CN108984726A - A method of the sLDA model based on extension carries out title annotation to image - Google Patents
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Abstract
The method that the present invention relates to a kind of sLDA model based on extension carries out title annotation to image, it is to solve existing annotation of images method and can encounter scalability issues, a small annotation vocabulary can only be handled, the shortcomings that lacking universal and ease for use and propose, it include: the image for input, the local feature of image is extracted, and obtains N number of visual vocabulary of image using K-means algorithm；The Posterior distrbutionp of given document hidden variable is indicated using LDA model；It introduces response variable and response variable distribution is defined as multivariable Bernoulli Jacob distribution；It is approximate that formula is carried out using the LDA reasoning variational algorithm based on convexity；Seek variational parameter value；Estimate model parameter；The distribution of predicated response variable.The present invention is suitable for image header annotation system.
Description
Technical field
The present invention relates to annotation of images method fields, and in particular to a kind of sLDA model based on extension marks image
The method that note is released.
Background technique
In in the past few decades, the problem of image and video frequency searching, is constantly in the forward position of computer vision research.To the greatest extent
For pipe in this way, since nearest largely picture and video can be found on the net, people are receiving a kind of efficient algorithm on a large scale
Search and the demand of navigation is concentrated also constantly to increase.Current state-of-the-art image search engine is depended critically upon using band annotation
Text or title identify and retrieve image.Although this method allows to carry out high-level semantic query, for being based on text
Search technique the vital heading message of success, usually obtain manually, and this process cannot be with current more
The ever-increasing scale of media corpus and extend.Therefore, it is necessary to automate this annotation procedure.Since it is to being related to digital matchmaker
The potential impact of the extensive application program of body archives, people are to the automatic chemical industry for designing and developing annotating images and video in recent years
The attention rate of tool is growing day by day.
In the case where no title, annotate algorithm task be by study image and text between association mode come
Predict the title of missing.The work in this field can be roughly divided into two groups in the past.In first group of work, annotation of images
Problem is converted into a supervised learning problem, and in this problem, annotation will be taken as concept class.For every in vocabulary
A word, class sigma-t be from learn in markd image.In annotation procedure, the posteriority point of class label is calculated
Then cloth uses the concept of maximum probability as the title of prediction.In practice, this method can encounter scalability issues,
And a small annotation vocabulary can only be handled, because each word must be learned by class sigma-t.
Another group, by modeling to the joint statistic correlation between two data types, is handled on the basis of more equality
Annotation and image data.These models use a potential variable frame, by assuming that each document have one group it is hiding
The factor controls the association between characteristics of image and corresponding heading, to understand the joint probability point of text and characteristics of image
Cloth.
Summary of the invention
The purpose of the present invention is to solve existing annotation of images methods can encounter scalability issues, can only handle one
A small annotation vocabulary, the shortcomings that lacking universal and ease for use, and propose a kind of sLDA model based on extension to image
The method for carrying out title annotation is capable of handling the multidimensional binary response variable of annotation data, comprising:
Step 1: extracting the local feature of image, and obtain the N of image using K-means algorithm for the image of input
A visual vocabulary wn, wherein wn∈{1,2...,N}。
Step 2: indicating the Posterior distrbutionp of given document hidden variable using LDA model.
Wherein α and β is model parameter, and z and θ are theme variable and theme ratio respectively.
Step 3: introducing the parameter η and δ of response variable y and response variable in step 2, while will and response be become
Amount distribution is defined as multivariable Bernoulli Jacob distribution, i.e., indicates formula (3) are as follows:
Step 4: according to the LDA reasoning variational algorithm based on convexity by formula (5) similar toWherein Di Li Cray parameter γ and multiple parameters (φ1,φ2,...,φN) it is freedom
Variational parameter；znFor n-th of descriptor；The desired difference of logp (θ, z, w | α, β, η, δ) and q (θ, z | γ, φ) is denoted as
L。
Step 5: seeking the variational parameter γ and φ that the lower bound of L can be made to reach maximum value.
Step 6: estimation model parameter ψ={ α, β, η, δ }.
Step 7: according to the distribution p of model parameter ψ and variational parameter γ and φ predicated response variable y (y | w).
