Generalizing Rao’s Contributions to Information Geometry in Future Quantum Computing Contexts: A (Very) Preliminary Analysis for Technology Experts.
Dr. Jonathan Kenigson, FRSA
C. R. Rao laid the foundation for modern Information Geometry in 1945 while a research student at Andhra and Calcutta – a task that involved the employment of Manifold Theory in conditions of divergence and entropy. The initial ideas employed in the paradigm involve: (1). Positive-Definite Information Matrices as Norms on Function Spaces; (2). Function Spaces that are restricted to the context(s) of spaces of manifolds of Distribution Functions. In particular, (2) assumes the form of spaces of Normal Distributions, whose global geometry is isomorphic to the “Upper-Half-Plane” Model of Hyperbolic Geometry with the classical Poincaré metric. In a “qualitative” sense, the distinction between two distributions may be interpreted as a proportion of the difference of their relative Information Entropies (IE). This interpretation lends itself well to modern computation and AI because such entropies, while long-known, have just begun to be explored in Quantum contexts. Information from network topologies and architectures explored from categorical contexts are devoid of this expression. In the former paradigm, entropy cannot be quantified in an integrable sense because of the lack of a measure-theoretic expression of the relevant categorical quantities. The Riemannian structure of the Information Manifold, however, renders it a Topological Space, and, more particularly, a “Metric Space,” whose distance function is positive-definite and derived from the concavity conditions implied in the “Fisher-Rao Information Matrix.” Any Information Manifold also possesses a “coherent” integral structure whose “Lebesgue-Measurable” sets are determined canonically by the “Ring of Outer Measures” on the manifold. The resulting Lebesgue Integral has properties that permit efficient computation as guaranteed by abstract Measure Theory – namely, arbitrary pointwise approximation by binary step functions and strong restrictions on information convergence demanded by such results as Fatou’s Lemma and the Lebesgue Dominated Convergence Theorem.
Entropies that generalize the Rényi entropy (like the Kullback-Leibler Divergence), may be constructed on the tangent spaces of an Information Manifold as Surface Integrals. The resulting paradigm permits novel applications of Ergodic metrization. Categorical and Measure-Theoretic notions may be introduced induced through integral Prokhorov metrization and then rendered algebraically by the introduction of Markov Transition Kernels between the manifolds. This approach retains the rich geometry of Rao’s paradigm without sacrificing the expediency and applicability of Categorical notions of transition functions. Given a countable sequence of random variables on a probability space (that are not necessarily identically distributed), one can employ Fisher-Rao theory to construct a sequence of Information Manifolds in the metric tensor defined by the Fisher Information. This manifold sequence is associated with a sequence of spaces of probability measures defined on the Borel subsets induced by each metric. Each of these is capable of Prokhorov Metrization. We introduce a modern measure theory to rigorously define transition kernels between the Fisher-Rao Manifolds, associating a generalized Markov process with each transition.
I present herein a paradigm for this attempt that could permit generalization of existing results to the study of Information Manifolds generated in Quantum Computing – in which, among other differences, the underlying manifolds are not comprised of Normal Distributions but of Wavefunctions from some underlying computational process that generates functional data capable of at least positive-semidefinite metrization in the manner of a Riemannian Manifold. Mathematically, the generalized Kolmogorov Central Limit Theorems are sufficiently robust to permit strong statements about Convergence in Law that generalize the classical CLT to situations in which the underlying random variables are not Independent and Identically Distributed (IID). Relevant resources for each step of the process are presented below. Initially, one begins with the notion of probability as a derivation on the Lebesgue Measures. Even though it is quite old, the Kolmogorov framework is still extremely useful. The best reference for this work available in modern translation is likely still Andrei Nikolajevich Kolmogorov’s 1950 masterwork, Foundations of the Theory of Probability (In Russian). The properties of a random variable on a measurable space are lucidly explored by Bert Fristedt and Lawrence Gray’s 1996 text, A modern approach to probability theory published by Birkhäuser in Boston. Convergence results should encapsulate the measure-theoretic properties of the base space in Lebesgue terms so that a general Smooth Orientable Manifold can be created from local spaces. It is the opinion of the current author that the most lucid account of convergence in such generality is furnished in Billingsley, Patrick (1999). Convergence of probability measures. 2nd ed. John Wiley & Sons. pp.1-28. Once an Information Manifold with a given metric tensor exists, one may set about to explore various divergences induced by: (1). Fisher Information and (2). Quasi-Canonical divergences that exist on the tangent spaces, like the Kullback-Leibler Divergence. It is the current author’s opinion that the best resources devoted to this approach are Nagaoka Hiroshi’s 2000 monograph, Methods of information geometry, Translations of mathematical monographs; v. 191, and Shun’ichi Amari’s 1985 introductory applied text Differential-geometrical methods in statistics published by Springer-Verlag in Berlin. Metrization of the spaces of Borel measures on a sequence of Information Manifolds is possible using so-called Prokhorov Processes. Billingsley’s (1999) treatment is particularly salient because convergence on Riemannian Manifolds also induces convergence on sequences of measures, which are themselves defined on the underlying rings of Sigma Algebras for each manifold’s integral structure.
