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FAQs
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How can I deactivate my Zerodha demat account?
Firstly, you need to make sure you do not have any debit (negative balance) in your trading account. You will have to clear your dues before you close your account.Also, you need to make sure there are no securities in your DEMAT account.Note: Closing your account is different from De-activation. Your Trading and DEMAT accounts will be closed permanently. You will have to re-do the account opening process if you want to resume trading with Zerodha.To close your account, download and print the account closure form, select 'Close' account, fill it out and send it to our head office.Alternatively, you can digitally sign (e-sign) the filled account closure form and submit it by raising a ticket below. The account closure form will be processed within 5-7 working days. The process of digitally e-signing is as below -Fill up the closure form - using MS Word or any other text editorSign up/ Log in to Digio - https://www.digio.inClick on Add document and upload the closure form.Click on sign icon.Enter your VID and once you do you will receive an OTP.Once you enter OTP digital signature will be affixed to account modification form, Once the signature is affixed you need download it and submit it by raising a ticket below.It is important to note that, In case of account 'closure cum transfer' you will have to send us a hard copy of the account closure form. Along with it, you'll have to send us the CMR (Client Master Report) of the target DEMAT account. The CMR has to be a hard copy with seal & stamp for NSDL account (or a digitally signed CMR for CDSL clients)Note: The Clients account shall be closed upon a specific request from the client. The closure shall be effective only after a period of one month has elapsed from the date of application/intimation or the date of settlement of account whichever is later.Settlement of account shall mean that there is no outstanding balance of shares or funds in the books of the client and Zerodha and the same is confirmed by the client. The date of confirmation shall be the effective date of settlement. As far as dormant accounts are concerned, we do not close such accounts but mark the same as “Inactive” till further action by the concerned client.Source: How do I close my Zerodha account?
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Will China become an innovator?
Yes. Start with the following chart.Innovation is neatly correlated with per capita GDP. That is because for innovation to thrive, you need low borrowing costs, a stable and moderately advanced economy where the risk adjusted returns of doing conventional things are low, consumer sophistication to generate demand for better goods and a conducive business environment. You just need to take a walk down Nanshan's kejiyuan to see how strong China's innovation system is as a subset of it's conventional investment led economy. Consumer sophistication has pushed China's electronics manufacturers to evolve from shanzhai to innovative brands. Chinese firms are investing heavily in R&D. Huawei, Midea, Haier etc compete internationally not on cost but on product quality. The chart also shows you that China is punching way above its weight when it comes to its performance in innovation relative to per capita GDP, that it is already a better innovator than other countries were at a similar level of economic development. Another way to look at innovation in China is to not consider it monolithic. McKinsey identified four key archetypes of innovation and assessed China's performance in each of them. *Revenue share of Chinese companies relative to China's proportion of global GDP. A score more than one shows that the respective cluster of firms have a global share of more than 14 percent. It is clear that Chinese firms have perform really well in efficiency driven and consumer focused innovation. In engineering based innovation, Chinese firms perform well in sectors where government mandated technology transfer helped them in rapid catch up with established players. China lags signNowly in science based innovation. It is perfectly reasonable to make the case that with proper investments and capital accumulation, China will emerge as an innovator in the leagues of say Japan or South Korea in the coming decades. I suggest you read McKinsey Global Institute's China Effect on Global Innovation to read more on innovation in China.The middle income trap is really just a theory based on the growth trajectory of Latin American countries. If you look in disaggregate terms, a lot of areas in China are high income economies. Beijing, Tianjin, Shanghai, Zhejiang and Shenzhen's GDP per capita is more than $22,000. Per capita GDP of Nanshan, Shenzhen's high technology district $49,000, ahead of Hong Kong and several OECD countries. There is no reason to believe that China as a whole will be become stuck in the middle income trap.
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How and why did silent letters emerge in English?
The following in reddit explains a lot.The question:What is the history behind the use of "silent letters" in words?I've often wondered why a few languages in the world developed "silent letters" instead of simply writing the word without the letter. For example, pterodactyl... but the p is silent. why not spell it terodactyl. knife... but the k is silent.. why not spell it nife? hola... but the h is silent. why not spell it ola? There are many, many cases of words like this in English and Spanish. Perhaps other languages too.So it begs the universal question. Why? Why "silent letters" were ever created? What are the rules for their use when creating words? Why use them at all? Are there any languages that don't use "silent letters?" Has there ever been a historical attempt to rid the language of these words?Of course you have words like debt, witch, numb, castle... which have "silent letters" within the word itself. why not spelled det, wich, num, or casle? But my primary focus is really on the silent letter at the beginning of a word.(as an amusing sidenote, you'd be surprised how many times auto-correct changed the words in this message forcing to type over and over again...)The answer:Usually, when a word's spelling doesn't match its pronunciation, it's because originally it did but the pronunciation changed while the spelling stayed the same. In order to avoid massive confusion, we keep using standardized spellings, even unintuitive ones, long after they cease to reflect the word's sounds. This explanation accounts for most of your examples. The word "knife", for example, had the [k] pronounced in Old English, but over time we dropped the [k] from all such words.English in particular has changed in pronunciation drastically since the time when the modern consensus was established regarding which spellings should be standard, so it's one of the worst languages around in this regard. (Probably not as bad as Irish, though!) This is both because English has changed a lot in just the last few centuries, and also because its spelling system is older than those of many other languages.The other main way that this kind of mismatch happens is when we borrow a word from a foreign language and we retain a spelling that reflects the source language even though such a pronunciation is not possible in the borrowing language. This accounts for "pterodactyl": it reflects the Greek spelling (πτεροδάκτυλος, according to Wikipedia), but syllable initial [pt] from Greek is not a consonant cluster that exists in English words, so English speakers have adapted the word by dropping the [p].The actual "rules" (or, rather, patterns) regarding how sounds get introduced or removed from words and from languages are quite complex, too much so for me to get into here. These sound changes are one of the main focuses of the field of historical linguistics. As for why languages change, that is an open question that is hotly debated.Another Answer:There are definitely languages which don't have silent letters - the obvious one being the various Chinese languages. Since their writing is ideographic, rather than alphabetic, when pronunciation changes over time, it's not visible in the written language.In terms of alphabetic languages, those with the fewest silent letters would be those which enact spelling reforms on a regular basis to ensure that official spellings match standard pronunciation. French recently announced a big spelling reform (notoriously removing the silent 'g' from oignon). My German isn't great, but as far as I'm aware they enact regular spelling reforms, and therefore in terms of pronunciation, what you see is very much what you get. This doesn't only affect silent letters, but also the irregular pronunciation that makes English so horrible for non-natives to speak (like the fact that 'though', 'cough' and 'bough' don't rhyme at all, even though they should based on how they're written).And the reason it seems to be a universal practice, is that it's a process which affects all languages, although it's effects are different based on how conservative its orthographic traditions are.Source: r/AskHistorians - What is the history behind the use of "silent letters" in words?Thanks & regards,Gitangshu AdhikaryWorld NetworksThe Writing InertiaWN EntertainmentWalt NetverkPlease visit, subscribe, follow, like & share the above to stay connected and know more.God Bless.