Further, step 3 specifically:
It utilizesη and δ generates response variable y, whereinIf the distribution of response variable y meets generalized linear
Model:
WhereinThen formula (3) can be expressed as
Wherein
Further, step 4 specifically:
It is approximately by formula (5) by following formula
Enable L (γ, φ；α, β) expression (8) the right, formula (8) is expressed as
Logp (w | α, β)=L (γ, φ；α,β)+D(q(θ,z|γ,φ)||p(θ,z|w,α,β)) (9)
L is write by formula (10) by using the factorization of p and q:
Further, step 5 specifically:
Step 5 one, in formula (13), utilize φniMaximize the lower bound of L, φniIndicate n-th of visual vocabulary by hidden
The probability that theme i is generated is hidden, thereforeAnd φ is included by separationniItem and add Lagrange multiplier appropriate
To form Lagrangian:
ψ (x) is double gamma functions；
It calculates about φniDerivative:
Wherein βivIt indicates for suitable v'sV is v-th of word of dictionary；
It further obtains in the case where response variable obeys Bernoulli Jacob's distribution occasion, parameter phinMore new formula
Step 5 two utilizes γiMaximize above formula, γiIndicate i-th of component part of posteriority Di Li Cray parameter；Include
γiItem:
To γiDerivation:
Enabling derivative is zero:
Iterative equation (16) to (19) is restrained until boundary, and then obtains the variation that the lower bound of L can be made to reach maximum value
Parameter γ and φ.
Further, step 6 specifically:
Step 6 one, the formula for acquiring parameter beta are as follows:
Step 6 two, the process for acquiring parameter alpha are as follows: for formula (22),
Derivation is carried out to obtain
The value of α is sought by Newton iteration method to formula (23)；
Step 6 three acquires parameter η and σ2Process are as follows:
Wherein μ ()=EGLM[Y|·]；
To σ2Derivation,Upper assessment
By calculating, parameter estimation result is finally obtained:
By parameter alphai、βij、ηiAnd δiIt is combined and model parameter ψ={ α, β, η, δ } can be obtained.
Further, step 7 specifically:
To not have headed new document w as input, task is to be inferred to most probable heading, utilizes φnWith q (θ)
Come approximate solution conditional probability p (y | w), as follows:
WhereinP (y | w) for inferring the new most probable heading of document w.
The invention has the benefit that
1, the present invention is made that adjustment to the structure of Corr-LDA, has deleted variable x, image subject is used directly for pre-
Mark topic, is integrated (and this is the step of Corr-LDA needs) without the posterior probability to title.And to sLDA into
Row extension, so that model is capable of handling multivariable binary responses variable, a response variable can only be handled by eliminating sLDA
Deficiency, for image annotation in further detail, therefore image retrieval is also more convenient and accurate.
2, in the case where number of topics, vocabulary number are more, predictablity rate of the invention is apparently higher than Corr-LDA model,
Averagely it is higher by 0.04.
Detailed description of the invention
Fig. 1 is the graphical model structure chart of sLDA-bin；
Fig. 2 is the graphical model structure chart of Corr-LDA；
Fig. 3 is the error curve diagram between the response of prediction and the observation of sLDA-bin and LDA；
The heading prediction graph of Corr-LDA and sLDA-bin when Fig. 4 is K=30；
The heading prediction graph of Corr-LDA and sLDA-bin when Fig. 5 is N=256；
The heading prediction graph of Corr-LDA and sLDA-bin when Fig. 6 is N=512；
Fig. 7 is the accuracy rate curve graph that Corr-LDA and sLDA-bin annotates partial objects.
Specific embodiment
The method that sLDA model based on extension of the invention carries out title annotation to image, may be simply referred to as sLDA-bin,
Include:
Step 1: extracting the local feature of image, and obtain the N of image using K-means algorithm for the image of input
A visual vocabulary wn, wherein wn∈{1,2...,N}。
Step 2: indicating the Posterior distrbutionp of given document hidden variable using LDA model:
Wherein α and β is model parameter, and z and θ are theme variable and theme ratio respectively.
Step 3: introducing the parameter η and δ of response variable y and response variable in step 2, while will and response be become
Amount distribution is defined as multivariable Bernoulli Jacob distribution, i.e., indicates formula (3) are as follows:
Step 4: according to the LDA reasoning variational algorithm based on convexity by formula (5) similar toWherein Di Li Cray parameter γ and multiple parameters (φ1,φ2,...,φN) it is freedom
Variational parameter；znFor the theme variable of n-th of word；By the desired difference of logp (θ, z, w | α, β, η, δ) and q (θ, z | γ, φ)
Value is denoted as L, and determines the representation of L lower bound.
