Hello, my name is Justin Watkins. I'm Professor of Burmese at SOAS in the University of London and I'm going to give you a short talk today on the WAH language, also known as Barao, a language of China and I've been invited to do so by Nathan Hill, Director of the Trinity College Dublin Mission Studies. Okay, so WAH is an Austro-Asian language and as such is in the minority, most of the languages spoken in China are Tibetan Burmese languages, relatives of Chinese and Tibetan. Monkmeh is a language family where the largest concentration of languages are found in Peninsula, Southeast Asia as you can see on this map and the WAH languages are the larger and orange blob in that small collection of languages, one branch of Austro-Asian of Monkmeh known as the Palauanian languages and most of the Palauanian languages are spoken in Myanmar and China, Northern Laos as well. Okay, so WAH speakers are located in the in what default called the WAH corridor, so the area, the geographical area between the Salwin and Nekong River, this area straddles the southwestern Chinese province of Yunnan, the Shans state of the Northeast of Burma, and the WAH are thought to be the autochthonous inhabitants of the area, it's likely that speakers of Northern Monkmeh languages were settled in the area earlier than speakers of Tibetan Burma and Taikadao, so we go back to the map and the impression you get is that Monkmeh languages, certainly the smaller Monkmeh languages, so leaving aside Cambodian and Vietnamese, are the remnants of the population, a larger population or more widespread population speaking Monkmeh languages who have been displaced by the arrival of speakers of Thai, Thai languages, speakers of Myanmar and other Tibetan Burma languages. So let's go on. The standard variety of WAH is known as Barak and is based on, and that's based on the variety spoken in the village of Yen Shui, in Chinese, Aishwai, in Burin WAH. There are about 1.2 million speakers, including Barak and other varieties in the language, and it has to be said that WAH is a complex cluster of many dialects, perhaps 40 or more, many of them undescribed or poorly described, lumped together or split wrongly, it's a confusing picture, but taking a slightly more lumping rather than splitting approach, two-thirds of the speakers of WAH are in China, about one-third in Myanmar, so a little over a million in total. So it's a significant language in Yunnan, most of the speakers are down near the border, but the WAH get around, and I won't be talking much about WAH society and politics in this talk, but if you go and read up, you'll find that there are interesting people with an amazing history, fiercely independent, they have a semi-independent state nestled next to the Chinese border, and in many ways are politically more affiliated to China than they are to Myanmar, but it's a complex and extraordinary picture. Recent migrations have seen WAH villages established in northern Thailand, there's WAH presence in Yangon, in Damji, Mandalay and other big cities in Myanmar, and also in Kunming, and across Yunnan province, there are WAH in Beijing, so they get around. Okay, I'm going to get linguistic and a bit more technical, straight away. So first of all, a little bit about how WAH has been written down. First thing is to say that it's not written all that much, and that there is a split orthography, so two parallel orthographies have been developed for WAH, one, and the two that I'm going to focus on are the two found in the WAH dictionary that I spent many years compiling with colleagues, which was published in 2013. And in that dictionary, we used official, the official WAH orthography, which is approved by the Literacy Committee of the United WAH State Party, so the authority that's in control in WAH state, the autonomous WAH area in Northern Shan State. And that was developed by improving on and developing the defective orthography used in an early 20th century translation of the New Testament, done by Vincent Young with WAH colleagues and others. And Vincent Young was unable to hear lots of the phonological contrast in WAH. They didn't mark breathy aspirates, you'll see a bit more about the sound system of WAH. Surely he couldn't hear the register contrast, so a quasi-tonal contrast of breathy and modal phonation. So that's not marked. And final glottal consonants and final h's, final glottal stops not marked. So there's a lot of syllables that are, that sound different, but written the same in the original New Testament orthography. And over the years, the WAH on the Nymar side of the border where this is, both this orthography is used, and Christian WAH and people associated with the descendants of that translation of the New Testament, then cast around to find improvements. So ways of marking final glottal consonants, ways of marking the register contrast, we'll see a bit more about that later. Anyway, on the other side of the border in China, after the establishment of the People's Republic of China in 49, so in the 1950s and onwards, linguist anthropologists were commissioned, they were sent from Beijing to the four corners of the Republic, the People's Republic to describe and develop writing systems for many of China's minority languages, including WAH. And what they did after a slightly unusual prototype writing system had been developed, which was more suitable for linguists, really, they developed a writing system for WAH, which aligns nicely in its treatment of various sounds with the Pinyin transcription used for standard Chinese. And this WAH orthography used in China is phonologically accurate, but it's not in wide use, it's used for formal