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Did Oscar Tay do all that interesting research himself? Or are they available in ordinary books for any student of doctorate in
rubs hands togetherThis is a long story.To illustrate how I would write an answer, I’ve chosen the first question to show up on my “Answer” tab, How do you say "king" in Proto-Indo-European, and does modern English have a cognate to it?, and I’ll be going through the process step by step.First, here’s what I know off the top of my head, without having to consult any sources:The English word “king” is from Old English cyning, from Proto-Germanic *kuningaz, which is itself literally “kin-ing” or “kin-person”. It has no direct Proto-Indo-European equivalent.The Latin rex (also “king”), however, does: it’s from a PIE word something like *regs, meaning “ruler” or “chief” or, roughly, “king”; it’s cognate with Hindi raj, English “rich”, German reich, and a Celtic word something like rix.The Celtic word for “bear” was ~artu, related to Greek arktos, also “bear”, whence “arctic”, literally “up where the bears are”. One Celtic name you’re familiar with was Artu-rix, “the bear-king” - the original Celtic name of King Arthur. This is the interesting sort of nugget I’d include in the final answer.From rex we get plenty of kingly words, including “regal” and “reign”, and plenty of others I’m not aware of or just can’t remember.It wouldn’t have been “king” in the same sense, though. The Proto-Indo-Europeans were a semi-nomadic society with individual groups or tribes ruled by a chief, so not quite the degree of power you’d be dealing with in a typical monarchy.To answer the question, while nothing like English “king” because of its derivation from “kin”, the PIE word was something like *regs, which is where we get Latin rex from, and from there English gets “regal”, “reign”, and some assorted others.I could post that as a semi-functioning answer. Fortunately, I would not do that, because a.) it’s not really complete, b.) it doesn’t do a good job of explaining the subject matter, and c.) it’s not all that interesting. It leaves open more questions:Why is the English word the way it is?Why didn’t the PIE root come down to English in the way it did to Latin or Sanskrit?Are there any other related words from other languages?What does this say about the different Indo-European societies?And so on. All those questions are rather boring for the average reader, so I’d have to find a way to make them more interesting, but that comes later, in the bit where I write the answer. Before I do that, I have to make sure what I’ve written here is correct.Etymonline and Wiktionary are the etymological dictionaries I usually consult. I’ve also got a copy of the compact edition of the Oxford English Dictionary (the two-volume set that comes with a magnifying glass), the American Heritage Dictionary of Indo-European roots (which picks up where the OED leaves off), and assorted others. Etymonline and Wiktionary are my main sources, though, because they are themselves a concise collection of all the etymological information in the OED or AHD. I’d only use the physical dictionaries if a word’s entry on either page were incomplete or if I needed examples of the word’s early use.The Etymonline page on “king” agrees with the etymology in my memory:a late Old English contraction of cyning “king, ruler” (also used as a title), from Proto-Germanic *kuningaz (source also of Dutch koning, Old Norse konungr, Danish konge, Old Saxon and Old High German kuning, Middle High German künic, German König).As does the Wiktionary page:From Middle English king, kyng, from Old English cyng, cyning (“king”), from Proto-Germanic *kuningaz, *kunungaz (“king”), equivalent to kin + -ing.Etymonline and Wiktionary agree here, but sometimes they don’t. When that happens, I’ll have to do a little more digging on the internet or in my physical dictionaries, with a hierarchy of sources I trust going from Etymonline down. (I would put the OED above Etymonline, but Etymonline takes its information from the OED as well as many other dictionaries, so it either has the same etymological information as the OED or better.)What I’d thought was correct, then. Good. As for rex, which I’m less sure on, Etymonline says:“a king,” 1610s, from Latin rex (genitive regis) “a king,” related to regere “to keep straight, guide, lead, rule,” from PIE root *reg- “move in a straight line,” with derivatives meaning “to direct in a straight line,” thus “to lead, rule” (source also of Sanskrit raj- “king”; Old Irish ri “king,” genitive rig).“Rex” is also an English word, borrowed directly from Latin. This is lucky. Etymonline is a source of English etymologies and English etymologies only. If I wanted the etymology of, say, Latin regalis, “regal”, it wouldn’t show up. I’d either have to consult Wiktionary, which has many languages’ etymologies, or I’d use Etymonline, but tricky this time.Etymonline’s got every English word’s etymologies, and it traces that etymology back to the source language. If a word in a language other than English happens to exist in some derived form in English, then I can go to that derived form for the word’s etymology. In the case of regalis, it exists in English as “regal”, so I can go to the etymology of “regal” for the etymology of regalis.Returning to the etymology of “rex”, where we can go straight to Latin this time, Wiktionary says:From Proto-Italic *rēks, from Proto-Indo-European *h₃rḗǵs (“ruler, king”). Cognates include Sanskrit राजन् (rājan, “king”) and Old Irish rí (“king”).Well, now we’ve got something of a disagreement. Etymonline says rex is from PIE *reg-, “to move in a straight line”, while Wiktionary claims it’s from PIE *h₃rḗǵs, “ruler”. Which one’s right? How do we know?Lucky again. Following the Wiktionary link on *h₃rḗǵs tells us it’s from *h₃reǵ-, also meaning “straight”, “direct”, “to move in a straight line”.It may seem there’s still a disagreement: Etymonline says *reg-, Wiktionary says *h₃reǵ-. But they’re the same word. Etymonline takes its etymologies from a few different sources, including the AHD. The AHD writes its PIE words in a simpler, friendlier-looking way that’s closer to how PIE would have been spoken in its later stages. For example, here’s Schleicher’s fable, the standard PIE story, in a form of that friendlier notation:Owis, jesmin wl̥nā ne ēst, dedork’e ek’wons woghom gʷr̥um weghontn̥s - bhorom meg'əm, monum ōk’u bherontn̥s. Owis ek’wobhos eweukʷet: K’erd aghnutai moi widn̥tei g’hm̥onm̥ ek’wons ag’ontm̥. Ek’woi eweukʷont: K’ludhi, owi, k’erd aghnutai dedr̥k'usbhos: monus potis wl̥nām owiōm temneti: sebhei ghʷermom westrom - owibhos kʷe wl̥nā ne esti. Tod k’ek’luwōs owis ag’rom ebhuget.*Reg-, with its basic Latin characters, fits nicely.Wiktionary does not do this. Wiktionary takes most of its Indo-European etymologies from the Brill Etymological Dictionaries, the most up-to-date comprehensive Indo-European etymological dictionaries. These use the most modern technical notation for PIE. Here’s the same text as above in this notation:h2áwey h1yosméy h₂wl̥h₁náh₂ né h₁ést, só h₁éḱwoms derḱt. só gʷr̥Húm wóǵʰom weǵʰed; só méǵh₂m̥ bʰórom; só dʰǵʰémonm̥ h₂ṓḱu bʰered. h₂ówis h₁ékʷoybʰyos wewked: “dʰǵʰémonm̥ spéḱyoh₂ h₁éḱwoms-kʷe h₂áǵeti, ḱḗr moy agʰnutor”. h₁éḱwōs tu wewkond: “ḱludʰí, h₂owei! tód spéḱyomes, n̥sméy agʰnutór ḱḗr: dʰǵʰémō, pótis, sē h₂áwyes h₂wl̥h₁náh₂ gʷʰérmom wéstrom wept, h₂áwibʰyos tu h₂wl̥h₁náh₂ né h₁esti.” tód ḱeḱluwṓs h₂ówis h₂aǵróm bʰuged.Wiktionary’s *h₃reǵ- is of this variety. It’s full of superscript characters and accents and h’s with numbers after them. It’s a lot uglier, but it’s more accurate, and so I prefer to use it in my answers.There you have it: Etymonline’s better for facts, Wiktionary for scope and technical bits. They complement one another wonderfully.And now we’ve got our two key etymologies:English king, from Old English cyning, from Proto-Germanic *kuningaz, common to all Germanic languages, all meaning “king”Latin rex, from Proto-Italic *rēks, from Proto-Indo-European *h₃rḗǵs, all meaning “king”, from *h₃reǵ-, “to move in a straight line”, “to be straight”, and by extension “to be just”, “to be correct”; as mentioned in Oscar Tay's answer to Why are right angles not called left angles?, the “rect” in “correct” is from *h₃reǵ-I could leave it at that. Fortunately - or not, depending on your opinion of my answers - I won’t. There are too many etymologies circling around *h₃rḗǵs to leave it at what the question’s after. What about Artu-rix? Raj? “Rich”? No, this seems like it ought to be an epic etymological wander-about. Rex and friends are scattered in odd places everywhere in Indo-European. I’ve never written about them on Quora. I may as well take this question to do so.You may have noticed that none of that research was original. There’s a lot of research, but all from other sources. I drew from my own knowledge, with Etymonline and Wiktionary for the specifics and to make sure I wasn’t making anything up.I don’t do original research. A linguist is someone who does do original research in linguistics, and since I don’t do that, I’m not a linguist. I’m a language teacher. That’s a totally different set of skills. As Gareth Roberts says in his answer to this question, I simply explain to non-linguists what linguists know.To answer your question, this is all available in ordinary books and websites and the like. Anyone could do what I’ve done to this point if they had a basic linguistics education. Hell, anyone could do this if they’d read enough books on etymology. (Which, I suppose, I have.) This is not that hard if you know where to look. I’ve just given out exactly where you need to look, what you need to do, and how to go about doing it.What comes next, however, is harder. It’s the most difficult part of writing any good answer on Quora. So