More abstractly, the current author finds that the Fisher Information can be defined on the tangent space S of arbitrary Radon measures using the Radon-Nikodym Theorem. The clearest and most concise existing resource for this treatment may be found in Mitsuhiro Ito and Yuichi Shishido’s 2008 article, “Fisher information metric and Poisson kernels” in the journal Differential Geometry and Its Applications. 26 (4): 347 – 356. The notion of “Poisson Kernel” is intended in the singular sense of a classical Normal Distribution rather than the sense of Kernel-Superpositions that arise in (for instance) Jacobi Theta representations of pure imaginary arguments. From this vantage, one defines an arbitrary f-Divergence on each Statistical Manifold S(X) in the manner of Coeurjolly, J-F. & Drouilhet, R. (2006). “Normalized information-based divergences”.arXiv:math/0604246. A Categorical view of the Markov Kernels and their transition states may be derived intuitively or read from F. W. Lawvere (1962). “The Category of Probabilistic Mappings” by any available publisher. This resource is free in the public domain in the USA. Further submissions will further elucidate the details of the proposed paradigm.
Best way to Activate PrimeVideo/MyTV Using Amazon.com
Before we talk about PrimeVideo/MyTV, let’s take a look at Amazon it is one of the largest eCommerce platforms. Here, people tend to buy goods and items from kitchen accessories to garden needful without any hassle. It will not be wrong to say amazon is the marketplace for all sorts of consumers. That’s the reason why, after research, it was declared that 90% of the consumers of the world rely on amazon.com.
Interestingly, this eCommerce platform is not just for purchasing items. Here, you can make use of a streaming service that offers content of all types. Let it be web series, TV shows, or movies; you will access all of it against a very nominal subscription fee.
The rise of users however increased in the previous two years because of covid 19. Since people were not allowed to go to the cinema, filmmakers made sure to release web series, shows, and movies on PrimeVideo. That’s how an audience like you and me were not deprived of watching something substantial.
Keeping all the basic information aside, you should know the best thing about Amazon Prime – the site is compatible with many devices other than smartphones and laptops. The devices that can easily get activated through Amazon Prime website include streaming devices, gaming consoles, and TV screens.
Note: As mentioned above, to use Amazon Prime, you have to activate the device.
How to Activate PrimeVideo/MyTV?
If you are still reading this article and want to know the process of activating PrimeVideo/MyTV. We would say, you are definitely reading the right blog post. Why? Well, it is because in this blog we have discussed everything that you should know regarding amazon.com/MyTV.
See, let us inform you that the activation process of www amazon com MyTV is not very easy. It is indeed a tad on the trickier side. Also, the steps mentioned here for the activation process can be baffling. However, we have tried our best to ensure simplicity in each step.
What to check before activating amazon.com/MyTV?
You must check the device’s compatibility with amazon.com. It goes out without saying that if it is not eligible, then the site will not function.
How would you know if the site is eligible for the device or not? Well, it is easy. All you are required to do is to follow the steps mentioned below.
Step 1: Visit the app store of the device
Step 2: Look for Amazon Prime or Amazon Prime Video
Step 3: Install or download the app
Note: In case you don’t see the app on the channel store. Sorry, the app is not eligible for the device.
If you are able to install/download the app. You must follow the steps for the activation process.
Step 1: First thing first, please connect internet and turn on the device.