Step 5: seeking the variational parameter γ and φ that the lower bound of L can be made to reach maximum value.
Step 6: estimation model parameter ψ={ α, β, η, δ }.
Step 7: according to the distribution p of model parameter ψ and variational parameter γ and φ predicated response variable y (y | w).
The principle and process of present embodiment is specifically described below, it should be noted that footmark is attached to the variable of n and not attached
Variable meaning with n is identical, and difference is that footmark is attached to the variable of n and emphasizes ordinal number n, that is, emphasizes that this variable is N number of word
In n-th of corresponding parameter.For example, variable z and znIt is identical meaning, difference is znIt is emphasised that the theme of n-th of word
Variable, z do not emphasize this point, and they be meant that it is identical.
Step 1: data indicates, N number of local feature of image is extracted, then N number of feature is clustered with K-means,
Known k initial average point m1,...,mk, alternately according to following two steps:
Distribution: each observation point is assigned in cluster, so that quadratic sum reaches minimum in organizing.Because this quadratic sum is just
Euclidean distance after being square, so very intuitively observation is assigned to from its nearest average point
Wherein each xpAll only it is assigned to a determining clusterIn, although theoretically it may be assigned to 2
A or more clusters.
It updates: for each cluster obtained in the previous step, with the center of fiqure of observation in cluster, as new average point
It has converted images into after the document being made of visual vocabulary, has carried out document annotation using LDA model.LDA is
A kind of typical bag of words, i.e. a, it considers that document is a set being made of one group of word, without suitable between word and word
Sequence and successive relationship.One document may include multiple themes, each word is given birth to by one of theme in document
At.
Step 2: the mode of a document structure tree is as follows in LDA model:
Sampling generates the theme distribution θ of document i from Di Li Cray distribution αi, from the multinomial distribution θ of themeiMiddle sampling
Generate the theme z of j-th of word of document ii,j, sampling generates theme z from Di Li Cray distribution βi,jWord distributionFrom word
The multinomial distribution of languageMiddle sampling generates vocabulary wi,j
In order to use LDA, the crucial reasoning problems for needing to solve are the posteriority point for giving a document calculations hidden variable
Cloth:
Step 3: utilizingη and δ generates response variable y, whereinThis variable, which can be, gives film
Star quantity downloads the number of online article or the classification of document.We combine modeling document and response, most can be pre- to find
Survey the potential theme of the response variable of the following unmarked document.
SLDA adapts to various types of responses using probability mechanism identical with generalized linear model: unrestricted reality
Actual value, the restrained true value (such as fault time) being positive, orderly or unordered class label, nonnegative integer (such as count number
According to) and other types.The distribution of response variable is a generalized linear model (GLM):
GLM frame improves flexibility, as long as the distribution of response variable can be write as form above, so that it may model.
The corresponding different h (y, δ) of different distribution and
Then (3) formula becomes
Wherein
But combination of the θ and β on potential theme is difficult to directly calculate, using the LDA reasoning variational algorithm based on convexity.
In order to handle the multivariable binary responses variable of annotation data, response variable can be distributed and be modeled as multivariable
Bernoulli Jacob is distributed, and its probability is defined with logical connection function.Wherein yi∈{0,
1 },It is logical function,{ai,μiBe vocabulary i regression coefficient.
Step 4: variation calculates.The basic thought of variation reasoning based on convexity is to utilize the acquisition pair of Jensen inequality
The adjustable lower bound of number likelihood.This race is characterized as being following variation distribution:
Wherein Di Li Cray parameter γ and multiple parameters (φ1,φ2,...,φN) it is all free variational parameter.
It is one optimization problem of setting in next step after specifying simplified family of probability distribution, which determines that variation is joined
The value of number γ and φ.The requirement for finding the most compact lower bound of log-likelihood is converted into following optimization problem
By minimizing the opposite entropy minimization between variation distribution and true Posterior distrbutionp p (θ, z | w, α, β), completion is above-mentioned
Target.
