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SEO Keyword summary for en.wikipedia.org/wiki/expectation%e2%80%93maximization_algorithm
Keywords are extracted from the main content of your website and are the primary indicator of the words this page could rank for. By frequenty count we expect your focus keyword to be mathbf
Focus keyword
Short and long tail
Short Tail Keywords mathbf theta boldsymbol |
long Tail Keywords (2 words) boldsymbol theta em algorithm mid boldsymbol mathbf z maximum likelihood |
long Tail Keywords (3 words) mid boldsymbol theta zdisplaystyle mathbf z displaystyle boldsymbol theta theta mid boldsymbol pmathbf x mid x mid boldsymbol boldsymbol theta t |
en.wikipedia.org On-Page SEO Scan
Descriptive Elements
The <head> element of a en.wikipedia.org/wiki/expectation%e2%80%93maximization_algorithm page is used to inform the browser and visitors of the page about the general meta information. The head section of the page is where we place the page title, the definition of the HTML version used, the language of in which the page is written. In the head section we can also include JavaScript and CSS (markup) files for the page.
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expectationndashmaximization algorithm wikipedia
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wikipedia free encyclopedia displaystyle mathbf boldsymbol theta lboldsymbol pmathbf mid int dmathbf qboldsymbol toperatorname sim pcdot tleftlog right underset operatorname argmax log beginalignedlog sum tlog thboldsymbol tendaligned hboldsymbol tqboldsymbol tmid tgeq fqtheta qlog ltheta xzhq dmathrm big qparallel pzmid xcdot xtheta dkl qtoperatorname fqttheta widehat sigma frac nsum nzkwidehat nwidehat fwidehat ffrac ldots ximid mathcal ndboldsymbol tau prod fmathbf iboldsymbol jsigma nprod jtau jmathbb zij exp leftsum mathbb zijbig jtfrac jtop tjitoperatorname zijmid ximathbf itheta tfrac jtsigma jttau tsigma ttau beginalignedqtheta nltheta izioperatorname tsum nlog izisum noperatorname zimid xiximathbf pzijmid jmathbf ijsum tjitbig endaligned izi pzimid tji qtheta beginalignedboldsymbol tunderset leftleftsum itrightlog rightendaligned ntjitsum itt itfrac ntjit itlefttfrac top itmathbf isum ezmid tmathbf leq varepsilon wikimedia foundation powered mediawiki
Mobile SEO en.wikipedia.org/wiki/expectation%e2%80%93maximization_algorithm
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Domain Level SEO
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16 characters long
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algorithm found in path !
expectation found in path !
maximization found in path !
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statistics
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en.m.wikipedia.org |
en.wikipedia.org wikimedia foundation inc
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w httpsenwikipediaorgwindexphptitleexpectationmaximizationalgorithmoldid1244458420
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wiki machine learning
data mining
supervised learning
unsupervised learning
semisupervised learning
selfsupervised learning
reinforcement learning
metalearning
online learning
batch learning
curriculum learning
rulebased learning
neurosymbolic ai
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quantum machine learning
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temporal difference td
multiagent
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humanintheloop
coefficient of determination
confusion matrix
learning curve
roc curve
kernel machines
biasvariance tradeoff
computational learning theory
empirical risk minimization
occam learning
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statistical learning
vc theory
ecml pkdd
glossary of artificial intelligence
list of datasets for machinelearning research
list of datasets in computer vision and image processing
outline of machine learning
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iterative method
maximum likelihood
maximum a posteriori
parameters
statistical model
latent variables
likelihood function
normal distribution
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geyser
elki
a p dempster
laird nm
rubin donald b
cedric smith
ho hartley
per martinlf
anders martinlf
c f jeff wu
exponential family
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ordered subset expectation maximization
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g a barnard
g rasch
annals of statistics
hinton geoffrey
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hastie trevor
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SEO Advice for en.wikipedia.org
In this section we provide pointers on how you can to optimize your web page so it can be found more easily by search engines and how to make it rank higher by optimizing the content of the page itself. For each of the individual criteria the maximum score is 100%. A score below 70% is considered to be indication that the page is not complying with general SEO standards and should be evaluated and/or fixed. Not every factor is weighted the same and some are not as important as others. Relatively unimportant factors like meta keywords are not included in the overall score.