Step 2: The second step is to find Amazon Prime on the channel store or the app store.
Step 3: Then, you have to install or download the app on the device.
Step 4: After the installation process is completed. Log in to the Amazon Prime Video profile aka account. Please create an account if you don’t have any.
Step 5: After logging in to the account, you will find the activation code on a screen. The code which you see is for one-time use and is unique in its form.
Step 6: Open the search bar on your device and then type amazon.com/MyTV.
Step 7: As soon as amazon/MyTV is accessed, you have two options. The first one is to log in to the account. However, the second is to create a fresh account.
Step 8: If you have an account, sign in. Then, give the activation code you have been given. Kindly make sure you are entering the right code. Once done, click on the option “verify device”.
Step 9: The last step is to go to the device. It will be activated on its own. You will notice that the device is synced with the amazon.com/MyTV account automatically. It means you are ready to stream movies, shows, and web series.
PS: bear in mind that some devices will need a few extra steps to activate amazon.com/MyTV. However, the basic steps for all devices will remain the same. Contacting the support system of Amazon Prime Video will be the best solution if you feel stuck.
How to Create Profile on Amazon Prime Video?
Creating an Amazon Prime login doesn’t require a lot of understanding. The reason is, it is a very easy and simple process. For your help, we have explained each step clearly. So, all you need to do is to follow the steps without skipping any point.
Step 1: Visit primevideo/MyTV website.
Step 2: Click on the “Add new” option.
Step 3: To create a child’s profile press the kid’s profile toggle. Ignore the toggle if it is not meant for a child’s profile.
Step 4: Now that we assume you have entered the profile name. You have to click on “save changes”.
Tadaa! You are all set to start watching movies, shows, and web series on Amazon MyTV. Keeping the same procedure in mind, you can create as many profiles as you may like.
Create Amazon Prime Log-in on Smartphone
Many of us like to watch movies and shows no the smartphone. It can be because of the comfort and easiness they feel around themselves. Another reason – is they don’t possess a personal laptop. However, if you want to watch Amazon Prime Videos on a smartphone. You will need an app. Let’s check out how to do that…
Step 1: Open the app (Amazon Prime Video) and then click on “my stuff”.
Step 2: Now, select the most recent profile name.
Step 3: Then, “create profile”.
Step 4: In this step, you have to give a name to the profile.
Step 5: Press the option, “save”.
You are all good to start streaming content on Amazon Prime from your smartphone.
Installation Process of Amazon Prime Video on Apple TV
Did you know you can actually make use of Apple TV for Amazon Prime Video? If you had no idea while having an Apple TV. The news might have come as a surprise. Anyhow, here is what you have to do to install the app on the device.
Step 1: Turn on or rather switch the device. Then make your way towards the main menu.
Step 2: Find the App Store on the device. Find “Amazon Prime” or “Amazon Prime Video” application. If due to any reason, you are not able to locate the app. Simply type the app’s name on the search bar.
Step 3: In this step, press the icon where you see Amazon Prime’s logo.
Step 4: Installation is easier now because all you have to do is to press the option that says, “install”.
Step 5: App will pop up on the home screen of the Apple TV.
Enjoy Watching Amazon Prime Content on an iOS Device Using Apple TV
To access Amazon Prime on any iOS device, you must follow the steps in the same sequence. Please don’t skip any step. If you do so, there might be an issue and won’t be able to access the content of Amazon Prime Video.
Step 1: Connect both, the iOS device and the Apple TV with the same internet connection.
Step 2: Open the Amazon Prime application.
Step 3: Use log-in details to sign in to the account. This has to be done using your smartphone.
Step 4: Now, look for the content as in movie or show and tab on “play”. This has to be done on the app.
Step 5: You will see the “Airplay” choice on the right top side.
Step 6: In this step, you have to press “Airplay” and then select Apple TV.
Note: Amazon.com/MyTV is synced with almost every other smart TV device. However, you have to follow the steps mentioned above to access streaming.
Frequently Asked Questions about PrimeVideo/MyTV
Here are answers to some of the commonly asked questions. If your question is not answered here, don’t worry. Comment below and we will answer your query too.
Question 1: Where will I use or enter the activation code?