official translations of documents into WAH, but it's not used by all that many people, but there's quite a lot of printed stuff that you can find in WAH using it from China. So let's have a look. So here's a WAH sentence to give you a first bit of WAH language material. So the sentence, have you read my letter yet? And you can see that the prototype PRC WAH orthography here is one that uses some symbols borrowed from Russian and Russian transcription of Mongolian. And so a little bit outlandish, they use a cue for a final glottal stop. We see that in various places. So some redundant letters used, some new letters used. They're using the Cyrillic hard sign to mark clear register. And so that was the prototype. And then that morphed into what's described here is the PRC orthography, where a bar over the vowel is used for a breathy vowel and the absence of a bar used for clear register vowel. And the treatment of final palatal consonants, so Monkmaer languages have final palatal as you find them in Khmer. They're written using a G, so a vela with a preceding I, which kind of gives an impression of the I off glide before palatal consonants. And that word for already or the yet is pronounced hoi. So it's a palatal final, spelled I G in PRC orthography. In official WAH, they didn't want to use final palatal, so they didn't want to use a C or anything like that, the final palatal side of that spelled I T in official WAH. So you have to have your wits around you, depending on where your WAH text is from to work out who's written it when, which side of the border that splits the WAH between Myanmar and China. And in fact, since this material was produced, the Literacy Committee of the United WAH State Party has revised official WAH again. So there should be another nine in this table. Okay, so let's come back to the register contrast in WAH. So in WAH, instead of having a tonal split between high and low that you might find in other Mon Khmer languages, or you find sets of high tones versus low tones in Chinese or in Thai, Vietnamese, WAH has clear and breathy phonation contrasting rather than tones. And the register contrast is phonetically a complex of phonetic correlates. So fundamental frequency, register slightly higher in frequency than breathy register a vowel quality. So breathy vowels are a little bit more open phonation type breathy, breathy phonation in the breathy register and vowel duration.  What's interesting is that there's no phonation type contrast. There's no register contrast in syllables with an initial laryngeal consonant so or with a with aspiration in the initial consonant. But you do get the register contrast if there's final a final bottle consonant so like an H or a bottle stop. So here is a set of contrasting syllables. So the words sorry, I had there's a mistake. The second word for peach should be sweet. So there is peach and there with two dots under it means sweet. So there less is there and turn is there. Land is there and wager is there. Change the slide. So you can't see that in the second. Right. And here is, I mentioned earlier that on the Myanmar side of the border, there were various attempts to bring the spelling system used in the old New Testament translation of war to bring it up to up to par by adding things to represent the register contrast, which wasn't represented in earlier forms of the language. And what they did, I think, was look at neighboring languages. And they decided that you could mark a there we go. We've got sweet and peach the work the right way around. So sweet is there and peach is there. And the table and the clear register so the non breathy register is written with two dots after the syllables. T I E I E is the digraph used for the vowel air. The two dots tell you it's clear register, not breathy register. And I think they were looking at high tone in Burmese, which uses something that looks like two dots to mark a high tone. And the absence of that means it's breathy register, which is lower in pitch. So we've got some minimal pairs here, sweet, peach, and I'm overdoing it for illustrative purposes. The register contrast isn't phonetically quite as distinct as that. The words for head and work head is game work is game. And other ways of marking register contrast, if there's a final, final stop, you use the unvoiced syllable P in clear register, the voice syllable B in breathy register. So nap and nap are spelt with different final consonants. And that reflects a little bit the way that tie consonants have a link with tone. If you've learned your tie tone rules and do a bit of historical phonology on tie, you have different consonant letters having an effect on the tone, which is red, which we find in lots of authorities around the region. Okay, however, in this version of official wa, orthography, if there's a final bottle stop, and an X is used for a bottle stop in this version, which they borrowed from the Chinese orthography instantly, incidentally, they're both written the same. So debt to wager to that are both written spelled the same in official while spelling. And I think that's been fixed now in the latest version. Although at the time when I was compiling the dictionary, I had meetings with people from the literacy committee of the United States party, and they told me I was not allowed to fiddle with the official orthography. It was the way it was. It's since been amended. But at the time, I was not invited to offer my suggestions for improvement. But they have most certainly got there. Now, and it wasn't that they didn't understand they were very aware of the complications of writing while using Roman script, which is a