|
|
| Line 1: |
Line 1: |
| In [[statistics]], '''identifiability''' is a property which a [[statistical model|model]] must satisfy in order for precise [[statistical inference|inference]] to be possible. We say that the model is '''identifiable''' if it is theoretically possible to learn the true value of this model’s underlying parameter after obtaining an infinite number of observations from it. Mathematically, this is equivalent to saying that different values of the parameter must generate different [[probability distribution]]s of the observable variables. Usually the model is identifiable only under certain technical restrictions, in which case the set of these requirements is called the '''identification conditions'''.
| | e - Shop Word - Press is a excellent cart for your on the web shopping organization. Online available for hiring are most qualified, well knowledgeable and talented Wordpress developer India from offshore Wordpress development services company. Step-4 Testing: It is the foremost important of your Plugin development process. Dead links are listed out simply because it will negatively have an influence on the website's search engine rating. By using this method one can see whether the theme has the potential to become popular or not and is their any scope of improvement in the theme. <br><br> |
|
| |
|
| A model that fails to be identifiable is said to be '''non-identifiable''' or '''unidentifiable'''. In some cases, even though a model is non-identifiable, it is still possible to learn the true values of a certain subset of the model parameters. In this case we say that the model is '''partially identifiable'''. In other cases it may be possible to learn the location of the true parameter up to a certain finite region of the parameter space, in which case the model is [[set identifiable]].
| | Any business enterprise that is certainly worth its name should really shell out a good deal in making sure that they have the most effective website that provides related info to its prospect. WPTouch is among the more well known Word - Press smartphone plugins which is currently in use by thousands of users. With the free Word - Press blog, you have the liberty to come up with your own personalized domain name. By purchasing Word - Press weblogs you can acquire your very own domain title and have total command of your web site. You can also get a free keyword tool that is to determine how strong other competing sites are and number of the searches on the most popular search sites. <br><br>ve labored so hard to publish and put up on their website. When a business benefits from its own domain name and a tailor-made blog, the odds of ranking higher in the search engines and being visible to a greater number of people is more likely. Possibly the most downloaded Word - Press plugin, the Google XML Sitemaps plugin but not only automatically creates a site map linking to everyone your pages and posts, it also notifies Google, Bing, Yahoo, and Ask. Nonetheless, with stylish Facebook themes obtainable on the Globe Broad Internet, half of your enterprise is done previously. If you loved this post and you would like to get a lot more details relating to [http://9ja.in/wordpress_dropbox_backup_628791 backup plugin] kindly visit our site. Purchase these from our site, or bring your own, it doesn't matter, we will still give you free installation and configuration. <br><br>The next thing I did after installing Wordpress was to find myself a free good-looking Wordpress-theme offering the functionality I was after. And, make no mistake,India's Fertility Clinics and IVF specialists are amongst the best in the world,and have been for some time. Websites that do rank highly, do so becaue they use keyword-heavy post titles. A whole lot worse, your site will likely be useless as well as your merchandise won't sell if no one has the endurance to wait for the web pages to load. Now all you have to do is log into your Word - Press site making use of the very same username and password that you initially had in your previous site. <br><br>Instead, you can easily just include it with our bodies integration field in e - Panel. s ability to use different themes and skins known as Word - Press Templates or Themes. Useful Plugins Uber - Menu Top Megamenu Now it is the time of sticky Top navbar. You should stay away from plugins that are full of flaws and bugs. 95, and they also supply studio press discount code for their clients, coming from 10% off to 25% off upon all theme deals. |
| | |
| ==Definition==
| |
| Let ℘ = {''P<sub>θ</sub>'': ''θ''∈Θ} be a [[statistical model]] where the parameter space Θ is either finite- or infinite-dimensional. We say that ℘ is '''identifiable''' if the mapping ''θ'' ↦ ''P<sub>θ</sub>'' is [[bijection|one-to-one]]:<ref>{{harvnb|Lehmann|Casella|1998|loc=Definition 1.5.2}}</ref>
| |
| : <math>
| |
| P_{\theta_1}=P_{\theta_2} \quad\Rightarrow\quad \theta_1=\theta_2 \quad\ \text{for all } \theta_1,\theta_2\in\Theta.