far, I’ve made sure I have all the information I need. I’ve even started on explaining some of it. But I still have one very important piece to add:I have to make it interesting.That is: I have to make a complicated, confusing, and ultimately unimportant field, historical linguistics - historical linguistics! It almost sounds comic in its boringosity - interesting for an audience that potentially knows nothing about the field. They also have a limited attention span and a slew of much more interesting information at their fingertips. I have to, if only for a few minutes, hold that attention with only words and the odd picture.In the form of an essay.With a niche subject, remember. Comically niche.Where my main source of humour is an etymological dictionary.Without sacrificing facts for entertainment.Furthermore, I have to make sure they read the entire thing, learn everything I’ve lain out in said thing, remember that information, and then click the little blue button that says they’ve enjoyed it. Bonus points if they actually enjoyed it. Bonus bonus points if they laugh, or have their day otherwise improved, or share that information with a friend.This is - if you’ll forgive the hyperbole - pretty hard. Most people can’t do this part. Anyone can read a dictionary, but how many of us can make a friend want to read the dictionary? Or an essay about the dictionary? How about tens of thousands of strangers? How about making them enjoy the experience?Yet - if I may be permitted this - I am horribly, disgustingly lucky. I’ve got a thing wrong with my noggin that, among other things, makes this part intuitive. I can make students enjoy learning grammar. On Quora, I can make twenty thousand of them enjoy it at once. And I can get them to laugh while they learn it.I can imagine no higher compliment than for you, O Matías Damián, O original poster of this question, to say “interesting research”. “Research”? Sure, anyone can do that. I am honoured to be able to do interesting research.And, because I can explain anything in an interesting way, I can also explain how to explain anything in an interesting way in an interesting way. It is, I hope, contagious. Here’s how it works.I have my information: the etymologies, as stated above. Now we have to make people care about them. You may ask, how do you make them care about etymologies? But you don’t ask this yet. You want your audience to begin the story with curiosity. On Quora, all stories begin with a question, literally. That’s good. That creates curiosity. That tells your reader what to wonder, and what they’re going to be learning.You have to keep this curiosity going. If you start the story with the etymology as it is, that may answer any curiosity generated by the question, but if you want to make sure they keep reading, you have to create a hook.A hook. Not clickbait. A hook. Clickbait screams for attention; a hook merely suggests it. If you were in a market and had to choose between one vendor yelling at you to buy their product and another, quiet, friendly vendor with an interesting product you wanted to know more about, which would you go to?You want to make sure the question, or if not on Quora then the title, is friendly. Our question, again, is this:How do you say "king" in Proto-Indo-European, and does modern English have a cognate to it?The average person on Quora won’t know two of these words: “Proto-Indo-European” and “cognate”. A friendly question (or title) is one your audience will understand. How can we rephrase this question to be friendly? “Proto-Indo-European” is a proper noun, a name for a thing, and there’s no way to easily rephrase that. We’ll get to how to fix “Proto-Indo-European” in a moment.“Cognate” is easier. “Cognate” means “a word that’s related to another word”. This is an easy concept with an easy layman’s translation. We can edit this question to its friendlier variant:How do you say "king" in Proto-Indo-European, and does modern English have any words that are related to it?And there we are!: How do you say "king" in Proto-Indo-European, and does modern English have any words that are related to it?. To someone who doesn’t know what Proto-Indo-European is, the curiosity will translate to this: “I wonder how you’d say ‘king’ in this language, and if English has any words that are related to it.” Still not great, but this is a question someone else has asked, so we can’t mess with it more than clarification requires.We’re fighting with lukewarm curiosity now. There’s a few ways you could go about this. One method I used early on on Quora was to recount an abbreviated version of the story of Proto-Indo-European, as in this answer or this one, as an extension of the hook.It gets tedious if I have to do that for every mention of the language. Instead, what I’ve done is written a reference answer for PIE, Oscar Tay's answer to How do we know that a Proto-Indo-European language really existed? What is the evidence?, which I link to as early on in the answer as possible.Virtually all my followers, who form much of the readership of any given answer and the majority of views and upvotes and whatnot early on in the answer’s publication lifetime, have read this answer and/or have a good-enough understanding of PIE, so for them the question in question is a very friendly one indeed. For everyone else, I’ll mention Proto-Indo-European first thing or nearly first thing in the answer, link to that answer, and then carry on.And now the question, with some help from the answer, is friendly.How should we do the hook? Remember, not like clickbait. A clickbait introduction would be like this:Here’s ten word origins you won’t believe! [Include short paragraphs, gratuitous emboldened text and capital letters, and tangentially related stock photographs for each word.]Don’t do this. You’re writing a story, not a tabloid or BuzzFeed article, so write it like a story. The hook should give some hint about what this story is about while not answering any questions just yet. Or, if it does answer the question, then in a somewhat mysterious way; or, if it answers the question quite clearly, then it creates more questions.This time, I’ll begin this one like this:How do you say "king" in Proto-Indo-European, and does modern English have any words that are related to it?We have perhaps too many.Short, pithy, sort-of-answers the question in a vague enough way, and, most importantly, it builds curiosity. It tells the reader that this will be an etymological soup, with words connecting to one another in unexpected and seemingly impossible ways, but saves them. It’ll be good, it says, but you’ll have to read it.Lead into an introductory paragraph. This is where I’d put the link to the Proto-Indo-European answer. If the hook’s good enough, this is also where you’d put the boring expository information. Answer some easier part of the question, correct assumptions, preface one part or another, et cetera.The word is *h₃rḗǵs. It meant “king” or “ruler”, but not in the way you’d imagine your typical monarchy. The Proto-Indo-Europeans were a semi-nomadicNo. Not that. Say “semi-nomadic” if you’ve got to, but say something cooler if you can.The word is *h₃rḗǵs. It meant “king” or “ruler”, but not in the way you’d imagine your typical monarchy. The Proto-Indo-Europeans lived on horseback, herding animals from one bit of grassland to the next, building new shelters along the way or staying as another tribe’s guests for a time.Much better. You can squeeze more information into the flow there. Notice also that I’ve linked to other answers instead of explaining their matter again: the one about the laryngeals at the “3”; the one on the guest-host relationship at the end. (I’ve been told this has the side effect of leading to a rabbit-hole/web of answers one can fall into.)The word is *h₃rḗǵs. It meant “king” or “ruler”, but not in the way you’d imagine your typical monarchy. The Proto-Indo-Europeans lived on horseback, herding animals from one bit of grassland to the next, building new shelters along the way or staying as another tribe’s guests for a time. Different tribes were ruled by different tribal chiefs, and it’s these people who were called *h₃réǵes.I write in a style somewhere between formal/academic and informal/conversational, tending to one side or the other depending on the situation. I’ll switch from one to another from one paragraph to another. You don’t have to do this, but I find it’s a good balance between talking about a subject seriously and sounding like a human.Now that the boring expository first paragraph is out of the way, we can move to the interesting bits.Latin took *h₃rḗǵs, threw out the *h₃, and shortened the remaining regs to rex, “king” in its modern sense.This one’s the Latin branch. I usually start etymology stories with the Latin branch because it gives us the most words, and they tend to sound the same. The etymologies grow weirder as the answer progresses.Latin took *h₃rḗǵs, threw out the *h₃, and shortened the remaining regs to rex, “king” in its modern sense. It then took rex and pulled all sorts of new words out of it. The time when a king rules was his regnum, or règne in French, or “reign” in English; the rule itself is his regimen, which became “regimen” and “regime” and “realm”; the place where he ruled was a regionem, and that regionem lost its -em and fell on down as “region”. And the king himself? Regalis, “regal”, of course!The etymological derivation phrases never repeat. No “and this word comes from this word, and this word comes from this word, and this word comes from this word”; it goes “and Latin took this word and turned it into this word, and this word became this word, and this word was this word and then that word”. The sentences vary. If it’s the same sentence again and again, it’s not interesting.The best places to find these words