choice. Obviously, they will be a lot easier to write while using, for example, booming script. But for those reasons on the Chinese side, that's not something that's been suggested. Right, let's move on. Okay, so consonants and while we'll do a bit of phonetics phonology to start off, there's a four way voicing contrast in initial stops. So voiced and unvoiced, aspirated, unaspirated. Initial consonants, clusters are restricted. So this is sort of classic mainland Southeast Asian language in many ways. Final consonants are restricted to unreleased, label, alveolar, palatal, and vela, the same set of nasals and glottals. And then you get a final H, final IH, which is the reflex of an S and sounds like an S. So that's a bit like what we find in Khmer, where a written S at the end of a syllable is pronounced as an H, except in certain dialects. So that's probably a fairly recent change. And there are a large number of breathy, aspirated voice segments. Okay, so here's your little table of consonants. So we've got a P and a B and an aspirated P, often pre-nasalized the voiced aspirates. And then you've also got breathy, aspirated nasals. So ma and ma, na, na, niya, niya. You'd also get ma and ma. Okay, you have breathy, aspirated, voiced fugitives. So you can have va and va and ra and ra, la, la. And here are some examples of the vowels in wa. So there are nine contrasting unitary vowels, so nine vowel contrasts, so e, e, e, a, a down the front or or mu up the back, and then ʊ and ʊ. So high back unrounded vowels. Again, lots of men in South East Asia have those. And then lots of diphthongs that have ʊ, ʊ, ʊ, ʊ, ʊ, ʊ and ʊ. Right, there we go. There are some vowel quadratials with those vowels on them. Okay, so wa has a very large number of sesku syllables, so syllables with a minor pre-syllable, a minor initial syllable with restricted vowel contrasts appended to a major syllable, which has the full set of possible vowel contrasts and the registered contrast. And you often get, as you do in Burmese and lots of other languages around, which like this pattern, so mon in particular, and other mon premier languages, you often get bisyllables where the first syllable gets reduced to a minor syllable. So you might find the word suso, meaning muddled up and pronounced as suso, or suso, or suso, or suso, and some other examples of that. And a very common pre-syllable is just an s with a sort of indistinct schwa in between the s and the major syllable.  And while the morphological system of prefixation, which existed in earlier forms of the language, is all but gone, we leave out whether just a few prefixes with hard to discern, hard to recover, and a range of functions. This pre-syllable s is by far the most common. According to Schurter, probably results from the generalisation in almost all prefixial contexts of a prefix, which originally were corresponded to those with an initial s in Balao and Rihang Lang. And if you read up in the difluor, difluor, the wa languages you can find reference to. So here are some, there's some evidence of prefixation in earlier forms of the language, so lah and blah, burn and hearth, lang, glang, long, this long, grout, grout, sorry, grout, deep, and this deep, there's this sort of demonstrative stative adjective, stative verbs that describe, if you're explaining to someone, it's this deep, grout, so you add a prefix to the word deep. Voicing of initial stop, so you get related pairs of words, booh, meaning thick, booh, thickness, bing, big, bing, sides, gip, thick, and sgip, thickness. So these are hard to formalise, but there's also evidence that this went on in wa in a previous stage of the language. Okay, onto a bit of morphology, so compounding, wa is a supremely isolating language, the syntax is pretty similar to Thai and Khmer, mon, languages like that in mainland Southeast Asia. So we've got words for clothes, where you often get two synonyms put together in a compound, so ground, cry, clothes, twice, meaning clothing, things, goods, possessions, and you can elaborate on that, so ground, cry, thai, dah, is clothing or clothes, so two generic words, skirt and shirt, altogether clothing and clothes, a bit more about those sorts of forms later. Words for a noun associated with the second noun, so weapons is war equipment, equipment war that way around, so why compounds are, why phrases are left-headed? Well, nyet, so relatives of the house, I remember the family, gan nyet, work to do with the house, nyet jia, so a bee hive, a bee house, and the word for toilet is nyet ang, so the shit house, as little in the way of euphemism in war, when we were compiling the diction we found that the Kedwin was pretty earthy, there's no well-developed system of politeness or euphemism, or avoiding speaking to the point in war. Right, compounding two, so you can form nouns using the relative isa ba, so the word for income or earnings is babon, the things that you get, that which you receive, was righteousness, that which is upright and straight, so it's to form nouns. A very large number of words in war formed, so what I think matters is called psycho-collocations, so expressing emotions in terms of the way a part of the body or an organ is, so if you think of in Burmese, if you want to say you're worried, you say that your mind, which is actually located where your heart is, your mind is hot, and we find similar things in war, so the word for heart is rom, and if your heart is salty, so a rom means that you're angry, a rom means that your head is hot, it means that you're upset or irritated, if your heart is spicy, so bright rom means angry, and there's a whole series of them here. Proverbs and