Answer: The QR code is used or entered at www.amazon.com/mytv/. The code will be of six digits and will be given for one-time use. Also, the code will be unique in nature. In addition, the code will be received only after completing the registration process.
Question 2: Can I make use of Samsung Smart TV for Amazon Prime?
Answer: Yes, you can for sure use older or recent versions of Samsung Smart TV. To access Amazon Prime, you have to go to Google Play Store. Then install the app on the device. Tadaa! You are all good to start streaming the content of your choice.
Question 3: What do I have to do to activate a website or app on a smart TV?
Answer: Open the app and then register the device. You will get a unique 6-digit code. Enter that in the website where it is asked. That’s it!
How to See Private Instagram Profiles without having to follow them
At the table, on the beach, at the cinema, on the bus, at school, in restaurants and even while walking or cycling … young people always have their phones in their hands at all times, and spend an average of up to six hours a day connected to their smartphones, so much so that it has earned the unflattering appellation of “generation with a bowed head”.
What is ever so urgent to see on your phone all the time? Whom are they chatting with at any time of day or night? Raise your hand if, seeing their children or partner far away and with a perpetually fixed gaze on the Instagram feed, at least once has not had the temptation to check who they chat with within Direct, who they follow, and what content they like and comments. Have you noticed, however, that their profile is set as private and is, in fact, inaccessible? We will come to your aid. We have created this practical guide just to show you how to see Instagram activity without following them.
How to make your Instagram profile private?
By setting your Instagram profile is private, only your followers will be able to see the posts, photos, videos, and stories you publish. Users who are not on your follower list, on the other hand, will not be able to see practically anything. Under the profile picture and the biography the message “This account is private will appear in a clear and rounded form. Follow to see photos and videos.
When a user sends you a follow request, it is up to you to choose whether to accept or reject it. By accepting the request, the user will become your follower and will be able to view the photos, videos, stories you publish and the detailed list of the profiles you follow and follow you. By refusing it, however, you will continue to deny them the ability to see your content. Making your profile private therefore allows you to choose from time to time, who can see what is published.
Is it possible to see private Instagram accounts without being a follower?
Try searching on search engines for “how to see private Instagram profile” or “access Instagram from another device”. You will get hundreds of millions of results, which means that, yes, it is possible to bypass the system and view the contents of any Instagram account, even if it is set as private. However, do not trust everything you find on the net. Most of the results lead to ineffective, and sometimes fraudulent, harmful and potentially damaging solutions. Keep your eyes peeled. Very often, these are scam sites that boast of offering you in a few moments the possibility of accessing someone else’s Instagram account remotely, in the face, however, of requests that should immediately put you on alert. It happens that in order to obtain login credentials, you are required to click on suspicious links, download programs, fill in unspecified surveys, and sometimes even provide bank details and credit card details.
The simplest, most direct and immediate way to see private Instagram profiles is to become a follower of the users in question. However, since it is a private profile, a request must be sent, which, at the discretion of the holder, can be accepted or rejected. It can also happen that the request is ignored and left pending.
Therefore, sending a request to follow Instagram to the person in question represents the immediate way to examine his profile, see what he publishes, which profiles he follows and who his followers are, but, if the request is not accepted, the possibility of viewing the contents is, in fact, denied. Furthermore, if the user in question has blocked you, you cannot interact in any way with his profile, let alone send a new follow request. If you cannot or do not want to follow the user in question on Instagram, or your follow request has been rejected, you can try to resort to crosscutting strategies that, with a little luck, could allow you to view the published content.
To see private Instagram accounts you can try using other social media, such as Facebook. Since the famous video photo-sharing community was bought by the big blue social network, publishing content simultaneously on the two platforms has become child’s play. Most users, in fact, set by default the automatic sharing of Instagram contents on their Facebook profile, without having to repeat the upload operation from time to time. To keep an eye on what a person is doing, therefore, you can try taking a look at their Facebook profile. Sometimes users also share their posts on other platforms, such as Twitter, Pinterest, and LinkedIn. In short, arm yourself with patience, do some research online and keep an eye on the various social profiles of the person you want to monitor. If the user in question only uses Instagram, this method is unfortunately not applicable. It goes if your Facebook profile is armored.
How to delete all contacts on iphone?