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PageTitle | 100% | Far too many sites lack a page title. A page title is the first thing that shows in the search results so always use the title element. | |
Title relevance | 43% | A title should reflect the contents of a site. This site has a 33 % match | |
Title Length | 100% | Limit your title to anywhere between 40 and 70 characters. Your title was 53 characters long | |
Meta Description | 0% | A meta description is the second element that shows in the search results so always use the meta description. | |
Meta description length | 0% | The meta description should be between 145 and 160 characters. This meta description is 1 characters long. | |
Meta description relevance | 0% | Meta Description should reflect the contents of a site. This site has a 0 % match | |
Number of internal links | 30% | Linking to internal pages makes pages easier to find for search engines. Try to keep the number of links on your page roughly below 100. There are 402 internal links on this page. | |
Folder structure | 100% | We found a folder structure in the links on your page. A good folder structure makes a site easier to navigate. We found 3 level 1 folders and 6 folders above or in the first level of navigation. | |
Headings | 16% | Headers should reflect the contents of a site. This site has a 7 % match | |
Links | 8% | Link anchors should to some degree reflect the contents of a site. This site has a 4 % match | |
Image alt tags | 22% | Image alt tags should to some degree reflect the contents of a site. This site has a 8 % match | |
Bold and italic | 27% | Bold and italic tags should reflect the contents of a site to some degree. This site has a 9 % match | |
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Keywords in Domainname | 30% | There are no important keywords in your domain name | |
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Structured Data | 100% | Structured data makes it easier for search engines to index your website | |
Inline css | 0% | Do not use inline css declarations. Inline css will slow down the rendering of the website. We detected 241 inline style declarations ( <a style="color:green">) with a size of 8956 bytes | |
Excessive use of the same words | 100% | There is no indication that there are one or more keywords that are used excessively. | |
Frames or iframes | 100% | Perfect, detected not (i)frames on your webpagina | |
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Css | 30% | We detected too much (2) CSS files on your page. Css files block the loading of a webpage. | |
Javascript | 100% | Perfect, we did not detect too many blocking JavaScript files | |
Mobile Website | 100% | Perfect, we found a responsive design for mobile users | |
Most important heading | 100% | Perfect, we detected a correct use of the most important (h1) heading! | |
Normalized headings | 40% | We dit not font a normalized heading structure. A heading 2 (h2) for example should be followed by a heading of an equal level (h2), a child heading (h3) or even a aprent heading (h1). |
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en.wikipedia.org images and descriptions
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http://en.wikipedia.org/static/images/icons/wikipedia.png height: 50 width: 50 description: no alt description found |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/33b025a6bf54ec02e65c871dc3e5897c921419cf height: height attribute not set width: width attribute not set description: {\displaystyle {\boldsymbol {\theta }}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/d4a774eb5746942b5a8bb44632c83f9a50bf39ef height: height attribute not set width: width attribute not set description: {\displaystyle l({\boldsymbol {\theta }};\mathbf {x} ,\mathbf {z} )=p(\mathbf {x} ,\mathbf {z} \mid {\boldsymbol {\theta }})} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/74ab2a5b6429f6d17632aedd043e6e177c9ca01c height: height attribute not set width: width attribute not set description: {\displaystyle l({\boldsymbol {\theta }};\mathbf {x} )=p(\mathbf {x} \mid {\boldsymbol {\theta }})=\int p(\mathbf {x} ,\mathbf {z} \mid {\boldsymbol {\theta }})\,d\mathbf {z} =\int p(\mathbf {x} \mid \mathbf {z} ,{\boldsymbol {\theta }})p(\mathbf {z} \mid {\boldsymbol {\theta }})\,d\mathbf {z} } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/24d245ae1f551d986964bdcfa59ef4bad70079cb height: height attribute not set width: width attribute not set description: {\displaystyle q({\boldsymbol {\theta }}\mid {\boldsymbol {\theta }}^{(t)})} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/ceb93467728ae61d8611ec69c73ca0dc3a89dbd9 height: height attribute not set width: width attribute not set description: {\displaystyle q({\boldsymbol {\theta }}\mid {\boldsymbol {\theta }}^{(t)})=\operatorname {e} _{\mathbf {z} \sim p(\cdot |\mathbf {x} ,{\boldsymbol {\theta }}^{(t)})}\left[\log p(\mathbf {x} ,\mathbf {z} |{\boldsymbol {\theta }})\right]\,} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/ec0b048c1c3ec3993dda22954d5024c87c268c7d height: height attribute not set width: width attribute not set description: {\displaystyle {\boldsymbol {\theta }}^{(t+1)}={\underset {\boldsymbol {\theta }}{\operatorname {arg\,max} }}\ q({\boldsymbol {\theta }}\mid {\boldsymbol {\theta }}^{(t)})\,} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/02e39e41a9408b3c868929e556216402e17bef76 height: height attribute not set width: width attribute not set description: {\displaystyle {\boldsymbol {\theta }}^{(t+1)}={\underset {\boldsymbol {\theta }}{\operatorname {arg\,max} }}\operatorname {e} _{\mathbf {z} \sim p(\cdot |\mathbf {x} ,{\boldsymbol {\theta }}^{(t)})}\left[\log p(\mathbf {x} ,\mathbf {z} |{\boldsymbol {\theta }})\right]\,} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/4921aead43c89a940741fbce1a43b20ee9f40aa2 height: height attribute not set width: width attribute not set description: {\displaystyle p(\mathbf {z} \mid \mathbf {x} ,{\boldsymbol {\theta }})} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/8799d6e07102db446f9348b163903df2eb51c0f7 height: height attribute not set width: width attribute not set description: {\displaystyle \log p(\mathbf {x} \mid {\boldsymbol {\theta }})=\log p(\mathbf {x} ,\mathbf {z} \mid {\boldsymbol {\theta }})-\log p(\mathbf {z} \mid \mathbf {x} ,{\boldsymbol {\theta }}).} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/a7c37375a0cfaacb347c7d16a264ff7556aa4607 height: height attribute not set width: width attribute not set description: {\displaystyle p(\mathbf {z} \mid \mathbf {x} ,{\boldsymbol {\theta }}^{(t)})} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/dea8d338c63fb03728cbc5188d3f7fe59d89fcee height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}\log p(\mathbf {x} \mid {\boldsymbol {\theta }})&=\sum _{\mathbf {z} }p(\mathbf {z} \mid \mathbf {x} ,{\boldsymbol {\theta }}^{(t)})\log p(\mathbf {x} ,\mathbf {z} \mid {\boldsymbol {\theta }})-\sum _{\mathbf {z} }p(\mathbf {z} \mid \mathbf {x} ,{\boldsymbol {\theta }}^{(t)})\log p(\mathbf {z} \mid \mathbf {x} ,{\boldsymbol {\theta }})\\&=q({\boldsymbol {\theta }}\mid {\boldsymbol {\theta }}^{(t)})+h({\boldsymbol {\theta }}\mid {\boldsymbol {\theta }}^{(t)}),\end{aligned}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/002e61b049741e19c9fb5b0c331b1ec093e334bd height: height attribute not set width: width attribute not set description: {\displaystyle h({\boldsymbol {\theta }}\mid {\boldsymbol {\theta }}^{(t)})} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/49af4819ab3c959837e17ed91605b592e863cf95 height: height attribute not set width: width attribute not set description: {\displaystyle \log p(\mathbf {x} \mid {\boldsymbol {\theta }}^{(t)})=q({\boldsymbol {\theta }}^{(t)}\mid {\boldsymbol {\theta }}^{(t)})+h({\boldsymbol {\theta }}^{(t)}\mid {\boldsymbol {\theta }}^{(t)}),} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/c8ad176b3f297b826026386893cbea4585678dfe height: height attribute not set width: width attribute not set description: {\displaystyle \log p(\mathbf {x} \mid {\boldsymbol {\theta }})-\log p(\mathbf {x} \mid {\boldsymbol {\theta }}^{(t)})=q({\boldsymbol {\theta }}\mid {\boldsymbol {\theta }}^{(t)})-q({\boldsymbol {\theta }}^{(t)}\mid {\boldsymbol {\theta }}^{(t)})+h({\boldsymbol {\theta }}\mid {\boldsymbol {\theta }}^{(t)})-h({\boldsymbol {\theta }}^{(t)}\mid {\boldsymbol {\theta }}^{(t)}).} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/507c40004ea5b6171086fd4b4c0cd25d82b8cf5c height: height attribute not set width: width attribute not set description: {\displaystyle h({\boldsymbol {\theta }}\mid {\boldsymbol {\theta }}^{(t)})\geq h({\boldsymbol {\theta }}^{(t)}\mid {\boldsymbol {\theta }}^{(t)})} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/8eb0a8b9d3317d3f24df360b699fd6da686ccbf4 height: height attribute not set width: width attribute not set description: {\displaystyle \log p(\mathbf {x} \mid {\boldsymbol {\theta }})-\log p(\mathbf {x} \mid {\boldsymbol {\theta }}^{(t)})\geq q({\boldsymbol {\theta }}\mid {\boldsymbol {\theta }}^{(t)})-q({\boldsymbol {\theta }}^{(t)}\mid {\boldsymbol {\theta }}^{(t)}).} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/f4e0798985f246154015deb375e1494dfa734922 height: height attribute not set width: width attribute not set description: {\displaystyle f(q,\theta ):=\operatorname {e} _{q}[\log l(\theta ;x,z)]+h(q),} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/57e86296710dd8d5759ee0d751bb662620b2c13c height: height attribute not set width: width attribute not set description: {\displaystyle f(q,\theta )=-d_{\mathrm {kl} }{\big (}q\parallel p_{z\mid x}(\cdot \mid x;\theta ){\big )}+\log l(\theta ;x),} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/ec388559464783fb600754a399c9a6172fa10bee height: height attribute not set width: width attribute not set description: {\displaystyle q^{(t)}=\operatorname {arg\,max} _{q}\ f(q,\theta ^{(t)})} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/696d87c8260b868730707884aa48c0db4bbaaae4 height: height attribute not set width: width attribute not set description: {\displaystyle \theta ^{(t+1)}=\operatorname {arg\,max} _{\theta }\ f(q^{(t)},\theta )} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/1beadd771d6bc57b9ea869b4310ebbff6bc4eba2 height: height attribute not set width: width attribute not set description: {\displaystyle {\widehat {\sigma }}_{v}^{2}={\frac {1}{n}}\sum _{k=1}^{n}{(z_{k}-{\widehat {x}}_{k})}^{2},} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/a18153ee206f0216802fb91fcaed7ba3a170490a height: height attribute not set width: width attribute not set description: {\displaystyle {\widehat {x}}_{k}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/a51cdfac24f8b95ed711f11ea9502da4087b6a24 height: height attribute not set width: width attribute not set description: {\displaystyle z_{k}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/d4cd67d275b9d98918febdc2bd14f4085d12dcaf height: height attribute not set width: width attribute not set description: {\displaystyle {\widehat {\sigma }}_{w}^{2}={\frac {1}{n}}\sum _{k=1}^{n}{({\widehat {x}}_{k+1}-{\widehat {f}}{\widehat {x}}_{k})}^{2},} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/74df0419c9808d257d9a4addec6ee67bb8b3d2f3 height: height attribute not set width: width attribute not set description: {\displaystyle {\widehat {x}}_{k+1}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/6136b1f02bee9d8cb762fc508fc951bfc46f49ef height: height attribute not set width: width attribute not set description: {\displaystyle {\widehat {f}}={\frac {\sum _{k=1}^{n}{({\widehat {x}}_{k+1}-{\widehat {f}}{\widehat {x}}_{k})}^{2}}{\sum _{k=1}^{n}{\widehat {x}}_{k}^{2}}}.} |