sayings in war, so you'll see in the references at the end, a collection of about 1,000 proverbs and sayings, and these generally pivot around the central rhyme, so we've got three examples here, I'll have a go at reading the first one, so ban dat nyat, lat dat groan, so lat is nyat and light, rhyme with each other around the two halves in the form of the proverbs, so rest in a frequent place, enter a hunter's hide, the joys of hunting, so a lot of the proverbs and sayings are about sort of food, life, how to live healthily, how to get pleasure, there's quite a few to do with sex and courting, and all those sorts of things, so prayer to ward off fire, sy'nach yin, brit nach grei, the war for war, so that's not great, so be romantic for a while, make a true effort once, so invest in long relationships, but for short relationships, just put in the effort, it's slightly earthy as it appears. Right, and elaborate expressions like Southeast Asian languages in general, and indeed Chinese we find the sort of structured four syllable, four morphine formulations which sound nice, which are a little bit more than the sum of their parts, so very often you have a sort of standard morphine, so here if the word for food is bret, and another more obscure or poetic morphine for food is brum, and they sort of fit together, they've got the same consonant cluster in the beginning, so bret, brum, standard word is bret, the elaborate sort of more poetic word is brum, so ngaw bret, ngaw brum means left over, where you could just say ngaw bret, but that would be boring, ngaw bret, ngaw brum, get this nice four morphine phrase which gives the whole thing a bit of twang. Okay, so to make food, so yaw bret is the standard word for making food, but yaw bret, yaw brum, sounds nicer, it sounds more, it has more impact as we're affecting, so another example of rom is heart, and ri is this slightly less common morphine for heart, so if you start a fresh, you have a change of heart, it's ngaw brum, but ngaw brum, ngaw ri, is more explicit, down we want. Large number of loan words from Chinese and Burmese, more Chinese than Burmese, but usage Mexican depends a lot on which side of the China and Myanmar border you're on, so we've got some examples here, so the word for car in Gua is motoc too, so too is for a sort of Yunanese, and most of the borings sound like Yunanese Chinese, obvious reasons, too is the word for vehicle, motoc is Burmese motto from the English word motor, so you've got a real international hybrid of a word there, a nice example here, the word for knowledge is uses the Chinese zhu yi, which comes out in Gua as zhu yi, and combined with the Burmese word pi nha, which means it's from Pali, banya knowledge, so zhu yi pi nha, again you've got four syllables, so that gives the whole thing a bit of twang, the Gua word for knowledge, zhu yi pi nha. And then we've got some examples of borings from Chinese, so jiao shi for classroom, which is appended to a head noun, a super ordinary noun, yi, so the Gua word for house, so house which is a classroom building, which is a classroom, zhan yi, jiao shi, yi, jiao shi, it comes out that way in Gua.  A shi dhuang, which is the Chinese word for suit, after the Gua word for clothes, so grang, se dhuang is a suit, bu, di se, so di se, the English word via Burmese, di se for diesel oil, and bu the Gua word for oil. So those words with those forms are a very common word for borings. Right, so a little bit about noun phrases, so they're all left-headed, you have a system of major words or classifiers, but with surface station rather than Chinese word order, so you have the noun, then the number, and then the classifier, so a quail, so jog is a quail, de is one, and mu is a classifier, and jog mu is a quail. Okay, so a very generous person, bui, dan ding dat, the gut, so gut is the classifier for people, so generous person, one classifier. Okay, pronouns in Gua, a nice set of basic Gua pronouns, and like other minority languages in Tbiliburma and minority languages spoken in Nomar, but less common around many in Southeast Asia, Gua contrasts, has some dual pronouns and inclusive, exclusive, and second person pronouns. There's the full set, kinship words as pronouns again, so like Burmese and other Southeast Asian languages, Gua uses words like grandfather, grandmother, brother-in-law, and uses some pronouns to refer to, plural pronouns to refer to singular addresses, but otherwise you don't find in Gua this sort of diglossic development of the language with a more basic and a more formal register. Tense mode aspect, so like Chinese, like other Southeast Asian languages, you have these pre-verbal morphemes, as I said in the video, so there's a perfected marker, a negative marker, experiential, remote past marker, so heart corresponds quite closely to the function of look in Chinese, so okay, modals, there are high frequency, partly grammaticalized verbs, so the verb to own is redeployed in grammaticalized form as a possessive to receive these uses can, so like Chinese, like pronouns, like time, but to see, to try and do something is see and do it, to this causative, benefactive verb to give that happens in Gua, the verb top, progressive with ought to live, so that's like, so lots in parallel with, as you'd expect, with neighboring languages of different language factors. Right, and that's it, so a little romp through some highlights of Gua for you, here are some references, do get in touch if you're interested in Gua, there is much more to read and learn about Gua and its dialects and related languages from Northern Monkmae, Palang languages and that's all, thank you very much for listening.