| |
| </math>
| |
| | |
| This definition means that distinct values of ''θ'' should correspond to distinct probability distributions: if ''θ''<sub>1</sub>≠''θ''<sub>2</sub>, then also ''P''<sub>''θ''<sub>1</sub></sub>≠''P''<sub>''θ''<sub>2</sub></sub>.<ref>{{harvnb|van der Vaart|1998|page=62}}</ref> If the distributions are defined in terms of the [[probability density function]]s, then two pdfs should be considered distinct only if they differ on a set of non-zero measure (for example two functions ƒ<sub>1</sub>(''x'')='''1'''<sub>0≤''x''<1</sub> and ƒ<sub>2</sub>(''x'')='''1'''<sub>0≤''x''≤1</sub> differ only at a single point ''x''=1 — a set of [[Lebesgue measure|measure]] zero — and thus cannot be considered as distinct pdfs).
| |
| | |
| Identifiability of the model in the sense of invertibility of the map ''θ'' ↦ ''P<sub>θ</sub>'' is equivalent to being able to learn the model’s true parameter if the model can be observed indefinitely long. Indeed, if {''X<sub>t</sub>''}⊆''S'' is the sequence of observations from the model, then by the [[strong law of large numbers]],
| |
| : <math>
| |
| \frac{1}{T} \sum_{t=1}^T \mathbf{1}_{\{X_t\in A\}} \ \xrightarrow{a.s.}\ \operatorname{Pr}[X_t\in A],
| |
| </math>
| |
| for every measurable set ''A''⊆''S'' (here '''1'''<sub>{…}</sub> is the [[indicator function]]). Thus with an infinite number of observations we will be able to find the true probability distribution ''P''<sub>0</sub> in the model, and since the identifiability condition above requires that the map ''θ'' ↦ ''P<sub>θ</sub>'' be invertible, we will also be able to find the true value of the parameter which generated given distribution ''P''<sub>0</sub>.
| |
| | |
| ==Examples==
| |
| | |
| ===Example 1===
| |
| Let ℘ be the [[normal distribution|normal]] [[location-scale family]]:
| |
| : <math>
| |
| \mathcal{P} = \Big\{\ f_\theta(x) = \tfrac{1}{\sqrt{2\pi}\sigma} e^{ -\frac{1}{2\sigma^2}(x-\mu)^2 }\ \Big|\ \theta=(\mu,\sigma): \mu\in\mathbb{R}, \,\sigma\!>0 \ \Big\}.
| |
| </math>
| |
| Then
| |
| : <math>\begin{align}
| |
| f_{\theta_1}=f_{\theta_2}\
| |
| &\Leftrightarrow\ \tfrac{1}{\sqrt{2\pi}\sigma_1}e^{ -\frac{1}{2\sigma_1^2}(x-\mu_1)^2 } = \tfrac{1}{\sqrt{2\pi}\sigma_2}e^{ -\frac{1}{2\sigma_2^2}(x-\mu_2)^2 } \\
| |
| &\Leftrightarrow\ \tfrac{1}{\sigma_1^2}(x-\mu_1)^2 + \ln \sigma_1^2 = \tfrac{1}{\sigma_2^2}(x-\mu_2)^2 + \ln \sigma_2^2 \\
| |
| &\Leftrightarrow\ x^2\big(\tfrac{1}{\sigma_1^2}-\tfrac{1}{\sigma_2^2}\big) - 2x\big(\tfrac{\mu_1}{\sigma_1^2}-\tfrac{\mu_2}{\sigma_2^2}\big) + \big(\tfrac{\mu_1^2}{\sigma_1^2}-\tfrac{\mu_2^2}{\sigma_2^2}+\ln\sigma_1^2-\ln\sigma_2^2\big) = 0
| |
| \end{align}</math>
| |
| This expression is equal to zero for almost all ''x'' only when all its coefficients are equal to zero, which is only possible when |''σ''<sub>1</sub>| = |''σ''<sub>2</sub>| and ''μ''<sub>1</sub> = ''μ''<sub>2</sub>. Since in the scale parameter ''σ'' is restricted to be greater than zero, we conclude that the model is identifiable: ƒ<sub>''θ''<sub>1</sub></sub>=ƒ<sub>''θ''<sub>2</sub></sub> ⇔ ''θ''<sub>1</sub>=''θ''<sub>2</sub>.