Carry out the log-likelihood of restricted document by using Jensen inequality first.For the sake of simplicity, parameter γ is omitted
And φ, have
Jensen inequality is that any variation is distributed the log-likelihood of q (θ, z | γ, φ) and provides a lower limit.Above formula
Left-right difference, that is, variation distribution and the really relative entropy between Posterior distrbutionp.The right first item is the joint to hiding and observation variable
The expectation of the logarithm of probability；Section 2 is the entropy of variation distribution, H (q)=- Eq[logq(θ,z)].Enable L (γ, φ；α, β) it indicates
The right of above formula, above formula become
Logp (w | α, β)=L (γ, φ；α,β)+D(q(θ,z|γ,φ)||p(θ,z|w,α,β)) (9)
This shows to maximize lower bound L (γ, φ by γ and φ；α, β) it is equivalent to minimize variation posterior probability and true
Relative entropy between posterior probability.
Lower bound is extended by using the factorization of p and q:
Then above formula is launched into the equation about model parameter (α, β) and variational parameter (γ, φ).First three items are
Variation distribution entropy be
Note that the variation objective function Section 4 of sLDA is different from LDA, that is, give the response variable of potential theme distribution
It is expected that log probability.
It can be seen that it is related with two expectations to calculate lower bound.First is contemplated to beSecond is contemplated to beIt can directly be calculated in some models, but usually require approximation.Specific mode wants the class of combining response
Type.
Here logical function it is expected that the calculating of E [logp (y | z, η, δ)] becomes complicated.Using convex duality and incite somebody to action
Logical function is expressed asChi square function a point supremum.More specifically,
Wherein,
Step 5: maximizing L lower bound by variational parameter γ and φ；
Step 5 one utilizes φniIt maximizes, φniIndicate the probability that n-th of visual vocabulary is generated by hiding theme i, thereforeIt include φ by separationniItem and add Lagrange multiplier appropriate to form Lagrangian:
For simplicity, the parameter of L is abandoned, and subscript φniIndicate that we those of only remain in L item, this is
φniFunction.It obtains about φniDerivative:
Wherein βivIt indicates for suitable v'sV is v-th of word of dictionary.As can be seen that for φni
Update rely onWhen some, such as Gauss or Poisson distribution, we can obtain accurate result,
He when, need the optimization method based on gradient.
It calculates in the case where response variable obeys Bernoulli Jacob's distribution occasion, parameter phinMore new formula
Step 5 two utilizes γiMaximize above formula, γiIndicate i-th of component part of posteriority Di Li Cray parameter.Include
γiItem:
To γiDerivation,
Enabling derivative is zero,
Since equation (19) depend on variation multinomial φ, complete variation reasoning needs between equation (16) and (19)
Alternately.Until boundary is restrained.
Step 6: parameter Estimation.
Step 6 one realizes that lower bound maximizes using condition multiple parameters β.It extracts the item containing β and Lagrange is added
Multiplier.
To β derivation and to enable derivative be zero, is obtained
Finally acquire
Step 6 two realizes that lower bound maximizes using Di Li Cray parameter alpha.Item containing α is as follows
To αiDerivation obtains
Derivative relies on αj, wherein i ≠ j, therefore the method that we must use iteration find qualified α.Particularly,
Its Hessian matrix can write H=diag (h)+1z1ΤForm,
Therefore linear session Newton iterative can be called.Then the update of α can write
αnew=αold-H(αold)-1g(αold) (25)
Wherein H (α) and g (α) is the Hessian matrix and gradient of point α respectively.
Multiplied by gradient, i-th of component is obtained:
Wherein
Step 6 three estimates GLM parameter.GLM parameter is η and σ2。
Wherein μ ()=EGLM[Y|·]。
To σ2Derivation,Upper assessment:
It can be with calculation equation (29), because exactly or approximately being had evaluated while being optimized to coefficient η
The summation of rightmost.According to h (y, δ) and its to the partial derivative of δ, we are availableIt can be closing form, can also
To be one dimensional numerical optimization.By calculating, the parameter estimation result of model is finally obtained:
Step 7: being predicted.Given one does not have headed new document w, and task is to be inferred to most probable title, i.e.,
Response variable y.In order to complete this task, we calculate using variation until convergence on document, and utilize φnCome with q (θ)
Approximate solution conditional probability p (y | w), it is as follows:
WhereinBy retaining title, we release φ from test imagen.And we compare
The performance of sLDA-bin and Corr-LDA.For the title quality of measurement model, we calculate title prediction as defined below
Probability
The protrusion effect of the present invention compared with the prior art can be specifically described according to Fig. 1 to Fig. 7:
Fig. 1 shows the graphical model structure of sLDA-bin, due to sLDA model heading is predicted it is outstanding
Performance, we use the structure of sLDA, it can be seen that sLDA-bin is made that adjustment to the structure of Corr-LDA, deletes
Variable, image subject are used directly for prediction title, are integrated (Corr-LDA need without the posterior probability to title
The step of wanting).And sLDA is extended, so that model is capable of handling multivariable binary responses variable, eliminate sLDA only
The deficiency that a response variable can be handled, for image annotation in further detail, therefore image retrieval is also more convenient and accurate.
