undirected layer. deep belief networks also share the acronym dbn with dynamic bayesian networks ( dean and kanazawa 1989, ), which are bayesian networks for representing markov chains. 662	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	677	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
chapter 20. deep generative models h ( 1 ) 1 h ( 1 ) 1 h ( 1 ) 2 h ( 1 ) 2 h ( 1 ) 3 h ( 1 ) 3 v1 v1 v2 v2 v3 v3 h ( 2 ) 1 h ( 2 ) 1 h ( 2 ) 2 h ( 2 ) 2 h ( 2 ) 3 h ( 2 ) 3 h ( 1 ) 4 h ( 1 ) 4 figure 20. 2 : the graphical model for a deep boltzmann machine with one visible layer ( bottom ) and two hidden layers. connections are only between units in neighboring layers. there are no intralayer layer connections. 20. 4 deep boltzmann machines a deep boltzmann machine or dbm ( salakhutdinov and hinton 2009a, ) is another kind of deep, generative model. unlike the deep belief network ( dbn ), it is an entirely undirected model. unlike the rbm, the dbm has several layers of latent variables ( rbms have just one ). but like the rbm, within each layer, each of the variables are mutually independent, conditioned on the variables in the neighboring layers. see ﬁgure for the graph structure. deep boltzmann 20	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	678	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
variables ( rbms have just one ). but like the rbm, within each layer, each of the variables are mutually independent, conditioned on the variables in the neighboring layers. see ﬁgure for the graph structure. deep boltzmann 20. 2 machines have been applied to a variety of tasks including document modeling ( srivastava 2013 et al., ). like rbms and dbns, dbms typically contain only binary units — as we assume for simplicity of our presentation of the model — but it is straightforward to include real - valued visible units. a dbm is an energy - based model, meaning that the the joint probability distribution over the model variables is parametrized by an energy function e. in the case of a deep boltzmann machine with one visible layer, v, and three hidden layers, h ( 1 ), h ( 2 ) and h ( 3 ), the joint probability is given by : p v h, ( 1 ), h ( 2 ), h ( 3 ) = 1 z ( ) θ exp −e, ( v h ( 1 ), h ( 2 ), h ( 3 ) ; ) θ. ( 20. 24 ) to simplify our presentation, we omit the bias parameters below	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	678	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
) = 1 z ( ) θ exp −e, ( v h ( 1 ), h ( 2 ), h ( 3 ) ; ) θ. ( 20. 24 ) to simplify our presentation, we omit the bias parameters below. the dbm energy function is then deﬁned as follows : e, ( v h ( 1 ), h ( 2 ), h ( 3 ) ; ) = θ −vw ( 1 ) h ( 1 ) −h ( 1 ) w ( 2 ) h ( 2 ) −h ( 2 ) w ( 3 ) h ( 3 ). ( 20. 25 ) 663	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	678	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
chapter 20. deep generative models h ( 1 ) 1 h ( 1 ) 1 h ( 1 ) 2 h ( 1 ) 2 h ( 1 ) 3 h ( 1 ) 3 v1 v1 v2 v2 h ( 2 ) 1 h ( 2 ) 1 h ( 2 ) 2 h ( 2 ) 2 h ( 2 ) 3 h ( 2 ) 3 h ( 3 ) 1 h ( 3 ) 1 h ( 3 ) 2 h ( 3 ) 2 v1 v2 h ( 2 ) 1 h ( 2 ) 1 h ( 2 ) 2 h ( 2 ) 2 h ( 2 ) 3 h ( 2 ) 3 h ( 1 ) 1 h ( 1 ) 1 h ( 1 ) 2 h ( 1 ) 2 h ( 1 ) 3 h ( 1 ) 3 h ( 3 ) 1 h ( 3 ) 1 h ( 3 ) 2 h ( 3 ) 2 figure 20. 3 : a deep boltzmann machine, re - arranged to reveal its bipartite graph structure. in comparison to the rbm energy function ( equation ), the dbm energy 20. 5 function includes connections between the hidden units ( latent variables ) in the form of the weight matrices ( w ( 2 ) and w ( 3 ) ). as we will	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	679	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
the rbm energy function ( equation ), the dbm energy 20. 5 function includes connections between the hidden units ( latent variables ) in the form of the weight matrices ( w ( 2 ) and w ( 3 ) ). as we will see, these connections have signiﬁcant consequences for both the model behavior as well as how we go about performing inference in the model. in comparison to fully connected boltzmann machines ( with every unit con - nected to every other unit ), the dbm [UNK] some advantages that are similar to those [UNK] by the rbm. speciﬁcally, as illustrated in ﬁgure, the dbm 20. 3 layers can be organized into a bipartite graph, with odd layers on one side and even layers on the other. this immediately implies that when we condition on the variables in the even layer, the variables in the odd layers become conditionally independent. of course, when we condition on the variables in the odd layers, the variables in the even layers also become conditionally independent. the bipartite structure of the dbm means that we can apply the same equa - tions we have previously used for the conditional distributions of an rbm to determine the conditional distributions in a dbm	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	679	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