At the beginning of the cell phone era, upgrading a device meant transferring your contacts. Contacts can easily be transferred to every new device you get with iCloud and email backups. Despite the convenience, you end up with old contacts you don’t need anymore and wondering How to delete all contacts on iPhone? You may want to purge some content from your Contacts list, even if these contacts aren’t harming your phone.
It is easy to become overwhelmed by the number of phone numbers, emails, and names of people you no longer wish to communicate with when you go through your contacts looking for someone you want to contact.
Getting rid of all of your contacts from your phone can be a fairly simple process. The process of choosing which ones go is a little more complicated if you want to pick and choose. You can also hide multiple contacts (or, in some cases, delete your entire contact list) to save you time.
How to Delete All Contacts on iPhone
You must first move all of your iPhone or iPad contacts to iCloud in order to easily delete them. Here are the steps to know How to delete all contacts on iPhone?
- On the iPhone, open the Settings app.
- Choose Accounts & Passwords in iOS 11 and later. Click iCloud.
- Select Mail on your iPhone running iOS 10 or earlier. Select iCloud from the Accounts menu.
- Contacts should be turned off.
- You can still keep your iPhone by choosing Keep on my iPhone instead of deleting it.
- Turn on Contacts again. Click Merge.
- The contacts in your iCloud account are now available to you anywhere. Turn off Contacts once more.
- Delete your iPhone by tapping the Delete button.
- You can now access your Accounts. There are several options for email providers, including Google Mail, Yahoo, and others.
- Delete Contacts for each account by turning off Contacts within each account.
- There should now only be contacts from Facebook left. Click Facebook from the Settings app and all those contacts will be deleted.
- You can turn off Contacts to remove them from Contacts.
Using this method, I have been able to delete all contacts from my iPhone without any problems. You can, however, simply reset your iPhone if you want to erase everything from it before selling or giving it away.
Frequently Asked Question
Can we delete all contacts from iPhone at once?
All contacts can be deleted at once from your iPhone. Just go to Settings > General > Reset > Reset All Settings. Your settings and data will be deleted, but not your media.
What is the best way to mass delete contacts on your iPhone?
Using the Contacts app on your iPhone is the fastest way to delete all your contacts.
Click the Edit button at the top-right corner of the Contacts app.
Press Delete on the lower-right corner to delete all items selected in the upper-left corner.
Is it possible to delete multiple contacts from an iPhone app?
Multiple contacts can be deleted in the iPhone app in a few different ways. You can delete contacts by swiping left on the contact list. Next, tap “Delete Contact”. You will be taken to a confirmation page, where you can choose “Delete Contacts” again if you wish to delete them all.
How do you delete multiple contacts on iPhone 2022?
To delete multiple contacts at one time, you can either delete all of them or select a few to delete. To delete all of them, go to the contact list and press the Edit button. In the top left corner, there is an option to select All Contacts. Once you have done this, you will be able to swipe from right to leftover each contact that you want to be deleted and they will be removed from your phone.
Why are all the contacts on my iPhone duplicated?
iPhone users often experience this problem. This issue may have been caused by an old iCloud account being synced with the iPhone and duplicate contacts appearing on the device. This problem can be resolved by removing all of the iCloud contacts and then adding them back one by one to determine which ones are duplicates.
Is it possible to delete all my contacts on iOS 14?
On your iPhone, start by opening the Settings app. In order to view all your contacts, you must tap “Mail, Contacts, and Calendars,” followed by “Contacts.” Once you have selected all your contacts, you must tap on the circles next to their names. At the bottom of this screen, you can tap the “Delete Contact” button.
What is the best way to delete multiple contacts at once?
If you want to delete multiple contacts at once, select all the contacts and swipe left.
How can I remove duplicate contacts from my iPhone?
It’s a pain to deal with duplicate contacts. You can remove duplicate contacts from your iPhone using several software programs available. My Contacts Backup Pro is one of the best. It’s simple to use and free of charge.
How do you select all contacts on iPhone?
An iPhone’s contact list can be accessed by selecting the “Select All” option in the upper left-hand corner.
Why can’t I delete contacts from my iPhone?
It is not possible to delete contacts on the iPhone since it saves them for your convenience. The information associated with a deleted contact will also be deleted. It is useful to delete a contact to make some room on your phone, but it may also pose a risk if it is a family member or coworker.
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