| |
| | |
| ===Example 2===
| |
| Let ℘ be the standard [[linear regression model]]:
| |
| : <math>
| |
| y = \beta'x + \varepsilon, \quad \operatorname{E}[\,\varepsilon|x\,]=0
| |
| </math>
| |
| (where ′ denotes matrix [[transpose]]). Then the parameter ''β'' is identifiable if and only if the matrix E[xx′] is invertible. Thus, this is the '''identification condition''' in the model.
| |
| | |
| ===Example 3===
| |
| Suppose ℘ is the classical [[errors-in-variables]] [[linear model]]:
| |
| : <math>\begin{cases}
| |
| y = \beta x^* + \varepsilon, \\
| |
| x = x^* + \eta,
| |
| \end{cases}</math>
| |
| where (''ε'',''η'',''x*'') are jointly normal independent random variables with zero expected value and unknown variances, and only the variables (''x'',''y'') are observed. Then this model is not identifiable,<ref name="riersol">{{harvnb|Reiersøl|1950}}</ref> only the product βσ²<sub style="position:relative;left:-.5em">∗</sub> is (where σ²<sub style="position:relative;left:-.5em">∗</sub> is the variance of the latent regressor ''x*''). This is also an example of [[set identifiability|set identifiable]] model: although the exact value of ''β'' cannot be learned, we can guarantee that it must lie somewhere in the interval (''β''<sub>yx</sub>, 1÷''β''<sub>xy</sub>), where ''β''<sub>yx</sub> is the coefficient in [[Ordinary least squares|OLS]] regression of ''y'' on ''x'', and ''β''<sub>xy</sub> is the coefficient in OLS regression of ''x'' on ''y''.<ref>{{harvnb|Casella|Berger|2001|page=583}}</ref>
| |
| | |
| If we abandon the normality assumption and require that ''x*'' were '''not''' normally distributed, retaining only the independence condition ''ε''⊥''η''⊥''x*'', then the model becomes identifiable.<ref name="riersol"/>
| |
| | |
| ==Software==
| |
| In the case of parameter estimation in partially observed dynamical systems, the profile likelihood can be also used for structural and practical identifiability analysis.<ref>
| |
| {{Citation
| |
| |last=Raue |first=A
| |
| |title=Structural and practical identifiability analysis of partially observed dynamical models by exploiting the profile likelihood
| |
| |url=http://bioinformatics.oxfordjournals.org/cgi/content/abstract/btp358?ijkey=iYp4jPP50F5vdX0&keytype=ref
| |
| |journal=Bioinformatics
| |
| |year=2009
| |
| |doi=10.1093/bioinformatics/btp358
| |
| |pmid=19505944
| |
| |last2=Kreutz
| |
| |first2=C
| |
| |last3=Maiwald
| |
| |first3=T
| |
| |last4=Bachmann
| |
| |first4=J
| |
| |last5=Schilling
| |
| |first5=M
| |
| |last6=Klingmüller
| |
| |first6=U
| |
| |last7=Timmer
| |
| |first7=J
| |
| |volume=25
| |
| |issue=15
| |
| |pages=1923–9
| |
| |postscript=.
| |
| }}</ref> An implementation of the [http://www.fdmold.uni-freiburg.de/~araue/Main/Software Profile Likelihood Approach] is available in the MATLAB Toolbox [[PottersWheel]].
| |
| | |
| ==See also==
| |
| * [[Parameter identification problem]]
| |
| | |
| ==References==
| |
| {{refbegin}}
| |
| * {{Citation
| |
| | last = Casella
| |
| | first = George
| |
| | coauthors = Roger L. Berger
| |
| | title = Statistical inference
| |
| | year = 2001
| |
| | edition = 2nd
| |
| | isbn = 0-534-24312-6
| |
| | ref = CITEREFCasellaBerger2001
| |
| | lccn = QA276.C37 2001
| |
| }}
| |
| * {{Citation
| |
| | last = Hsiao
| |
| | first = Cheng
| |
| | title = Identification
| |
| | year = 1983
| |
| | series = Handbook of Econometrics, Vol. 1, Ch.4
| |
| | publisher = North-Holland Publishing Company
| |
| }}
| |
| * {{Citation
| |
| | last = Lehmann
| |
| | first = E. L.