So this is the article that Stefan was mentioning. It's about monetary policy as a means for economic control. And the most important aspects I present here during this talk, and for going more into depth, I definitely can recommend checking out the article because I can just make a brief overview on the depth of where we're going to. So yeah, let's start. You all have heard the saying, money rules the world. And I would like you to think of when was the last time that you were thinking about that? So for me, this came up frequently, actually. And one very important aspect was when I see lobbyists that were able to influence politics with the money they have. But money essentially depends on the system that it is like built around. And the structure of the monetary system actually defines how money is used. And here, Bitcoin is a freedom technology. So I see that Bitcoin brings very different properties than fiat money, which is essentially a means of control. And this is what I would like to present to you today. And first, I will lay out why fiat money is a means of control. And then I will go into the antithesis to the system, namely Bitcoin. So first, I would like to go back in time with you to the Gold Standard. And I would like to show you how lending works during the Gold Standard. So here you have a bang. This is a balance sheet of a bank. Here you see the assets that the bank owns, which is in this case, only gold. So this bank was a gold custody only. And for the gold it held in custody, it was giving out warehouse receipts. And it turned out that the people could then also use these warehouse receipts as a medium of exchange because it was not dependent on the owner or on the name on it. But you can just take this receipt and go to the gold custody and get your gold out. So what happens when a loan is given out? So first, I would like to stress that the bank is then not only operating as a gold custody anymore, but also as an investment vehicle. So it is giving out gold to the one who was lending some gold. And this influences those who were giving the gold to the bank in the first place. So essentially, those that hold these warehouse receipts that gave the gold to the bank are an investor in an investment fund. So it's both a custody and an investment fund. And if an investment fund does their business honestly, they should check out with the investors and they should explain the risk of this loan. And this is how it would work in an ideal case. And now I would like to show you how it evolved. So on the left side, we have the same as before. And then a new loan was given out. And then the balance sheet was simply extended on both sides. So it was not that this gold was given out, but just a new loan was placed under the asset as an asset, essentially. And new warehouse receipts were created that were not backed with gold. And this is essentially fraudulent behavior because imagine everyone goes back to the bank and wants to get back their gold, then the last ones will not get the gold because there are unbacked receipts in the system. And you see that the balance sheet was just extending. And this is how it worked with fraud, essentially. And now we have a look how the current system works. And you see it is basically the same makeup. And you see we also have here an extension of the balance sheet on both sides. But I will guide you through this a little bit more. So this principle applies for both central banks and banks. And I would like to show you how it works for central banks. So here we have the balance sheet of the central bank. And here on the asset side, you have the claims on loans, which is an asset. And then you have the deficits from banks, many commercial banks, and the government. And then when the central bank gives us out a new loan, it is simply added on the asset sheet. I have to change that, right? It's added to the asset sheet. And it's also added to the asset side. And it's also added to the liability side. So the bank has now a new claim on the loan. And the money is simply created out of thin air and placed on the deficit account of the debt term. So nothing is transferred here. No money is transferred. It is simply created out of thin air. And when you see this approach is exactly the same as we had here during the lending of the gold standard where we had unbacked receipts. And what happens when we extend the monetary supply? This leads to a debasement. So every monetary account is less worth. And this is inflation, essentially. And Rothbard was saying something very important on that. So he's a very famous Austrian economist. And he says, it is, in fact, difficult to see the economic or moral difference between the issuance of pseudo receipts and the appropriation of someone else's property or outright embezzlement or more directly counterfeiting. Most present legal systems do not outlaw this practice. In fact, it is considered basic banking procedure. So this means that the issuance of pseudo receipts was essentially a dispossession of someone else's property. And he also says here, as we saw before, that most present legal systems do not outlaw this practice. It's completely legal. It is even much more than that. It's basic banking procedure. And this is not the case only for central banks, but also for banks. And this has great implications. So I go into this in much more depth in my article. But what are the mechanics on credit creation? And especially, when does a bank create credit? And this is very much influenced by the refinancing rates. So at which rate can a bank get money at the central bank? The bank needs central bank money to process transactions, essentially. So when a bank has a reduced refinancing rate, the banks can borrow more at central banks, which they need for transactions. And so they can expand their business by giving out more debt, because they can much more easily refinance themselves. So and since banks are living in a competitive environment, they are reducing the interest on debt to attract new debtors. And then when the interest rate lowers, then there is more demand for debt. More people are willing to take on this debt. And so more debt is created. And this essentially leads to boom and bust cycles. So it is a boom because the interest rate was artificially reduced. And this eventually has to go into a bust because it's unsustainable. I can very much recommend the article by Ben Kaufmann, Bitcoin and the Business Psychic, who explains this mechanism in great depth. But now we continue and explain and I explain a little bit