even layers also become conditionally independent. the bipartite structure of the dbm means that we can apply the same equa - tions we have previously used for the conditional distributions of an rbm to determine the conditional distributions in a dbm. the units within a layer are conditionally independent from each other given the values of the neighboring layers, so the distributions over binary variables can be fully described by the bernoulli parameters giving the probability of each unit being active. in our example with two hidden layers, the activation probabilities are given by : p v ( i = 1 \| h ( 1 ) ) = σ w ( 1 ) i, : h ( 1 ), ( 20. 26 ) 664	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	679	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
chapter 20. deep generative models p h ( ( 1 ) i = 1 \| v h, ( 2 ) ) = σ vw ( 1 ) :, i + w ( 2 ) i, : h ( 2 ) ( 20. 27 ) and p h ( ( 2 ) k = 1 \| h ( 1 ) ) = σ h ( 1 ) w ( 2 ) :, k. ( 20. 28 ) the bipartite structure makes gibbs sampling in a deep boltzmann machine [UNK]. the naive approach to gibbs sampling is to update only one variable at a time. rbms allow all of the visible units to be updated in one block and all of the hidden units to be updated in a second block. one might naively assume that a dbm with l layers requires l + 1 updates, with each iteration updating a block consisting of one layer of units. instead, it is possible to update all of the units in only two iterations. gibbs sampling can be divided into two blocks of updates, one including all even layers ( including the visible layer ) and the other including all odd layers. due to the bipartite dbm connection pattern, given the even layers, the distribution over the odd layers is factorial and thus can be sampled	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	680	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
, one including all even layers ( including the visible layer ) and the other including all odd layers. due to the bipartite dbm connection pattern, given the even layers, the distribution over the odd layers is factorial and thus can be sampled simultaneously and independently as a block. likewise, given the odd layers, the even layers can be sampled simultaneously and independently as a block. [UNK] sampling is especially important for training with the stochastic maximum likelihood algorithm. 20. 4. 1 interesting properties deep boltzmann machines have many interesting properties. dbms were developed after dbns. compared to dbns, the posterior distribu - tion p ( h v \| ) is simpler for dbms. somewhat counterintuitively, the simplicity of this posterior distribution allows richer approximations of the posterior. in the case of the dbn, we perform classiﬁcation using a heuristically motivated approximate inference procedure, in which we guess that a reasonable value for the mean ﬁeld expectation of the hidden units can be provided by an upward pass through the network in an mlp that uses sigmoid activation functions and the same weights as the original dbn. distribution any q ( h ) may be used to obtain a variational lower bound on the	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	680	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
units can be provided by an upward pass through the network in an mlp that uses sigmoid activation functions and the same weights as the original dbn. distribution any q ( h ) may be used to obtain a variational lower bound on the log - likelihood. this heuristic procedure therefore allows us to obtain such a bound. however, the bound is not explicitly optimized in any way, so the bound may be far from tight. in particular, the heuristic estimate of q ignores interactions between hidden units within the same layer as well as the top - down feedback inﬂuence of hidden units in deeper layers on hidden units that are closer to the input. because the heuristic mlp - based inference procedure in the dbn is not able to account for these interactions, the resulting q is presumably far 665	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	680	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