are, again, Etymonline and Wiktionary, which have pages devoted to lists of words from a shared root. One more again, what I’m doing is not new research, but taking old, boring, dusty research and making it interesting for an audience who would not be able to read said research in the original.Go a further back from rex to *h₃rḗǵs and beyond and you’ve got *h₃reǵ-, “straight”, “to move in a straight line”, and by extension “to be just”, “to be correct”, as a king ought to be. As covered here, there’s plenty more from that route: “right” and “right” (the other kind) and the “rect” in “correct”; just one from that root in Latin is regula, “direct”, whose definition is echoed in all its descendants. From regula come “regulate”, and “rule”, and “ruler”, and “ruler”, even “rail”: tracks to direct a train, to rule where it goes.This one’s too long. Four to five, maybe six lines to a paragraph on Quora is what I try to stick to; a paragraph up, I left out regina, “queen”, because it would make the paragraph too long, it being already at six lines. This paragraph directly above has seven. Is there any way to break it up? If there isn’t, then we can leave it, because it’s not too far over the limit; but there is a logical spot where we can do this:Go a further back from rex to *h₃rḗǵs and beyond and you’ve got *h₃reǵ-, “straight”, “to move in a straight line”, and by extension “to be just”, “to be correct”, as a king ought to be. As covered here, there’s plenty more from that route: “right” and “right” (the other kind) and the “rect” in “correct”.Just one from that root in Latin is regula, “direct”, whose definition is echoed in all its descendants. From regula come “regulate”, and “rule”, and “ruler”, and “ruler”, even “rail”: tracks to direct a train, to rule where it goes.Flows better, looks nicer, less intimidating. More an aesthetics choice than a writing one, but aesthetics help any writing. Pictures every few paragraphs are great if you can fit them in and if they make sense. I forwent them in this answer because they wouldn’t make sense, but in others, where they do, it helps them be prettier.Away from Latin went *h₃rḗǵs as well, off to Germanic and Celtic and other branches of the family tree. In Germanic, it turned to *rīkiją, “kingdom” or “authority”, then Old English rīċe; stuck on the end of a word, it was the realm belonging to that word, as in “bishopric”. German, on the other hand, turned it to reich.Write it in storytelling-esque prose, too. Academics read papers. People read stories. Find some way to fit your subject into a story, whether or not there’s a real story there. Languages and words are not themselves alive, but I treat them as such and give them thoughts and opinions and wants and hopes and goals and so on, as for the suffix -ish in this answer.One way to make something into a story is to find a story already present within it and tell it instead. For instance, in Oscar Tay's answer to What is Linear B syllabary?, instead of saying “The Linear B syllabary was a writing system used predominantly in Crete in the latter half of the second millennium BC”, which sounds like an answer to an exam question, I told the story of its decipherment and the people involved. It’s a lot more interesting, fun, and likely to be remembered than stating just the facts of it.Then there was Proto-Germanic *rīkijaz, “rich” and “powerful” and “mighty” - kingly qualities, for sure. English whittled *rīkijaz to rīċe again, removed the e, swapped round some consonants, and made itself “rich”.That’s just a pun. I like puns.Then there was Proto-Germanic *rīkijaz, “rich” and “powerful” and “mighty” - kingly qualities, for sure. English whittled *rīkijaz to rīċe again, removed the e, swapped round some consonants, and made itself “rich”. The Romance languages liked that idea, so they looted the Germanic word-hoard and made off with ric or rico, as in Spanish Puerto Rico, literally “rich port”.More wordplay. Keeps it light while still being interesting and not sacrificing information for entertainment.Hardly content on staying within the realm of theAnother pun. I hide a lot of etymological puns in my answers. “Stellar”/“disaster” is my favourite.Hardly content on staying within the realm of the people-kings, *h₃rḗǵs ran over to Proto-Germanic again, dropped in as *rekô, and merged with *anadz, “duck”, to become *anadrekô: king of the ducks, manliest of the ducks, the *anadrekô, the male duck. Minus the first two syllables, we’ve still got it as “drake”.Making fun of the etymology. Always entertaining.In Celtic, our royal root decided it’d like to be rix. If you were an especially impressive king, with the strength of a bear, you might be known as the “bear-king”, artu-rix: Arthur.Wait. Is this true? I’ve heard it from somewhere, but I don’t know where. The name shows up on neither etymology page. I should check this first. As it turns out, this particular etymology is controversial and uncertain, so it wouldn’t do well to include it in the answer. Check to make sure your information is true, however much you’d like to include it.I’d also like to include mention of raj in this answer, but it would ruin how it works. If I added a paragraph on it at the end, since “raj” is the only English word from this root via the Indo-Iranian branch, it’d made the story anti-climactic. I need to lead into a section with more words than one in it.And I’ve got it: names. The root *h₃rḗǵs appears in name after name after name, given how cool it makes one’s name sound if you know what it means. “Richard” is boring and normal, unless you know it’s from *Rīkaharduz, literally “brave king” but with all the cultural associations of naming your kid “Emperor Fearless”.I may as well make that the paragraph. Some editing later, you’ve got this:And content less softly yet with just regular words, the root *h₃rḗǵs ran through name after name after name, given just how cool it makes one’s name sound if you know what it means. “Richard” sounds boring and normal - unless you know it’s from Proto-Germanic *Rīkaharduz, literally “brave king” but with all the cultural associations of naming your kid “Emperor Fearless”.Or Eiríkr, the Viking name, come to modern-day English as “Eric”. Definition: eternal ruler. “Henry”, too, from *Haimarīks, “king of the home”; and “Fred” from *Friþurīks, “king of peace”; and “Derek” from *Þeudarīks, “king of people”; and finally “Raj”, from Sanskrit rāj, from the same root, all meaning simply “king”.And I even managed to fit “raj” in there!Normally, this is where I would tie together all the loose questions. This answer isn’t like that. This is an etymology-abouting answer, not an essay on some piece of linguistic history. I can still answer those questions here, if there are any. If you’re smart, like I’m not, you can think ahead and leave some question to end on here.Luckily, I forgot the first etymology, so I can put it in here:But what about English “king”? It’s related to the word all Germanic languages use, from German König to Dutch koning to Finnish kuningas, wherein Finnish is not a Germanic language but stole the word anyway. That word in Proto-Germanic was *kuningaz. (Ten points to Finnish for preserving it so well while the languages it belonged to in the first place tossed syllables away as they so pleased.)*Kuningaz is not from the same root as rex, as you might have guessed. Rather than deriving from some word for justness or correctness, it comes from a very human word, which is to say it comes from a word more or less meaning “human”: *kunją, ancestor of our English “kin”. Using a since-lost sense of “-ing” to mean “belonging to”, *kunją-ingaz became *kuningaz, “of the people” or “ruler of the people”.And add some vaguely meaningful line to the end to finish the content:The king, according to the Proto-Germans, belongs to the people he rules.Finally, the sign-off:Thanks for asking!The final answer is available here: Oscar Tay's answer to How do you say "king" in Proto-Indo-European, and does modern English have any words that are related to it?For more essay-type answers like this one, I’ll tie everything back with a summary, so a.) no one can say I didn’t answer the question, b.) if they’re not interested in the content they can just read the summary, and c.) if I made it too confusing, then to reiterate the point and make sure we’re all thinking the same thing. Like so:To answer your question, I do research for my answers - the resulting answer to that question is only 750 words, and my longer ones approach 5000, so scale that up accordingly and you’ve got however long that takes - but none of it is original. Anyone could do the research I did. If you want reading material for historical linguistics, I’ve got a list here.Then summarize anything in the answer you may have said that was not directly related to the question:That isn’t really the point. My job is not to expand the boundaries of human knowledge, but to translate the sliver of human knowledge occupied by linguistics into something regular humans can understand, learn from, and enjoy.And add some vaguely meaningful paragraph or two to the end to finish the content:To look at it another way, anyone could do the research I did. If you’re the kind of person to go on Quora, you’ve probably got your own field of interest and/or expertise, and you can do the research in the way I’ve done it in linguistics for your chosen field.If you can make what you write interesting, and explain it well - and this is difficult, so it may take quite a bit of practice - then you can improve the world as much as any scientist can, just differently. If you haven’t tried writing about your interests on Quora before, or have but haven’t had success with it, I’d encourage you to try again with whichever of these nuggets of writing advice you’ve found helpful.Finally, the sign-off:Thanks for asking!