The structure for the Corr-LDA that Fig. 2 is shown, Corr-LDA is a kind of currently popular LDA model.Not with SLDA-bin
Together, Corr-LDA limits each heading associated with a specific image region.It can be seen that each heading is logical
It crosses and selects an image-region come what is generated first, be embodied in the x variable additionally having more in Fig. 2.
Estimated performance analysis under different themes number:
Fig. 3 illustrates the comparison of sLDA-bin model and unsupervised LDA model in prediction film review.We use five
Roll over the quality of cross validation assessment prediction.We measure the relevance between prediction and response variable.Relevance is stronger, illustrates to predict
Closer with response variable observation, forecast quality is higher.Figure it is seen that when number of topics is lower, unsupervised LDA's
Forecast quality is higher than sLDA-bin, but increasing along with number of topics, the forecast quality of sLDA-bin can be promoted rapidly, surpasses
More unsupervised LDA model.This is because sLDA-bin uses the structure of sLDA, the theme variable and response variable of model are straight
Connect relevant, therefore number of topics increases, and facilitates us and obtains the more information about response variable, for correctly predicted response
Variable has valuable help
Fig. 5 and Fig. 6 is respectively shown under vocabulary number N=256 and 512, the shadow of the prediction probabilities of two kinds of models by number of topics K
It rings.Five groups of data between we have randomly selected from K=5 to K=60 are compared.First it can be seen that, either N=256
Or N=512, the prediction curve of sLDA-bin are consistently higher than Corr-LDA, and good predictive ability is attributed to the fact that image by us
Direct correlation model between theme and corresponding title.By eliminating hidden variable, image subject is used directly for predicting
Title is integrated (step required for Corr-LDA) without the posterior probability to header topics.It is fixed in vocabulary number
In the case where, sLDA-bin is more sensitive to number of topics, and with the growth of number of topics, sLDA-bin prediction effect, which has, obviously to be mentioned
It rises, but if number of topics is excessive, will lead to prediction decline.This is because the response variable and theme of sLDA-bin have direct pass
Connection, excessive number of topics will lead to over-fitting, and prediction effect is bad instead.So sLDA-bin performance, which plays, depends on number of topics
With the cooperation of vocabulary number.
Estimated performance analysis under different vocabulary numbers:
As shown in Figure 4, fixed number of topics k=30, when vocabulary number N is respectively 128,256 and 512, sLDA-bin is
Give the prediction effect better than Corr-LDA model.Then we conclude that, as we increase view in visual dictionary
Feel vocabulary quantity, we also obtain better prediction probability, this point comparison diagram 5 and Fig. 6 it is also seen that.Because of vocabulary
Several increases, model can obtain the detailed information of more images, thus preferably know the content information of image, it is more preferable to carry out
The matching of image-title.
Image classification performance evaluation:
Fig. 7 shows that sLDA-bin model and Corr-LDA model, can be with to the annotation accuracy rate of part common objects
Find out, in each single item, the annotation accuracy rate of sLDA-bin is all higher than Corr-LDA.On the whole, the accuracy rate of sLDA-bin
Averagely exceed Corr-LDA about 0.04.Because Corr-LDA limits each heading associated with a specific image region.
In fact, some annotation words describe entire scene on the whole, it the use of this restrictive association model is to be grossly inaccurate.It is logical
Cross and recurrence operation carried out to theme ratio, sLDA-bin allow each heading by from all image-regions theme and spy
Determine the influence of image-region, this depends on corresponding regression coefficient.Therefore, correlation model of the invention is more general, accurately
How reflection generates the process really annotated.