chapter 20. deep generative models from optimal. in dbms, all of the hidden units within a layer are conditionally independent given the other layers. this lack of intralayer interaction makes it possible to use ﬁxed point equations to actually optimize the variational lower bound and ﬁnd the true optimal mean ﬁeld expectations ( to within some numerical tolerance ). the use of proper mean ﬁeld allows the approximate inference procedure for dbms to capture the inﬂuence of top - down feedback interactions. this makes dbms interesting from the point of view of neuroscience, because the human brain is known to use many top - down feedback connections. because of this property, dbms have been used as computational models of real neuroscientiﬁc phenomena (, ; series et al. 2010 reichert 2011 et al., ). one unfortunate property of dbms is that sampling from them is relatively [UNK]. dbns only need to use mcmc sampling in their top pair of layers. the other layers are used only at the end of the sampling process, in one [UNK] ancestral sampling pass. to generate a sample from a dbm, it is necessary to use mcmc across all layers, with every layer of the model participating in every markov chain transition	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	681	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
used only at the end of the sampling process, in one [UNK] ancestral sampling pass. to generate a sample from a dbm, it is necessary to use mcmc across all layers, with every layer of the model participating in every markov chain transition. 20. 4. 2 dbm mean field inference the conditional distribution over one dbm layer given the neighboring layers is factorial. in the example of the dbm with two hidden layers, these distributions are p ( v h \| ( 1 ) ), p ( h ( 1 ) \| v h, ( 2 ) ) and p ( h ( 2 ) \| h ( 1 ) ). the distribution over all hidden layers generally does not factorize because of interactions between layers. in the example with two hidden layers, p ( h ( 1 ), h ( 2 ) \| v ) does not factorize due due to the interaction weights w ( 2 ) between h ( 1 ) and h ( 2 ) which render these variables mutually dependent. as was the case with the dbn, we are left to seek out methods to approximate the dbm posterior distribution. however, unlike the dbn, the dbm posterior distribution over their hidden units — while complicated — is easy to approximate with a variational approximation ( as discussed	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	681	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
dbn, we are left to seek out methods to approximate the dbm posterior distribution. however, unlike the dbn, the dbm posterior distribution over their hidden units — while complicated — is easy to approximate with a variational approximation ( as discussed in section ), speciﬁcally a 19. 4 mean ﬁeld approximation. the mean ﬁeld approximation is a simple form of variational inference, where we restrict the approximating distribution to fully factorial distributions. in the context of dbms, the mean ﬁeld equations capture the bidirectional interactions between layers. in this section we derive the iterative approximate inference procedure originally introduced in salakhutdinov and hinton ( ). 2009a in variational approximations to inference, we approach the task of approxi - 666	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	681	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
chapter 20. deep generative models mating a particular target distribution — in our case, the posterior distribution over the hidden units given the visible units — by some reasonably simple family of dis - tributions. in the case of the mean ﬁeld approximation, the approximating family is the set of distributions where the hidden units are conditionally independent. we now develop the mean ﬁeld approach for the example with two hidden layers. let q ( h ( 1 ), h ( 2 ) \| v ) be the approximation of p ( h ( 1 ), h ( 2 ) \| v ). the mean ﬁeld assumption implies that q ( h ( 1 ), h ( 2 ) \| v ) = j q h ( ( 1 ) j \| v ) k q h ( ( 2 ) k \| v ). ( 20. 29 ) the mean ﬁeld approximation attempts to ﬁnd a member of this family of distributions that best ﬁts the true posterior p ( h ( 1 ), h ( 2 ) \| v ). importantly, the inference process must be run again to ﬁnd a [UNK] distribution q every time we use a new value of. v one can conceive of many ways of measuring how well q ( h v \| ) ﬁts p (	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	682	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