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What are some useful tech hacks for everyday use?
Here are some I find useful and fun:Instead of "Ctrl, Alt & Del", press "Ctrl, Shift and Escape" to get straight to the task-manager in Windows.If you want to download a Youtube video, just add "ss" to the URL between www. and Youtube.The program DeTune will transfer all of the songs from an iPod/iPhone and put them on you computer.If you search "do the harlem shake" on YouTube the page itself will do the harlem shake.(This one's just for fun)The "Hola Unblocker" extension on Google Chrome will allow you to access to UK version of Netflix; thus unlocking many more shows and movies.Need to focus on studying? Screen Time is an app that lets you limit the time you use on your iphone or ipad. Set the time, press start and when it expires it closes whatever you were doing.If you play YouTube videos through Safari you can still listen to them with your phone's screen turned off.Replace the "en" in a Wikipedia link with "simple" to strip away the complex and mostly irrelevant information on the page.Accidentally erase something you just typed on your iPhone? To undo that, just shake it!1. Go to Google and type in "50 most popular women" 2. Click on the first link 3. Check out #7 xDTo move frame by frame on Youtube, pause the video and then use J or L to go backward or forward respectively.Just in case something caught your attention then I am Rohan Bhatia.You could have gone anyway.
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Why do most fans say that Aegon is Blackfyre?
First off, before you get to the “Blackfyre” part, you need to get to the “Aegon isn’t actually Aegon” part.Way back in the second book, Dany sees a vision of a cloth dragon on a pole, being cheered by the crowd. This is afterward referred to as the “mummer’s dragon.” “Mummer’s [blank]” is used in the story to denote something fake, e.g. “mummer’s tears” are false tears. In other words, we’re being told to look out for a “fake dragon.” Not necessarily not a dragon, but a dragon that is pretending to be something it isn’t. The “false dragon” concept pops up again in the fifth book, with Moqorro telling Tyrion he sees dragons “true and false.”The Aegon cover story itself has some holes in it. For example, where did Varys find a Valyrian lookalike baby in Pisswater Bend on short notice? Why was only Aegon smuggled out? It’s lucky the baby’s face was destroyed — this would in theory prevent face-based identification — but that’s not something Varys could have foreseen until after the fact. The Pisswater baby was paid for with “Arbor gold,” which crops up repeatedly in the story to symbolize lies and deceit.In so many words, “Aegon is actually Aegon” is something you’d be a fool to take at face value.Now we can proceed to the Blackfyre part.If Aegon isn’t Aegon, narratively speaking, he would still be someone. To hit on the Blackfyre angle, consider all of this:GRRM has invested considerable ink in his supplementary material — namely the D&E stories and the WOIAF — in the Blackfyre backstory. Not only is this world-building in general, but it could also imply that this is done to set up a Blackfyre angle later in the main story.Blackfyre mentions are peppered in the main series, but pick up in frequency and specificity in the fifth book — the one in which we first see Aegon. There are also Blackfyre info-drops in out-of-context places (for example, we get one in Theon’s WoW chapter from Stannis, in a scene without any Blackfyre connections at all). The Blackfyres become more central backstory figures in correlation with Aegon’s own plotline.The dragon sign on the Quiet Isle can’t be unseen once you’ve seen it. A dragon head sign from an inn washes out black and comes back red. A Blackfyre “washes out” and comes back disguised as a Targaryen (a “red” dragon).The Golden Company, famous for never breaking a contract, breaks one to fight for Aegon. To paraphrase, “Some contracts are written in ink, others are written in blood.”We’re specifically told the Blackfyres are extinct in the male line, which means that female-line Blackfyres are still out there. So who are they?Once you consider the Blackfyre angle, Illyrio’s role in this all makes sense. His attitude toward Aegon is already much cosier than a detached political operator’s would be. We know he met his wife Serra in Lys. We know Varys came from Lys. We know Jon Connington spent part of his exile in Lys. We know Aerion Brightflame was exiled in Lys (the idea that the Brightflames joined up at some point with the Blackfyres is popular). We know Maelys Blackfyre killed his cousin Daemon to take over the Golden Company, and can guess based on his name that Daemon was a senior-line Blackfyre, giving his descendants the strongest Blackfyre-based claim. We have these discrete pieces that add up to something. So what is that? My guess: Serra and Varys are related to each other in some way and in turn are related to the deposed Daemon. After Maelys took over, he sold both of them into slavery to remove them as dynastic threats, or they fled him and ended up in slavery that way. Varys having Blackfyre blood explains why someone would want to use him in a blood magic ritual. Varys met up with Illyrio and convinced him to help his sister/relative Serra. Illyrio fell in love with her, married her, Aegon is their son (the Blackfyre scion in the female line), and Illyrio and Varys are sneaking Aegon, a Blackfyre, in through the back door disguised as a Targaryen — the Targaryen with, if he’s real, the best overall claim to the Iron Throne.Ta da.
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What are signs of a covert narcissist?