what the depth actually means for a normal person. So let's say I am getting a credit from a bank to buy a house. And then this house is a security for my credit. And when I cannot pay back my debt, then the bank may seize the house. Or they can reduce my risk rating. So essentially they have a lot of power over me when I take on a debt. And they can do so by just creating this money out of nowhere. And this goes even further with the money creation through asset purchase. So I will go back to the schema that I used before. But now I explain how money is created with asset purchase. So again, banks and central banks can create money by buying assets. Here we again have the balance sheet of the central bank. We have the asset side, which are claims on bonds, like the government bonds, which are then noted as an asset. And then you have the deposits as before with the commercial banks and the government account. And now let's assume that the central bank buys a government bond. This government bond is simply added to the asset side. And then the money is created out of thin air and placed in the deposit account of the seller of the government bond. So this is always working for commercial banks. Imagine a commercial bank is selling a government bond to the central bank. Then the commercial bank gets simply added a new number, the price for which sold the government bank to its deposit account. And the government bond is added as an asset on the balance sheet. So this means the Fed can essentially infinitely buy assets on the market. They can also exercise direct control on the economy through the selective purchase of assets. So they can decide which asset is bought more and which not. So they can decide who gets the funding. And yeah, essentially, ownership is also transferred to the Fed in the end. Relevant assets are primarily government bonds. So the central bank primarily buys government bonds, but also mortgage-backed securities, bonds from companies, and stocks. This happened most recently. They're buying ETFs, which is kind of a bundle stock. And here I would like to show you the standard employee index, what happened in the last month. We see that here was a great crash. It was in March. And you remember, it was during the lockdown. And the price of the stocks was just falling. And at one point, it magically turned upward. And yeah, maybe you can imagine already what happened. The central bank was buying up these stocks. So it was exactly on the 23rd of March, 2020, when the Fed announced that it's buying ETFs to stabilize the market, which actually means pumping up their markets, although we already have a bust. And this is a funny thing. Investors are already selling these stocks, but we're not seeing a crash. We're not seeing a complete bust because it is hidden through the program of the Federal Reserve buying up the standard and poor 500 ETF stocks. And what is happening thereby? The investors who actually did a small investment because they bought overvalued assets are bailed out because they can't just sell it to the Fed for high prices. And here the bank, of course, works as an intermediary. So I would like to rephrase the quote in the beginning or the saying in the beginning, money rules the world to those who create money out of an air rule the world. And now I show you where I actually have this pyramid from. It's on the US $1 note. And this pyramid actually explains quite nicely what we're seeing in the financial system. So I placed it like this. I consider the Fed even superior to the government and, of course, superior to banks. And I want to explain you why. But first, I would like to tell you that the Fed was actually created by many banks coming together and intending to create the Fed to get these bailouts so they can make actually more risky investments or risky procedures. So this is where the Fed is coming from. I can very much recommend the book, The Creature of Czech Island, which explains this very well. I'm just at the beginning, but it's very enlightening. And so this is actually a banking cartel. And as we saw, they are creating money out of thin air. And Rothbard calls this out as dispossession. And this is completely legalized. So it is government backed this banking cartel, essentially. So central banks exercised influence on the economy through dictating monetary policy. And this power is directly related to the ability to create money out of thin air. The banks and the central bank can both create money out of thin air. But it can only do so because it's legalized. So in theory, the government could come and say, OK, we don't want this banking procedure. Now let's go to Bitcoin. But it's not very realistic because the government is very much profiting from the system because they can easily finance themselves by just selling these government forms that are then bought up by the Federal Reserve. And so the government is very much dependent on the Federal Reserve. So I don't see this as coming. And here I have this from the Federal Reserve annual report. It actually says that the funds of the Federal Reserve are going to the Treasury. So they are working very much together. And you can see this in the annual report. Now for giving you a little bit of a context, why I thought that the Federal Reserve is actually superior to the government, I found this here. It's the hearings before the Senate and of the House of Representatives. So there was an investigation on the banking system. And this is what they found out. And I would like to just read out what they found. Quote, let us control the money of a country. And we cannot who makes its laws. This is the maxim of the House of Rochiles and the foundation principle of European banks. If a country and its people are mortgaged for the SS value of their property, and the bankers control the money, the bond holders are not the people on that country. It makes no difference whether you call it a republic or a monarchy. The people can never be free. It's the borough, it's the servant, to the lender. So what can we do with that? We now have an alternative. We have Bitcoin. And it is an alternative to the central banking system with Bitcoin. Money cannot be created out of thin air, but only through a process called mining. And Europe was calling Bitcoin a supermassive monetary gravitation object, which actually pulls in value. But I would like to think this a little bit further, because actually people are more and more coming into Bitcoin thinking about the whole theater system that we have right now, questioning this whole