). importantly, the inference process must be run again to ﬁnd a [UNK] distribution q every time we use a new value of. v one can conceive of many ways of measuring how well q ( h v \| ) ﬁts p ( h v \| ). the mean ﬁeld approach is to minimize kl ( ) = q p h q ( h ( 1 ), h ( 2 ) \| v ) log q ( h ( 1 ), h ( 2 ) \| v ) p ( h ( 1 ), h ( 2 ) \| v ). ( 20. 30 ) in general, we do not have to provide a parametric form of the approximating distribution beyond enforcing the independence assumptions. the variational approximation procedure is generally able to recover a functional form of the approximate distribution. however, in the case of a mean ﬁeld assumption on binary hidden units ( the case we are developing here ) there is no loss of generality resulting from ﬁxing a parametrization of the model in advance. we parametrize q as a product of bernoulli distributions, that is we associate the probability of each element of h ( 1 ) with a parameter. speciﬁcally, for each j, [UNK] ( 1 ) j = q (	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	682	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
we parametrize q as a product of bernoulli distributions, that is we associate the probability of each element of h ( 1 ) with a parameter. speciﬁcally, for each j, [UNK] ( 1 ) j = q ( h ( 1 ) j = 1 \| v ), where [UNK] ( 1 ) j ∈ [ 0, 1 ] and for each k, [UNK] ( 2 ) k = q ( h ( 2 ) k = 1 \| v ), where [UNK] ( 2 ) k ∈ [ 0 1 ],. thus we have the following approximation to the posterior : q ( h ( 1 ), h ( 2 ) \| v ) = j q h ( ( 1 ) j \| v ) k q h ( ( 2 ) k \| v ) ( 20. 31 ) = j ( [UNK] ( 1 ) j ) h ( 1 ) j ( 1 [UNK] ( 1 ) j ) ( 1−h ( 1 ) j ) × k ( [UNK] ( 2 ) k ) h ( 2 ) k ( 1 [UNK] ( 2 ) k ) ( 1−h ( 2 ) k ). ( 20. 32 ) of course, for dbms with more layers the approximate posterior parametrization can be extended in the obvious way, exploiting the	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	682	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
k ( 1 [UNK] ( 2 ) k ) ( 1−h ( 2 ) k ). ( 20. 32 ) of course, for dbms with more layers the approximate posterior parametrization can be extended in the obvious way, exploiting the bipartite structure of the graph 667	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	682	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
chapter 20. deep generative models to update all of the even layers simultaneously and then to update all of the odd layers simultaneously, following the same schedule as gibbs sampling. now that we have speciﬁed our family of approximating distributions q, it remains to specify a procedure for choosing the member of this family that best ﬁts p. the most straightforward way to do this is to use the mean ﬁeld equations speciﬁed by equation. these equations were derived by solving for where the 19. 56 derivatives of the variational lower bound are zero. they describe in an abstract manner how to optimize the variational lower bound for any model, simply by taking expectations with respect to. q applying these general equations, we obtain the update rules ( again, ignoring bias terms ) : [UNK] ( 1 ) j = σ i viw ( 1 ) i, j + k w ( 2 ) j, [UNK] ( 2 ) k, j [UNK] ( 20. 33 ) [UNK] ( 2 ) k = σ j w ( 2 ) j, [UNK] ( 1 ) j, k. [UNK] ( 20. 34 ) at a ﬁxed point of this system of equations, we have a local maximum of the variational lower bound l ( q ). thus these ﬁxed	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	683	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
( 2 ) j, [UNK] ( 1 ) j, k. [UNK] ( 20. 34 ) at a ﬁxed point of this system of equations, we have a local maximum of the variational lower bound l ( q ). thus these ﬁxed point update equations deﬁne an iterative algorithm where we alternate updates of [UNK] ( 1 ) j ( using equation ) and 20. 33 updates of [UNK] ( 2 ) k ( using equation ). on small problems such as mnist, as few 20. 34 as ten iterations can be [UNK] to ﬁnd an approximate positive phase gradient for learning, and ﬁfty usually [UNK] to obtain a high quality representation of a single speciﬁc example to be used for high - accuracy classiﬁcation. extending approximate variational inference to deeper dbms is straightforward. 20. 4. 3 dbm parameter learning learning in the dbm must confront both the challenge of an intractable partition function, using the techniques from chapter, and the challenge of an intractable 18 posterior distribution, using the techniques from chapter. 19 as described in section, variational inference allows the construction of 20. 4. 2 a distribution q ( h v \| ) that approximates the intractable p ( h v \| ). learning then proceeds	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	683	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
distribution, using the techniques from chapter. 19 as described in section, variational inference allows the construction of 20. 4. 2 a distribution q ( h v \| ) that approximates the intractable p ( h v \| ). learning then proceeds by maximizing l ( v θ, q, ), the variational lower bound on the intractable log - likelihood,. log ( ; ) p v θ 668	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	683	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