They need support your emotional support, all the time.They never smile with their eyes. It’s that creepy, dead eyed smile.When you ask them to support you emotionally their bodies become stiff and their eyes betray an almost scare look.Once they get used to you, they talk all the time. They talk about what they have, what they are going to get, what they want to do, where they’re going. Once they are done they look at you as if to say “ok agree.”They will let you talk about something you are passionate about, let you finish, and then start talking about something totally different. Or cut you off.They never put themselves in a position that could leave them looking less than you. They do not do anything that they know you can do better. They do not try anything they could lose at.Their idea of fun is bringing up endless Youtube videos with you, while watching you to see if you laugh at them. They will be watching you more than the video. You will feel the great need they have of you to laugh.They are very often positioning themselves above you by offering to help you. Yet when you ask for this help they will clearly be put off by it.They love to bring you along to things that they are very proficient in, so they can feel better than you.They love to find things wrong with whatever you are doing, but in a very subtle way.They love to minimize parts of your story when you are talking about something good happening to you.If you watch them when they aren’t paying attention you can see their real dead looking eyes. If they catch you noticing this, the look of fear is intense.They copy people’s ticks, laughs, movements, behaviors, tones, words, and phrases.If you pay attention they will attempt to copy your movements as you are doing them.They will take a story that was well received by your circle of friends, tell the same story to you, with them in your place, with a straight face.They attack you when they feel less than you.If you are not paying attention to them they will try to annoy you by touching, taking your things, or crowding your space.You will often catch them just staring at you. they will deny this.The look on their face when you are receiving positive attention from others is remarkable. I wanna say it’s a look of awe, but at the same time of sadness.If they hear of people praising you, if they tell you, it will sound like they don’t believe what they were hearing.Any praise they give you sounds forced.They become animated at your misfortune.They have a hard time not kicking you when you are down.A lot of their time is spent gossiping about other people.Upon realizing you are who they want, they will make a show of being generous by giving you something that you might want, but didn’t ask for.Their emotions are off. They sound unsure when they speak with a lot of emotion.Speaking joyously about anything will leave them with a face like they just smelled a fart.Everything is about power with them. They are constantly fighting Their feelings of inferiority by subtly positioning themselves above others. Any attempt on your part to even the playing field is usually met with silence and blank stares.
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What is 'exact inference' in the context of Probabilistic Graphical Models?
(This is the second answer in a 7 part series on Probabilistic Graphical Models ('PGMs').)Let’s define ‘inference’:The task of using a given PGM of a system, complete with fitted parameters, to answer certain questions regarding that system.For such questions, an exact answer exists. Exact inference algorithms calculate such answers exactly. It's an extremely difficult, well defined, class of problem. The algorithms designed are, consequently, mind-bending.Though, one realization relaxed me in the face of these precise beasts: they are all built to exploit factorization. You know, the act of turning [math]ac + ad + bc + bd[/math] into [math](a+b)(c+d)[/math]. Knowing this doesn't make these algorithms simple (they aren't) but it certainly demystifies them.Another trick is caching calculations for re-use. The monumental calculations involved often have a fair amount of redundancy - keeping track of past work substantially reduces compute time.But first, let's do a short review.Refresher (Skip this if you’ve read answers 1!)In the first answer, we discovered why PGMs are useful for representing complex system. We defined a complex system as a set, [math]\mathcal{X}[/math], of [math]n[/math] random variables (‘RVs’) with a relationship we'd like to understand. We assume there exists some true but unknown joint distribution, [math]P[/math], which govern these RVs. We take it that a 'good understanding' means we can answer two types of questions regarding this [math]P[/math]:Probability Queries: Compute the probabilities [math]P(\mathbf{Y}|\mathbf{e})[/math]. This means: what is the distribution of [math]\mathbf{Y}[/math] given we have some observation of [math]\mathbf{E}[/math]?MAP Queries: Determine [math]\textrm{argmax}_\mathbf{Y}P(\mathbf{Y}|\mathbf{e})[/math]. That is, determine the most likely values of some RVs given an assignment of other RVs.(Where [math]\mathbf{E}[/math] and [math]\mathbf{Y}[/math] are two arbitrary subsets of [math]\mathcal{X}[/math]. If this notation is unfamiliar, see the 'Notation Guide' from the first answer).The idea behind PGMs is to estimate [math]P[/math] using two things:A graph: a set of nodes, one for each RV in [math]\mathcal{X}[/math], and a set of edges between them.Parameters: objects that, when paired with a graph and a certain rule, allow us to calculate probabilities of assignments of [math]\mathcal{X}[/math].Depending on these two, PGMs come in two main flavors: Bayesian Networks ('BNs') and Markov Networks ('MNs').A Bayesian Network involves a graph, denoted as [math]\mathcal{G}[/math], with directed edges and no directed cycles. The parameters are Conditional Probability Tables ('CPDs' or 'CPTs'), which are, as the naming suggests, select conditional probabilities from the BN. They give us the right hand side of the Chain Rule, which dictates we calculate probabilities this way:[math]P_{B}(X_1,\cdots,X_n)=\prod_{i=1}^n P_{B}(X_i|\textrm{Pa}_{X_i}^\mathcal{G})\tag*{}[/math]where [math]\textrm{Pa}_{X_i}^\mathcal{G}[/math] is the set of parent nodes of [math]X_i[/math] in the graph.A Markov Network's graph, denoted as [math]\mathcal{H}[/math], is different in that its edges are undirected and it may have cycles. The parameters are a set of functions (called ‘factors’) which map assignments of subsets of [math]\mathcal{X}[/math] to nonnegative numbers. Those subsets, which we'll call [math]\mathbf{D}_i[/math]'s, correspond to complete subgraphs of [math]\mathcal{H}[/math]. We can refer to this set as [math]\Phi=\{\phi_i(\cdots)\}_{i=1}^m[/math]. With that, we say that the Gibbs Rule for calculation probabilities is:[math]P_M(X_1,\cdots,X_n) = \frac{1}{Z} \underbrace{\prod_{i = 1}^m \phi_i(\mathbf{D}_i)}_{\text{we call this }\tilde{P}_M(X_1,\cdots,X_n)}\tag*{}[/math]where [math]Z[/math] is defined such that our probabilities sum to 1.Lastly, we recall that the Gibbs Rule may express the Chain Rule. That is, we can always recreate the probabilities produced by a BN's Chain Rule with an another invented MN and its Gibbs Rule. Essentially, we define factors as those that reproduce looking up in a BN's CPDs. This equivalence allows us to reason solely in terms of the Gibbs Rule, while assured that whatever we discover will also hold for BNs. In other words, with regards to inference, if something works for [math]P_M[/math], then it works for [math]P_B[/math].Whew, OK, so, now... what is happening?To repeat, we are concerned with the task of inference. That task is to calculate those previously mentioned queries, given we have our graph and it's parameters (CPDs for BNs and factors for MNs) determined. In this answer, we discuss algorithms that do so exactly.The Gibbs TableThis whole time, I’d like you to picture something. That is the 'Gibbs Table' - it lists out all possible assignments of [math]\mathcal{X}[/math] and the product of factors associated with each.Let's consider an example. Let's say we have the system [math]\mathcal{X}=\{C,D,I,G,S\}[/math], where each RV can take one of two values. Let's also say we have these factors determined:[math]\begin{align} &\phi_1(C) \\ &\phi_2(C,D) \\ &\phi_3(I) \\ &\phi_4(G,I,D) \\ &\phi_5(S,I) \\ \end{align}\tag*{}[/math]By 'determined', we mean we know the nonnegative number each of these functions map to for any assignments of their RVs. For example, we might have:Now we can picture the Gibbs Table:In general, this table is exponentially large, so we want to think about it, but never actually write it down.Conditioning is just filtering the Gibbs TableWith this view, let's consider our probability query [math]P_M(\mathbf{Y}|\mathbf{e})[/math]. Bayes Theorem tells us: [math]P_M(\mathbf{Y}|\mathbf{e})=\frac{P_M(\mathbf{Y},\mathbf{e})}{P_M(\mathbf{e})}[/math]. Both the numerator and the denominator concern rows of the Gibbs Table where [math]\mathbf{E}=\mathbf{e}[/math]. Specifically, the denominator is the sum of all such rows and the numerator refers to a partitioning of these rows into groups where any two rows are in the same group if they have the same assignments to [math]\mathbf{Y}[/math]. I'll make this more clear with an example.But first, we should recognize a broader point. The act of conditioning makes all rows that don't agree with our observation irrelevant - we can throw them away! Said differently, any time we deal with a factor that involves [math]\mathbf{E}[/math], we plug in the [math]\mathbf{e}[/math]-values, so we don't care about the factor outputs for non-[math]\mathbf{e}[/math]-values. This means we can think of defining a new MN with [math]\mathbf{E}=\mathbf{e}[/math] baked in. This is called 'reducing' the Markov Network. The result is a new graph (call it [math]\mathcal{H}_{|\mathbf{e}}[/math]) where we delete the [math]\mathbf{E}[/math] nodes and any edges that involve them. Also, we get a new set of factors, (call it [math]\Phi_{|\mathbf{e}}[/math]) which are our original factors, but with the [math]\mathbf{e}[/math]-values fixed as inputs. The normalizer in this reduced network, [math]Z_{|\mathbf{e}}[/math], is just [math]P_M(\mathbf{e})[/math].So just think: conditioning means we filter the Gibbs Table to where [math]\mathbf{E}=\mathbf{e}[/math] and those rows can be considered their own MN.But I still owe you an example.An example of naively calculating [math]P_M(\mathbf{Y}|\mathbf{e})[/math]Let's say [math]\mathbf{Y}=\{D,I\}[/math] and [math]\mathbf{E}=\{G\}[/math], where we are dealing with the observation [math]G=g^0[/math]. So our probability query is 'What is the distribution of [math]\{D,I\}[/math] given we observe [math]G=g^0[/math]?' In other words, we want to calculate [math]P(D,I|g^0)[/math].Let's start simple and consider just the probability [math]P_M(d^0,i^1|g^0)[/math]. Following the previous argument, we start by filtering the table to rows where [math]G=g^0[/math]. If we sum these rows, we'll get our Bayes Theorem denominator, [math]P_M(g^0)[/math]. Of these rows, if we sum those where [math]D=d^0[/math] and [math]I=i^1[/math], we'll get our numerator, [math]P_M(d^0,i^1, g^0)[/math] and have all we need to calculate [math]P_M(d^0,i^1|g^0)[/math]. With a manageably sized table, you could do this in Excel in 60 seconds.From this, we can tell that the core process is filtering to rows that agree with some assignment and summing up the unnormalized probabilities. So lets refer to this as a 'row sum' function [math]rs(\cdots)[/math]. Specifically, [math]rs(d^0,i^1, g^0)[/math] would give us that numerator and [math]rs(g^0)[/math] would give the denominator. So we calculated:[math]P_M(d^0,i^1|g^0) = \frac{rs(d^0,i^1, g^0)}{rs(g^0)}\tag*{}[/math]But there's an issue. These summations could be over an exponential number of rows, making the summation impossible. Well then, we’ll have to be clever!The first easy compute saver.Ultimately, we care about all probabilities for all assignments of [math]D[/math] and [math]I[/math], not just [math]D=d^0[/math] and [math]I=i^1[/math]. In other words, we want to calculate all 4 of these:Well, because an observation of [math]g^0[/math] must be associated with one and only one assignment of [math]D[/math] and [math]I[/math], then those assignments partition the rows associated with [math]g^0[/math] into 4 groups. So we realize that those numerators sum up to the denominator, which is the same in all cases. That is:[math]rs(g^0) = \sum_D \sum_I rs(D,I, g^0)\tag*{}[/math]This means we never need to calculate [math]rs(g^0)[/math] directly. Just calculate the numerators first and add them up at the end to get the denominator.So, on a general note, we just need the ability to calculate this row sum function for any given assignment of whatever subset of [math]\mathcal{X}[/math]. Then we can answer the probability queries.Now, let’s see the big guns.Factoring and caching.We've reduced the problem to computing [math]rs(\cdots)[/math] efficiently. As an arbitrary example, let's consider the rows of [math]rs(c^1)[/math]:What a headache! No matter - we have algorithms for such ugliness.To get there, let's count the simple add/multiply calculations involved in calculating this naively. We have 4 multiplications per row and 15 summations, giving us [math]4 \times 16 + 15 = 79[/math] calculations. In the non-toy case, this number would be exponentially frightening. Hmm, can we get it down? Certainly!First of all, why are we multiplying [math]\phi_1(c^1)[/math] for each row when it's the same thing each time? If we factor that out, we'd save 15 calculations.What else can we do? Let's consider summing the first 2 rows (with [math]\phi_1(c^1)[/math] pulled out). That's this guy:This involves 7 calculations. Hmm, but if we look closely, most of this is the same - only [math]\phi_5(\cdots)[/math] changes. Oh, duh, it factors! It's the same as:Which is only 4 calculations. Nice!This silliness is to highlight something painfully important: factoring sums of products into products of sums saves us compute - big time!Now, let's see caching in action. Let's pretend we are back to naively calculating this sum. First, we multiply out the first row. We are right about to begin calculation then second row when we realize: we've already calculated the first four terms of that product! Ah, so if we could cache results and identify chunks we've already seen, we can save more compute.These two ideas, factoring and caching, sit at the foundation of every exact inference algorithm. So, I believe we're ready to discuss one.Variable Elimination (‘VE’)There's one simple factoring trick that enables all of VE. Since it's important, I'll write it out generally. Let's say we'd like to calculate [math]rs(\mathbf{x})[/math], corresponding to an assignment for some [math]\mathbf{X}[/math] from [math]\mathcal{X}[/math]. Let's also say that [math]\mathbf{Z}[/math] is all the other RVs not in [math]\mathbf{X}[/math] and [math]Z_0 \in \mathbf{Z}[/math] is a variable we've picked out. Further, let's also pretend there are only two factors, [math]\phi_1(\mathbf{D}_1)[/math] and [math]\phi_2(\mathbf{D}_2)[/math], where [math]\mathbf{D}_1 \cup \mathbf{D}_2 = \mathcal{X}[/math], and [math]Z_0[/math] is not in [math]\mathbf{D}_2[/math]. Then we'd like to calculate:[math]rs(\mathbf{x}) = \sum_{\mathbf{Z}} \phi_1(\mathbf{D}_1) \phi_2(\mathbf{D}_2)\tag*{}[/math](It's hidden from this notation, but if an [math]X_j[/math] from [math]\mathbf{X}[/math] is in one of these [math]\mathbf{D}_i[/math] sets, then that RV takes on the [math]x_j[/math] assignment in [math]\phi_i(\mathbf{D}_i)[/math]. All other RVs ([math]\mathbf{Z}[/math]) are summed over.)Since [math]Z_0[/math] is not in [math]\mathbf{D}_2[/math], then the algebra gods permit:[math]\begin{align} rs(\mathbf{x}) =& \sum_{\mathbf{Z}} \phi_1(\mathbf{D}_1) \phi_2(\mathbf{D}_2) \\ =& \sum_{\mathbf{Z}\setminus \{Z_0\}} \Big[\phi_2(\mathbf{D}_2) \sum_{Z_0} \Big[\phi_1(\mathbf{D}_1) \Big]\Big]\\ \end{align}\tag*{}[/math]This means you may 'push' summation signs inside product signs, so long as the expressions to the left don't involve the RV of that summation sign.