The present invention can also have other various embodiments, without deviating from the spirit and substance of the present invention, this field
Technical staff makes various corresponding changes and modifications in accordance with the present invention, but these corresponding changes and modifications all should belong to
The protection scope of the appended claims of the present invention.
Claims (6)
1. a kind of method that sLDA model based on extension carries out title annotation to image characterized by comprising
Step 1: extracting the local feature of image, and obtain N number of view of image using K-means algorithm for the image of input
Feel vocabulary wn, wherein n ∈ { 1,2..., N }, N are positive integer；
Step 2: indicating the Posterior distrbutionp of given document hidden variable using LDA model:
Wherein α and β is model parameter, and z and θ are theme variable and theme ratio respectively；
Step 3: introducing the parameter η and δ of response variable y and response variable in step 2, while will and response variable be divided
Cloth is defined as multivariable Bernoulli Jacob distribution, i.e., indicates formula (3) are as follows:
Step 4: according to the LDA reasoning variational algorithm based on convexity by formula (5) similar toWherein Di Li Cray parameter γ and multiple parameters (φ1,φ2,...,φN) it is freedom
Variational parameter；znFor the theme variable of n-th of word；By the desired difference of logp (θ, z, w | α, β, η, δ) and q (θ, z | γ, φ)
Value is denoted as L；
Step 5: seeking the variational parameter γ and φ that the lower bound of L can be made to reach maximum value；
Step 6: estimation model parameter ψ={ α, β, η, δ }；
Step 7: according to the distribution p of model parameter ψ and variational parameter γ and φ predicated response variable y (y | w).
2. the method that the sLDA model based on extension carries out title annotation to image according to claim 1, it is characterised in that:
Step 3 specifically:
It utilizesη and δ generates response variable y, whereinIf the distribution of response variable y meets generalized linear model:
WhereinThen formula (3) can be expressed as
Wherein
3. the method that the sLDA model based on extension carries out title annotation to image according to claim 2, it is characterised in that:
Step 4 specifically:
It is approximately by formula (5) by following formula
Enable L (γ, φ；α, β) expression (8) the right, formula (8) is expressed as
Logp (w | α, β)=L (γ, φ；α,β)+D(q(θ,z|γ,φ)||p(θ,z|w,α,β)) (9)
L is write by formula (10) by using the factorization of p and q:
4. the method that the sLDA model based on extension carries out title annotation to image according to claim 3, it is characterised in that:
Step 5 specifically:
Step 5 one utilizes φniMaximize the lower bound of L, φniIndicate the probability that n-th of visual vocabulary is generated by hiding theme i,
ThereforeAnd φ is included by separationniItem and add Lagrange multiplier appropriate to form Lagrangian:
ψ (x) is double gamma functions；
It calculates about φniDerivative:
Wherein βivIt indicates for suitable v'sV is v-th of word of dictionary；
It further obtains in the case where response variable obeys Bernoulli Jacob's distribution occasion, parameter phinMore new formula
Step 5 two utilizes γiMaximize the lower bound of L, γiIndicate i-th of component part of posteriority Di Li Cray parameter；Include
γiItem:
To γiDerivation:
Enabling derivative is zero:
Iterative equation (16) to (19) is restrained until boundary, and then obtains the variational parameter that the lower bound of L can be made to reach maximum value
γ and φ.
5. the method that the sLDA model based on extension carries out title annotation to image according to claim 4, it is characterised in that:
Step 6 specifically:
Step 6 one, the formula for acquiring parameter beta are as follows:
Step 6 two, the process for acquiring parameter alpha are as follows: for formula (22),
Derivation is carried out to obtain
The value of α is sought by Newton iteration method to formula (23)；Wherein M indicates the number of documents of training set；Footmark d indicates d
Piece document；
Step 6 three acquires parameter η and σ2Process are as follows:
Wherein μ ()=EGLM[Y|·]；
To σ2Derivation,Upper assessment
By calculating, parameter estimation result is finally obtained:
By parameter alphai、βij、ηiAnd δiIt is combined and model parameter ψ={ α, β, η, δ } can be obtained.
6. the method that the sLDA model based on extension carries out title annotation to image according to claim 5, it is characterised in that:
Step 7 specifically:
To not have headed new document w as input, and utilize φnCome approximate solution conditional probability p (y | w) with q (θ), as follows:
WhereinP (y | w) for inferring the new most probable heading of document w.
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