chapter 20. deep generative models for a deep boltzmann machine with two hidden layers, is given by l l ( ) = q, θ i j viw ( 1 ) i, [UNK] ( 1 ) j + j k [UNK] ( 1 ) j w ( 2 ) j, [UNK] ( 2 ) k− h log ( ) + z θ ( ) q. ( 20. 35 ) this expression still contains the log partition function, log z ( θ ). because a deep boltzmann machine contains restricted boltzmann machines as components, the hardness results for computing the partition function and sampling that apply to restricted boltzmann machines also apply to deep boltzmann machines. this means that evaluating the probability mass function of a boltzmann machine requires approximate methods such as annealed importance sampling. likewise, training the model requires approximations to the gradient of the log partition function. see chapter for a general description of these methods. dbms are typically trained 18 using stochastic maximum likelihood. many of the other techniques described in chapter are not applicable. techniques such as pseudolikelihood require the 18 ability to evaluate the unnormalized probabilities, rather than merely obtain a variational lower bound on them. contrastive divergence is slow for	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	684	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
other techniques described in chapter are not applicable. techniques such as pseudolikelihood require the 18 ability to evaluate the unnormalized probabilities, rather than merely obtain a variational lower bound on them. contrastive divergence is slow for deep boltzmann machines because they do not allow [UNK] sampling of the hidden units given the visible units — instead, contrastive divergence would require burning in a markov chain every time a new negative phase sample is needed. the non - variational version of stochastic maximum likelihood algorithm was discussed earlier, in section. variational stochastic maximum likelihood as 18. 2 applied to the dbm is given in algorithm. recall that we describe a simpliﬁed 20. 1 varient of the dbm that lacks bias parameters ; including them is trivial. 20. 4. 4 layer - wise pretraining unfortunately, training a dbm using stochastic maximum likelihood ( as described above ) from a random initialization usually results in failure. in some cases, the model fails to learn to represent the distribution adequately. in other cases, the dbm may represent the distribution well, but with no higher likelihood than could be obtained with just an rbm. a dbm with very small weights in all but	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	684	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
, the model fails to learn to represent the distribution adequately. in other cases, the dbm may represent the distribution well, but with no higher likelihood than could be obtained with just an rbm. a dbm with very small weights in all but the ﬁrst layer represents approximately the same distribution as an rbm. various techniques that permit joint training have been developed and are described in section. however, the original and most popular method for 20. 4. 5 overcoming the joint training problem of dbms is greedy layer - wise pretraining. in this method, each layer of the dbm is trained in isolation as an rbm. the ﬁrst layer is trained to model the input data. each subsequent rbm is trained to model samples from the previous rbm ’ s posterior distribution. after all of the 669	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	684	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
chapter 20. deep generative models algorithm 20. 1 the variational stochastic maximum likelihood algorithm for training a dbm with two hidden layers. set, the step size, to a small positive number set k, the number of gibbs steps, high enough to allow a markov chain of p ( v h, ( 1 ), h ( 2 ) ; θ + ∆θ ) to burn in, starting from samples from p ( v h, ( 1 ), h ( 2 ) ; θ ). initialize three matrices, [UNK], [UNK] h ( 1 ) and [UNK] h ( 2 ) each with m rows set to random values ( e. g., from bernoulli distributions, possibly with marginals matched to the model ’ s marginals ). while not converged ( learning loop ) do sample a minibatch of m examples from the training data and arrange them as the rows of a design matrix. v initialize matrices [UNK] h ( 1 ) and [UNK] h ( 2 ), possibly to the model ’ s marginals. while not converged ( mean ﬁeld inference loop ) do [UNK] h ( 1 ) ←σ v w ( 1 ) + [UNK] h ( 2 ) w ( 2 ). [UNK] h ( 2 ) ←σ [UNK] h	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	685	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