system, questioning centralized authorities, and more and more realizing the importance of property and freedom and the ability to be safe, which is now possible in a highly inflating currency. So I see we have a confrontation of different value systems, namely the centralized version around the Federal Reserve, which is a lot about dependence, centralization, and essentially serfdom. And then we have Bitcoin, which brings in self-sorrenty, decentralization, and freedom, because we can use these Bitcoins as a property to save for the future, and then even maybe buy a house and be completely self-dependent. Or we can think this even further, that thinking about the monetary system, this just set the stone rolling and going much more further into organizational structures. So we are questioning this whole organizational structure we have right now, and we think of alternatives. And the role of centralized authority is more and more challenged. Do we really need it? Or is a free market maybe much more reasonable? And here, Austrian economists also really much comment a plane. So I truly think that we have parallel societies forming around Bitcoin and central banking. So some people will be rather stuck in this whole system, work in these whole mechanics, bureaucracy, tax, everything. And more and more people want to have a Bitcoin Citadel, which is essentially a free private city, where only the most basic things are provided by the structure, which is defense, inner security, and the provision of, let's say, an organization to settle fights. Kind of a law thing, but competition amongst these is also very much wanted. And there is no healthcare because everyone can do this on their own. But yeah, if you don't have to pay so much to your health insurance, which is not voluntary here, but what you have to do, then you can think of alternative means, which are, for example, not supported here in Germany where I'm living. So I would be definitely better off in such a system. So I hope to see you in the Bitcoin Citadel. And I would like to end this talk with don't trust, verify, and I hope I could give you some inspirations. And I would like to invite you to really go into the depth to understand why the structure is the way I showed you and challenge everything, challenge everything that I say. And you can follow me on Twitter with at Stephanie B. Jane. Thank you.


David, you drew against Gladiura Benjamin. How did the game go? Okay, after the opening, I was slightly worse. Then I equalized and when I avoided Ufra repetition, I became slightly worse again and he offered me a draw, so I gladly accepted it. I wanted to ask you because the position that he offered you in a draw, a draw end, do you feel that that position is equal because I thought White still had chances there? I would have played it with White, but okay, if Black defends correctly, it should not be seriously worse, just slightly worse. How's the experience in Gibraltar been so far for you? It's quite a nice tournament, only I need to get accustomed to less space than I'm used to in round-turbine tournaments. But okay, it's good that there are so many people here and the tournament is very nice and many people are playing here therefore. So that's, in a way, it shows that the tournament is quite well organized. Right, absolutely. We look forward to seeing your games in the remaining rounds. Good luck for it. Thank you very much. Thanks.


Guys, please take your seats. OK, so we'll start. I realize you just had an exam and are perhaps a bit exhausted. So we'll just slowly ease our way into the hierarchical Gaussian filtering that we're doing. So what we did at the end of yesterday was to take apart this learning rate at the outcome level in the standard HDF hierarchical Gaussian filter. And we saw that all three kinds of uncertainty that we knew have to be taken account of were actually taken account of here. So we have outcome uncertainty, the irreducible kind of uncertainty, risk in economic terms. We had informational uncertainty, our ignorance about the state of the world. But this we can learn by making more observations. And then we had the uncertainty stemming from the environment changing, states moving around. And all of this was represented here in the adaptive learning rate that we wanted to have. This is what we set out to get. And now we've got it. Then a first little toy application to a financial time series. We can see volatility at the second level. At the first level, it's simply the time series of exchange rates that we're looking at. At the second level, it's the volatility of that time series. And we can see that at interesting points when interesting stuff happens, this volatility shoots up. And even the volatility of the volatility at the third level shoots up. And what this means is that the model realizes that it's somehow lost its grip on reality, that the underlying reality has changed. Old information has to be thrown away. And the way to do that is to increase the learning rate. And you can see this dramatic spike in the precision weights applied to the updates at the different levels. It's most extreme at the second level, where you see the precision weight shooting up. But then after the model has caught up with reality, you see the precision weight dropping again, because now it would be maladaptive to throw information away too quickly. So we want to slow down. In the next few slides, you will see the update equations. And what I was projecting there, I promised always to go over here, because apparently people can more easily see it over here. Or maybe the best way to do it is to use the mouse. So the precision weight on the prediction error, this is the prediction error, and this is the precision weight. And this adjusts automatically in response to an increase in prediction error. So what drives this is an increase in prediction errors. And this, in turn, leads to higher precision weights. And this leads to a greater effect of prediction errors. So there's a kind of self-reinforcing weight on prediction errors. So once the system says, these prediction errors are more than I expected, you can actually see the algebra here. This is the update. Now if this is, if i is 1, this is the update we have in mu 1 in the mean of the belief on x1. So basically the mean of this red line here. Then if the update here is large, this leads to a large quantity here. This is the volatility prediction error. The prediction error on how quickly stuff is changing. So this increases. And if this is large for i minus 1 equals 1, so