| |
| | coauthors = G. Casella
| |
| | title = Theory of point estimation
| |
| | edition = 2nd
| |
| | year = 1998
| |
| | publisher = Springer
| |
| | isbn = 0-387-98502-6
| |
| | ref = CITEREFLehmannCasella1998
| |
| }}
| |
| * {{Citation
| |
| | last = Reiersøl
| |
| | first = Olav
| |
| | year = 1950
| |
| | title = Identifiability of a linear relation between variables which are subject to error
| |
| | journal = Econometrica
| |
| | volume = 18
| |
| | issue = 4
| |
| | pages = 375–389
| |
| | jstor = 1907835
| |
| | doi = 10.2307/1907835
| |
| | publisher = The Econometric Society
| |
| }}
| |
| * {{Citation
| |
| | last = van der Vaart
| |
| | first = A.W.
| |
| | title = Asymptotic Statistics
| |
| | year = 1998
| |
| | publisher = Cambridge University Press
| |
| | isbn = 978-0-521-49603-2
| |
| | ref = CITEREFvan_der_Vaart1998
| |
| }}
| |
| {{refend}}
| |
| | |
| ===Notes===
| |
| {{reflist}}
| |
| | |
| [[Category:Estimation theory]]
| |
e - Shop Word - Press is a excellent cart for your on the web shopping organization. Online available for hiring are most qualified, well knowledgeable and talented Wordpress developer India from offshore Wordpress development services company. Step-4 Testing: It is the foremost important of your Plugin development process. Dead links are listed out simply because it will negatively have an influence on the website's search engine rating. By using this method one can see whether the theme has the potential to become popular or not and is their any scope of improvement in the theme.
Any business enterprise that is certainly worth its name should really shell out a good deal in making sure that they have the most effective website that provides related info to its prospect. WPTouch is among the more well known Word - Press smartphone plugins which is currently in use by thousands of users. With the free Word - Press blog, you have the liberty to come up with your own personalized domain name. By purchasing Word - Press weblogs you can acquire your very own domain title and have total command of your web site. You can also get a free keyword tool that is to determine how strong other competing sites are and number of the searches on the most popular search sites.
ve labored so hard to publish and put up on their website. When a business benefits from its own domain name and a tailor-made blog, the odds of ranking higher in the search engines and being visible to a greater number of people is more likely. Possibly the most downloaded Word - Press plugin, the Google XML Sitemaps plugin but not only automatically creates a site map linking to everyone your pages and posts, it also notifies Google, Bing, Yahoo, and Ask. Nonetheless, with stylish Facebook themes obtainable on the Globe Broad Internet, half of your enterprise is done previously. If you loved this post and you would like to get a lot more details relating to backup plugin kindly visit our site. Purchase these from our site, or bring your own, it doesn't matter, we will still give you free installation and configuration.
The next thing I did after installing Wordpress was to find myself a free good-looking Wordpress-theme offering the functionality I was after. And, make no mistake,India's Fertility Clinics and IVF specialists are amongst the best in the world,and have been for some time. Websites that do rank highly, do so becaue they use keyword-heavy post titles. A whole lot worse, your site will likely be useless as well as your merchandise won't sell if no one has the endurance to wait for the web pages to load. Now all you have to do is log into your Word - Press site making use of the very same username and password that you initially had in your previous site.
Instead, you can easily just include it with our bodies integration field in e - Panel. s ability to use different themes and skins known as Word - Press Templates or Themes. Useful Plugins Uber - Menu Top Megamenu Now it is the time of sticky Top navbar. You should stay away from plugins that are full of flaws and bugs. 95, and they also supply studio press discount code for their clients, coming from 10% off to 25% off upon all theme deals.