model ’ s marginals. while not converged ( mean ﬁeld inference loop ) do [UNK] h ( 1 ) ←σ v w ( 1 ) + [UNK] h ( 2 ) w ( 2 ). [UNK] h ( 2 ) ←σ [UNK] h ( 1 ) w ( 2 ). end while ∆w ( 1 ) ←1 mv [UNK] h ( 1 ) ∆w ( 2 ) ←1 m [UNK] h ( 1 ) [UNK] ( 2 ) for do l k = 1 to ( gibbs sampling ) gibbs block 1 : [UNK], j, [UNK] i, j sampled from p ( [UNK], j = 1 ) = σ w ( 1 ) j, : [UNK] ( 1 ) i, :. [UNK], j, [UNK] ( 2 ) i, j sampled from p ( [UNK] ( 2 ) i, j = 1 ) = σ [UNK] h ( 1 ) i, : w ( 2 ) :, j. gibbs block 2 : [UNK], j, [UNK] ( 1 ) i, j sampled from p ( [UNK] ( 1 ) i, j = 1 ) = σ [UNK], : w ( 1 ) :, j + [UNK] h ( 2 ) i, : w ( 2 ) j, :. end for ∆w ( 1 ) ←∆w ( 1 ) −1	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	685	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
i, j = 1 ) = σ [UNK], : w ( 1 ) :, j + [UNK] h ( 2 ) i, : w ( 2 ) j, :. end for ∆w ( 1 ) ←∆w ( 1 ) −1 mv [UNK] h ( 1 ) ∆w ( 2 ) ←∆w ( 2 ) −1 m [UNK] h ( 1 ) [UNK] h ( 2 ) w ( 1 ) ←w ( 1 ) + ∆w ( 1 ) ( this is a cartoon illustration, in practice use a more [UNK] algorithm, such as momentum with a decaying learning rate ) w ( 2 ) ←w ( 2 ) + ∆ w ( 2 ) end while 670	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	685	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
chapter 20. deep generative models rbms have been trained in this way, they can be combined to form a dbm. the dbm may then be trained with pcd. typically pcd training will make only a small change in the model ’ s parameters and its performance as measured by the log - likelihood it assigns to the data, or its ability to classify inputs. see ﬁgure 20. 4 for an illustration of the training procedure. this greedy layer - wise training procedure is not just coordinate ascent. it bears some passing resemblance to coordinate ascent because we optimize one subset of the parameters at each step. the two methods [UNK] because the greedy layer - wise training procedure uses a [UNK] objective function at each step. greedy layer - wise pretraining of a dbm [UNK] from greedy layer - wise pre - training of a dbn. the parameters of each individual rbm may be copied to the corresponding dbn directly. in the case of the dbm, the rbm parameters must be modiﬁed before inclusion in the dbm. a layer in the middle of the stack of rbms is trained with only bottom - up input, but after the stack is combined to form the dbm, the layer will have both bottom - up and top	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	686	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
##ed before inclusion in the dbm. a layer in the middle of the stack of rbms is trained with only bottom - up input, but after the stack is combined to form the dbm, the layer will have both bottom - up and top - down input. to account for this [UNK], salakhutdinov and hinton 2009a ( ) advocate dividing the weights of all but the top and bottom rbm in half before inserting them into the dbm. additionally, the bottom rbm must be trained using two “ copies ” of each visible unit and the weights tied to be equal between the two copies. this means that the weights are [UNK] doubled during the upward pass. similarly, the top rbm should be trained with two copies of the topmost layer. obtaining the state of the art results with the deep boltzmann machine requires a modiﬁcation of the standard sml algorithm, which is to use a small amount of mean ﬁeld during the negative phase of the joint pcd training step ( salakhutdinov and hinton 2009a, ). speciﬁcally, the expectation of the energy gradient should be computed with respect to the mean ﬁeld distribution in which all of the units are independent from each other. the	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	686	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
( salakhutdinov and hinton 2009a, ). speciﬁcally, the expectation of the energy gradient should be computed with respect to the mean ﬁeld distribution in which all of the units are independent from each other. the parameters of this mean ﬁeld distribution should be obtained by running the mean ﬁeld ﬁxed point equations for just one step. see ( ) for a comparison of the performance of centered goodfellow et al. 2013b dbms with and without the use of partial mean ﬁeld in the negative phase. 20. 4. 5 jointly training deep boltzmann machines classic dbms require greedy unsupervised pretraining, and to perform classiﬁcation well, require a separate mlp - based classiﬁer on top of the hidden features they extract. this has some undesirable properties. it is hard to track performance during training because we cannot evaluate properties of the full dbm while training the ﬁrst rbm. thus, it is hard to tell how well our hyperparameters 671	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	686	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