(Going forward, I'll drop those big brackets. They're to demonstrate that the right summation is a term inside the left summation. It's not the product of two summations.)The thing to look out for is that [math]\sum_{Z_0} \phi_1(\mathbf{D}_1)[/math] is a new function that doesn't involve [math]Z_0[/math]. What remains is a function of [math]\mathbf{D}_1 - \{Z_0\}[/math]. In effect, we've 'eliminated' [math]Z_0[/math]. After the summation is done, you could say we've replaced [math]\phi_1(\cdots)[/math] with a new, slightly simpler, factor. We'll use [math]\tau_{Z_0}(\cdots)[/math] to refer to the factor you get when you eliminate [math]Z_0[/math]. We'll call them 'elimination factors'.Now you might be thinking: "Uh, but there aren't two factors in general." Yes, but the principle still works. Just think of [math]\phi_1(\mathbf{D}_1)[/math] as the product of all factors that involve [math]Z_0[/math] and [math]\phi_2(\mathbf{D}_2)[/math] as the product of all those that don't.With that, you can do this repeatedly, pushing sums inside products, eliminating everything in [math]\mathbf{Z}[/math] and obtaining your answer.But what about caching? That plays a role, but it'll be easier to see in an example.An example of Variable EliminationLet's compute that familiar [math]rs(c^1)[/math] guy, which is the sum of the rows of the above Gibbs Table. It may be written:Let's arbitrarily decide to eliminate [math]S[/math], then [math]I[/math], then [math]D[/math] and then [math]G[/math] (this is called the 'elimination order'). At this point, I blindly follow the rule for manipulating symbols: push sums inside products until all factors to the right involve the RV of that summation sign. It's easy - give it a shot. You'd get:Now let's point out all those elimination factors.(The omission of '()' in front of [math]\tau [/math]means this function is a single number.)To see why caching helps, imagine coding this up. You tell the computer that so-and-so elimination factor is a sum across this-and-that factors, just as you see in the above cascade. Then you say, 'OK computer, calculate [math]\phi_1(c^1) \tau_G[/math]' and it begins calculating this recursive mess. But in doing so, it'll sometimes come across calculations it's seen before. For example, consider that'll we'll have to calculate [math]\tau_I(g^0,d^0)[/math], [math]\tau_I(g^0,d^1)[/math], [math]\tau_I(g^1,d^0)[/math] and [math]\tau_I(g^1,d^1)[/math]. All of these rely on knowing the numbers [math]\tau_S(i^0)[/math] and [math]\tau_S(i^1)[/math]. Well, the moment we figured out [math]\tau_I(g^0,d^0)[/math], we must have calculated those numbers. So when it comes to the next calculation involving [math]\tau_S(I)[/math] (which might be [math]\tau_I(g^0,d^1)[/math]), don't recalculate them. Since we're assuming they're cached, just look them up.Also, this gives us a means for identifying things we've calculated before. It's just the input-output mappings of our eliminations factors.So we see, VE exploits factorization and caching.What about the MAP queries?Fortunately, we can leverage our previous knowledge to substantial shorten the length of this explanation.First, we restrict our attention to MAP queries where [math]\mathbf{E}[/math] and [math]\mathbf{Y}[/math] together make up the whole of [math]\mathcal{X}[/math]. If this isn't the case, that means there are some RVs [math]\mathbf{Z} = \mathcal{X} - \{\mathbf{E},\mathbf{Y}\}[/math] and our query really is:[math]\textrm{argmax}_\mathbf{Y}P_M(\mathbf{Y}|\mathbf{e})=\textrm{argmax}_\mathbf{Y}\sum_\mathbf{Z}P_M(\mathbf{Y},\mathbf{Z}|\mathbf{e})\tag*{}[/math]This mixture of maxes and sums makes the problem much harder - so hard, efficient algorithms for it don't even exist. These are called 'marginal-MAP queries' and our plan of attack will be to run away.So we'll consider the case where there are no RVs in [math]\mathbf{Z}[/math]. In this case, Bayes Theorem tells us that this conditional MAP assignment maximizes their joint assignment:[math]\textrm{argmax}_\mathbf{Y}P_M(\mathbf{Y}|\mathbf{e}) = \textrm{argmax}_\mathbf{Y}P_M(\mathbf{Y},\mathbf{e})\tag*{}[/math]Further, determining the answer for the unnormalized product gives us the same answer:[math]\textrm{argmax}_\mathbf{Y}P_M(\mathbf{Y},\mathbf{e}) = \textrm{argmax}_\mathbf{Y}\tilde{P}_M(\mathbf{Y},\mathbf{e})\tag*{}[/math]This means that our task is to filter the Gibbs Table to rows consistent with [math]\mathbf{e}[/math] and to find the row (assignment) with the maximum [math]\tilde{P}(\mathbf{y},\mathbf{e})[/math]. We'll assume that if we determine a maximum, we'll keep track of the assignment[1]. That way, we may speak in terms of the max-value. Since that value is a product, this is often called a 'max-product' algorithm[2].To keep in line with our previous explanation, let's call this function [math]mr(\cdots)[/math] for 'max-row':[math]mr(\mathbf{e}) = \textrm{max}_\mathbf{Y}\tilde{P}_M(\mathbf{Y},\mathbf{e})\tag*{}[/math]At this point, some algorithms diverge in their approach - let’s go with a familiar one.Variable Elimination for MAP queriesThere is an analogous factor tricking in the case of MAP queries.Let's say [math]Y_0[/math] is a variable from [math]\mathbf{Y}[/math] we'd like to eliminate. Once again, let's pretend there are only two factors, [math]\phi_1(\mathbf{D}_1)[/math] and [math]\phi_2(\mathbf{D}_2)[/math], where [math]\mathbf{D}_1 \cup \mathbf{D}_2 = \mathcal{X}[/math], and [math]Y_0[/math] is not in [math]\mathbf{D}_2[/math]. With this setup, we are determining:[math]mr(\mathbf{e}) = \textrm{max}_\mathbf{Y} \phi_1(\mathbf{D}_1) \phi_2(\mathbf{D}_2)\tag*{}[/math]Once again, the algebra gods provide a gift:[math]\begin{align} mr(\mathbf{e}) & = \textrm{max}_\mathbf{Y} \phi_1(\mathbf{D}_1) \phi_2(\mathbf{D}_2) \\ &= \textrm{max}_{\mathbf{Y}\setminus \{Y_0\}} \Big[\phi_2(\mathbf{D}_2) \textrm{max}_{Y_0} \Big[\phi_1(\mathbf{D}_1) \Big]\Big]\\ \end{align}\tag*{}[/math]And from here, we may proceed exactly as we did in the case of conditional probabilities. It's literally as simple as replacing sums with maxes! Nice!It gets complicatedWith the details of VE understood, we can briefly discuss an otherwise terrifying algorithm: the Junction Tree algorithm.Typically, we aren't concerned with only one probability/map query, which would be specific to one observation [math]\mathbf{e}[/math] and one set of RVs [math]\mathbf{Y}[/math]. If we were to use VE to address many queries, we'd have to rerun VE for each one. However, it's likely that many of these queries involve identical elimination factors. The Junction Tree algorithm is a super clever way of organizing our graph into something that allows us to identify beforehand which calculations are common to which queries. That something is called a Clique Tree. It's a tree whereby nodes correspond to groups of RVs and edges correspond to intersections of RVs between these groups. We compile this tree before hand, incurring the cost of approximately one VE run, and then we can answer a wider range of probability queries. In effect, it's another layer of caching results for later reuse.What's next?The issue with exact inference is that it's exact and such perfection comes with a price. Fortunately, there are approximate methods which make that trade. Those are explained here:3. What is Variance Inference in the context of Probabilistic Graphical Models?4. How are Monte Carlo methods used to perform inference in Probabilistic Graphical Models?But maybe you're satisfied and would like to see how we learn parameters from data. I have two more options for you:5. How are the parameters of a Bayesian Network learned?6. How are the parameters of a Markov Network learned?Footnotes[1] Algorithmically, this means we need to set up a 'traceback'.[2] Since you could take the log of this product and maximize that sum to yield the same answer, algorithms that accomplish this are often called 'max-sum'.Sources[1] Koller, Daphne; Friedman, Nir. Probabilistic Graphical Models: Principles and Techniques (Adaptive Computation and Machine Learning series). The MIT Press. I owe the effective notation of this answer to that book, along with much of my understanding of this subject.[2] Murphy, Kevin. Machine Learning: A Probabilistic Perspective. The MIT Press. The idea to show VE as that cascade of equations came from this book.
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