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https://upload.wikimedia.org/wikipedia/commons/thumb/0/09/clusteranalysis_mouse.svg/400px-clusteranalysis_mouse.svg.png height: 161 width: 400 description: no alt description found |
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https://upload.wikimedia.org/wikipedia/commons/thumb/a/a7/em_old_faithful.gif/240px-em_old_faithful.gif height: 240 width: 240 description: no alt description found |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/61e9345a73e716032dfe920da2ef9d9265e6226b height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {x} =(\mathbf {x} _{1},\mathbf {x} _{2},\ldots ,\mathbf {x} _{n})} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b height: height attribute not set width: width attribute not set description: {\displaystyle n} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab height: height attribute not set width: width attribute not set description: {\displaystyle d} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/394a19c373675e608117295cd6415e84f06eaaeb height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {z} =(z_{1},z_{2},\ldots ,z_{n})} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/e88d94caf64a6cc0cd17d33538f6dbf959fa023b height: height attribute not set width: width attribute not set description: {\displaystyle x_{i}\mid (z_{i}=1)\sim {\mathcal {n}}_{d}({\boldsymbol {\mu }}_{1},\sigma _{1})} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/f9513f8d71de99a9b05902f3cffd3f3b731c7c93 height: height attribute not set width: width attribute not set description: {\displaystyle x_{i}\mid (z_{i}=2)\sim {\mathcal {n}}_{d}({\boldsymbol {\mu }}_{2},\sigma _{2}),} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/9ee708039dc8c18644060681372da6736ecec2ba height: height attribute not set width: width attribute not set description: {\displaystyle \operatorname {p} (z_{i}=1)=\tau _{1}\,} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/69d5d20010a5d405950bcddfeee65fe0e4720e85 height: height attribute not set width: width attribute not set description: {\displaystyle \operatorname {p} (z_{i}=2)=\tau _{2}=1-\tau _{1}.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/bc9e5920c91b1e981bfd29ff579a9015aed036e2 height: height attribute not set width: width attribute not set description: {\displaystyle \theta ={\big (}{\boldsymbol {\tau }},{\boldsymbol {\mu }}_{1},{\boldsymbol {\mu }}_{2},\sigma _{1},\sigma _{2}{\big )},} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/d298ae52eb7b86ed22202288fbf66da466994d60 height: height attribute not set width: width attribute not set description: {\displaystyle l(\theta ;\mathbf {x} )=\prod _{i=1}^{n}\sum _{j=1}^{2}\tau _{j}\ f(\mathbf {x} _{i};{\boldsymbol {\mu }}_{j},\sigma _{j}),} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/8992a646a56612c1c52a38302e6a84ccceeb949a height: height attribute not set width: width attribute not set description: {\displaystyle l(\theta ;\mathbf {x} ,\mathbf {z} )=p(\mathbf {x} ,\mathbf {z} \mid \theta )=\prod _{i=1}^{n}\prod _{j=1}^{2}\ [f(\mathbf {x} _{i};{\boldsymbol {\mu }}_{j},\sigma _{j})\tau _{j}]^{\mathbb {i} (z_{i}=j)},} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/c7c30db649a3d79fca184a9ba9485609ff6ddf62 height: height attribute not set width: width attribute not set description: {\displaystyle l(\theta ;\mathbf {x} ,\mathbf {z} )=\exp \left\{\sum _{i=1}^{n}\sum _{j=1}^{2}\mathbb {i} (z_{i}=j){\big [}\log \tau _{j}-{\tfrac {1}{2}}\log |\sigma _{j}|-{\tfrac {1}{2}}(\mathbf {x} _{i}-{\boldsymbol {\mu }}_{j})^{\top }\sigma _{j}^{-1}(\mathbf {x} _{i}-{\boldsymbol {\mu }}_{j})-{\tfrac {d}{2}}\log(2\pi ){\big ]}\right\},} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/8205f06e0d279689ed04a1ac04a3d9c249c637df height: height attribute not set width: width attribute not set description: {\displaystyle \mathbb {i} } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61 height: height attribute not set width: width attribute not set description: {\displaystyle f} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/7554fde3a46aa69efabd1e64538c9e5de68c36f7 height: height attribute not set width: width attribute not set description: {\displaystyle \mathbb {i} (z_{i}=j)} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/cbd5cf1ea52696745ea83c2e67493e9e41d8b67f height: height attribute not set width: width attribute not set description: {\displaystyle t_{j,i}^{(t)}:=\operatorname {p} (z_{i}=j\mid x_{i}=\mathbf {x} _{i};\theta ^{(t)})={\frac {\tau _{j}^{(t)}\ f(\mathbf {x} _{i};{\boldsymbol {\mu }}_{j}^{(t)},\sigma _{j}^{(t)})}{\tau _{1}^{(t)}\ f(\mathbf {x} _{i};{\boldsymbol {\mu }}_{1}^{(t)},\sigma _{1}^{(t)})+\tau _{2}^{(t)}\ f(\mathbf {x} _{i};{\boldsymbol {\mu }}_{2}^{(t)},\sigma _{2}^{(t)})}}.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/c4c12cc7f548a84bb750a77d2faf71cb6f06dfd4 height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}q(\theta \mid \theta ^{(t)})&=\operatorname {e} _{\mathbf {z} \mid \mathbf {x} =\mathbf {x} ;\mathbf {\theta } ^{(t)}}[\log l(\theta ;\mathbf {x} ,\mathbf {z} )]\\&=\operatorname {e} _{\mathbf {z} \mid \mathbf {x} =\mathbf {x} ;\mathbf {\theta } ^{(t)}}[\log \prod _{i=1}^{n}l(\theta ;\mathbf {x} _{i},z_{i})]\\&=\operatorname {e} _{\mathbf {z} \mid \mathbf {x} =\mathbf {x} ;\mathbf {\theta } ^{(t)}}[\sum _{i=1}^{n}\log l(\theta ;\mathbf {x} _{i},z_{i})]\\&=\sum _{i=1}^{n}\operatorname {e} _{z_{i}\mid x_{i}=x_{i};\mathbf {\theta } ^{(t)}}[\log l(\theta ;\mathbf {x} _{i},z_{i})]\\&=\sum _{i=1}^{n}\sum _{j=1}^{2}p(z_{i}=j\mid x_{i}=\mathbf {x} _{i};\theta ^{(t)})\log l(\theta _{j};\mathbf {x} _{i},j)\\&=\sum _{i=1}^{n}\sum _{j=1}^{2}t_{j,i}^{(t)}{\big [}\log \tau _{j}-{\tfrac {1}{2}}\log |\sigma _{j}|-{\tfrac {1}{2}}(\mathbf {x} _{i}-{\boldsymbol {\mu }}_{j})^{\top }\sigma _{j}^{-1}(\mathbf {x} _{i}-{\boldsymbol {\mu }}_{j})-{\tfrac {d}{2}}\log(2\pi ){\big ]}.\end{aligned}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/c9d7c15b71808c0aa9700ac4abbea1ac788071e6 height: height attribute not set width: width attribute not set description: {\displaystyle \log l(\theta ;\mathbf {x} _{i},z_{i})} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/a6cb63d21c4798790355502371133afb825970d3 height: height attribute not set width: width attribute not set description: {\displaystyle p(z_{i}\mid x_{i}=\mathbf {x} _{i};\theta ^{(t)})} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/57d2ef3df60acdb53bdf90535264041fea7231cd height: height attribute not set width: width attribute not set description: {\displaystyle \mathbf {x} _{i}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/896bd3433b93d00916b90aa6493c84434596c232 height: height attribute not set width: width attribute not set description: {\displaystyle t_{j,i}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/c8b54e65c55fb22de728162c5e1ddd7314976e9b height: height attribute not set width: width attribute not set description: {\displaystyle q(\theta \mid \theta ^{(t)})} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/38a7dcde9730ef0853809fefc18d88771f95206c height: height attribute not set width: width attribute not set description: {\displaystyle \tau } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/3364e5b9ac90ca67273d7d3debc0a2e9c0ae125a height: height attribute not set width: width attribute not set description: {\displaystyle ({\boldsymbol {\mu }}_{1},\sigma _{1})} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/f921e9edd14dec88d45fcbafa7981ddb274b2958 height: height attribute not set