chapter 20. deep generative models d ) a ) b ) c ) figure 20. 4 : the deep boltzmann machine training procedure used to classify the mnist dataset ( salakhutdinov and hinton 2009a srivastava 2014, ; et al., ). train an rbm ( a ) by using cd to approximately maximize log p ( v ). train a second rbm that models ( b ) h ( 1 ) and target class y by using cd - k to approximately maximize log p ( h ( 1 ), y ) where h ( 1 ) is drawn from the ﬁrst rbm ’ s posterior conditioned on the data. increasek from 1 to 20 during learning. combine the two rbms into a dbm. train it to approximately ( c ) maximize log p ( v, y ) using stochastic maximum likelihood with k = 5. delete ( d ) y from the model. deﬁne a new set of features h ( 1 ) and h ( 2 ) that are obtained by running mean ﬁeld inference in the model lacking y. use these features as input to an mlp whose structure is the same as an additional pass of mean ﬁeld, with an additional output layer for the estimate of y	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	687	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
2 ) that are obtained by running mean ﬁeld inference in the model lacking y. use these features as input to an mlp whose structure is the same as an additional pass of mean ﬁeld, with an additional output layer for the estimate of y. initialize the mlp ’ s weights to be the same as the dbm ’ s weights. train the mlp to approximately maximize log p ( y \| v ) using stochastic gradient descent and dropout. figure reprinted from (, ). goodfellow et al. 2013b 672	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	687	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
chapter 20. deep generative models are working until quite late in the training process. software implementations of dbms need to have many [UNK] components for cd training of individual rbms, pcd training of the full dbm, and training based on back - propagation through the mlp. finally, the mlp on top of the boltzmann machine loses many of the advantages of the boltzmann machine probabilistic model, such as being able to perform inference when some input values are missing. there are two main ways to resolve the joint training problem of the deep boltzmann machine. the ﬁrst is the centered deep boltzmann machine ( montavon and muller 2012, ), which reparametrizes the model in order to make the hessian of the cost function better - conditioned at the beginning of the learning process. this yields a model that can be trained without a greedy layer - wise pretraining stage. the resulting model obtains excellent test set log - likelihood and produces high quality samples. unfortunately, it remains unable to compete with appropriately regularized mlps as a classiﬁer. the second way to jointly train a deep boltzmann machine is to use a multi - prediction deep boltzmann machine ( goodfellow 2013	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	688	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)
. unfortunately, it remains unable to compete with appropriately regularized mlps as a classiﬁer. the second way to jointly train a deep boltzmann machine is to use a multi - prediction deep boltzmann machine ( goodfellow 2013b et al., ). this model uses an alternative training criterion that allows the use of the back - propagation algorithm in order to avoid the problems with mcmc estimates of the gradient. unfortunately, the new criterion does not lead to good likelihood or samples, but, compared to the mcmc approach, it does lead to superior classiﬁcation performance and ability to reason well about missing inputs. the centering trick for the boltzmann machine is easiest to describe if we return to the general view of a boltzmann machine as consisting of a set of units x with a weight matrix u and biases b. recall from equation that he energy 20. 2 function is given by e ( ) = x −xux b − x. ( 20. 36 ) using [UNK] sparsity patterns in the weight matrix u, we can implement structures of boltzmann machines, such as rbms, or dbms with [UNK] numbers of layers. this is accomplished by partitioning x into visible and hidden units and zeroing	/home/ricoiban/GEMMA/mnlp_chatsplaining/RAG/Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org).pdf	688	Deep Learning by Ian Goodfellow, Yoshua Bengio, Aaron Courville (z-lib.org)