width: width attribute not set description: {\displaystyle ({\boldsymbol {\mu }}_{2},\sigma _{2})} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/56c54a77e3b770e7602caa4fffd1a4282fbc4684 height: height attribute not set width: width attribute not set description: {\displaystyle \tau _{1}+\tau _{2}=1} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/f3d2b6d3ae054fad990836865c545ef1d4f1fffd height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}{\boldsymbol {\tau }}^{(t+1)}&={\underset {\boldsymbol {\tau }}{\operatorname {arg\,max} }}\ q(\theta \mid \theta ^{(t)})\\&={\underset {\boldsymbol {\tau }}{\operatorname {arg\,max} }}\ \left\{\left[\sum _{i=1}^{n}t_{1,i}^{(t)}\right]\log \tau _{1}+\left[\sum _{i=1}^{n}t_{2,i}^{(t)}\right]\log \tau _{2}\right\}.\end{aligned}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/d2f22acb3fec62d449d1d6651beb6f5ba048d33c height: height attribute not set width: width attribute not set description: {\displaystyle \tau _{j}^{(t+1)}={\frac {\sum _{i=1}^{n}t_{j,i}^{(t)}}{\sum _{i=1}^{n}(t_{1,i}^{(t)}+t_{2,i}^{(t)})}}={\frac {1}{n}}\sum _{i=1}^{n}t_{j,i}^{(t)}.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/e395f0bf406a91c497ad82b273c9d6b86759e394 height: height attribute not set width: width attribute not set description: {\displaystyle {\begin{aligned}({\boldsymbol {\mu }}_{1}^{(t+1)},\sigma _{1}^{(t+1)})&={\underset {{\boldsymbol {\mu }}_{1},\sigma _{1}}{\operatorname {arg\,max} }}\ q(\theta \mid \theta ^{(t)})\\&={\underset {{\boldsymbol {\mu }}_{1},\sigma _{1}}{\operatorname {arg\,max} }}\ \sum _{i=1}^{n}t_{1,i}^{(t)}\left\{-{\tfrac {1}{2}}\log |\sigma _{1}|-{\tfrac {1}{2}}(\mathbf {x} _{i}-{\boldsymbol {\mu }}_{1})^{\top }\sigma _{1}^{-1}(\mathbf {x} _{i}-{\boldsymbol {\mu }}_{1})\right\}\end{aligned}}.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/45f3e73f50d396aadc98182709eee0c0d513aa6b height: height attribute not set width: width attribute not set description: {\displaystyle {\boldsymbol {\mu }}_{1}^{(t+1)}={\frac {\sum _{i=1}^{n}t_{1,i}^{(t)}\mathbf {x} _{i}}{\sum _{i=1}^{n}t_{1,i}^{(t)}}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/a92651be432155520db19dc0b4da807039d96eb0 height: height attribute not set width: width attribute not set description: {\displaystyle \sigma _{1}^{(t+1)}={\frac {\sum _{i=1}^{n}t_{1,i}^{(t)}(\mathbf {x} _{i}-{\boldsymbol {\mu }}_{1}^{(t+1)})(\mathbf {x} _{i}-{\boldsymbol {\mu }}_{1}^{(t+1)})^{\top }}{\sum _{i=1}^{n}t_{1,i}^{(t)}}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/939349990d3a58d89689a8872f23fbd27c828a43 height: height attribute not set width: width attribute not set description: {\displaystyle {\boldsymbol {\mu }}_{2}^{(t+1)}={\frac {\sum _{i=1}^{n}t_{2,i}^{(t)}\mathbf {x} _{i}}{\sum _{i=1}^{n}t_{2,i}^{(t)}}}} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/88bd69bf8025781f8cad373b62fbdc322e5c2b0e height: height attribute not set width: width attribute not set description: {\displaystyle \sigma _{2}^{(t+1)}={\frac {\sum _{i=1}^{n}t_{2,i}^{(t)}(\mathbf {x} _{i}-{\boldsymbol {\mu }}_{2}^{(t+1)})(\mathbf {x} _{i}-{\boldsymbol {\mu }}_{2}^{(t+1)})^{\top }}{\sum _{i=1}^{n}t_{2,i}^{(t)}}}.} |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/1936cdfb30674d85c6029aed6c4b3e571300374b height: height attribute not set width: width attribute not set description: {\displaystyle e_{z\mid \theta ^{(t)},\mathbf {x} }[\log l(\theta ^{(t)};\mathbf {x} ,\mathbf {z} )]\leq e_{z\mid \theta ^{(t-1)},\mathbf {x} }[\log l(\theta ^{(t-1)};\mathbf {x} ,\mathbf {z} )]+\varepsilon } |
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https://wikimedia.org/api/rest_v1/media/math/render/svg/a30c89172e5b88edbd45d3e2772c7f5e562e5173 height: height attribute not set width: width attribute not set description: {\displaystyle \varepsilon } |
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