here we have i and here we have i minus 1. So if this is large at the first level, and we're updating the second level, this is delta 1 for the update of mu 2. And the weight that drives the update on mu 2 is then in turn also increased. But this prediction error gets more weight. And what you see is this spike here at the second level. This is the weight applied to the updates at the second level. This shoots up because the volatility prediction error here, the delta 1, the volatility prediction error at the first level, this is a prediction error that tells you how much you were off in your prediction of how much your x1 will move. So we can parse this term by term. So the main thing to remember is down here, we have predicted variance. And this has two components. The first is our ignorance about x1. This would be sigma 1 here. And then we have the environment driving changes in x1. This is from the level above. So this would be mu 2. If this is the prediction error on the volatility prediction error on the first level, this would be delta 1 and this would be mu 2. So if mu 2 is driving this, then this is the expected amount of change in x1. Now in the numerator, we have the observed amount of change in x1. So we have our new, our updated uncertainty about x1. This has time index k minus 1. This is before we make the update, before we make the observation. Sorry. And this is after we've made the observation. And performed our update. So this is the new, the updated uncertainty about x1. And this is the update in the mean of the belief about x1. And it is squared. So we don't care in what direction it went. It's just the amount we updated. So in a sense, this is observed uncertainty about x1. And this is predicted uncertainty about x1. And now because this works at the log scale, because in the definition of the HDF, if we go back, back, back, back, here, we have an exponential. So the levels are linked by a volatility that is situated at the log level. So our prediction error is not a difference. It is a ratio. So if you take the log of this fraction, you get a difference. You know that. So it is observed uncertainty divided by expected or predicted uncertainty down here. And now this is a prediction error driving an update, driving this update in mu2. So you see that if our observed uncertainty is exactly like our predicted uncertainty, then this fraction will be 1 minus 1 gives you 0. If the volatility is exactly as expected, we do not need to update mu2. This will be 0. And mu2 will be like the predicted mu2. No volatility-driven update. However, if the observed uncertainty is greater than the predicted uncertainty, the fraction will be greater than 1. And subtracting 1 will give you a positive prediction error, positive volatility prediction error. And this will push up the estimate in mu2, the estimate of x2. Mu2 is the mean of the estimate of x2. And conversely, if the observed uncertainty is less than the predicted uncertainty, the fraction will be less than 1. If we subtract 1, it will be left with a negative prediction error. And the estimate on mu2, the estimate on x2, will go down. And this is exactly what we see happening here. So where you see, I'll use the mouse, where you see this spike, at first, the volatility prediction error is always positive.  The updates in mu2 are all positive. And then it comes down again. So the updates in mu2 are going to come down again. So this is what drives mu2 up. And then it comes slowly down again as the system understands or settles back into a belief that it knows what's going on so it can decrease its learning ratio. Yes? When I'm iterating these up, yes. So what we do here, and the reason is one efficiency and we also want to anchor this in the brain. So what we do here is we use this one-shot update. So we use the update equation once. But we can also do this until convergence. So I haven't actually tried to prove that it'll converge or when it will exactly converge. In the years ago, I programmed an application where we actually do the updates through the system until convergence. And the difference it made with the one-shot update was at least in the examples I looked at, minimal. But it's a good question. So if you'd like to do a research project on that, that would be a good idea. Because actually, I mean, we like to keep things simple because we think the simpler they are, the more likely that the brain can do them or collection of neurons can do them. But it is actually not implausible to assume that different populations of neurons keep passing messages between each other until they're at a certain equilibrium. So this is what actually might happen. So computational efficiency, we decided to stick with one-shot updates because in the cases where I looked at, the difference between waiting until convergence and doing one-shot, there wasn't an appreciable difference. But it's a really good question. So how do we get the update equations? We get the update equations exactly. And we'll do this next week. I first want to give you an overview of what's possible with this. And then we'll go into the details and the derivations of this. So as we saw this week, in exactly the way I showed you, we derive the variational energy here. We fill in the model and use the definition of the variational energy at the i-th level. And for each level, we have a variational energy. So for generic levels, it looks like this. And then using the equations we had here, here. So using these equations and the variational energy that you saw here, we derive the update equations for our new here variance. Or if you take the inverse, then the precision. And for the new mean, that gives us our update equations. And this is what you have here. These are the update equations for a generic HDF level. We shall see that these are modified if you change the way the HDF works. In a concrete case, we'll see some examples of this. So why is this practically useful? So first, the updates have a general and interpretable structure. So the whole story I told you about this volatility prediction error, which is sort of the main feature of this filtering scheme that we have. We do not only have a prediction error on the amount that we're predicting and that we have an error on. We also have an undirected prediction error about the volatility of the time series we're filtering. Other schemes do not have that. And the great thing is this drops out of the derivation of the update equations. And yet, even though you've derived it, you will do this on the blackboard, or at least partly we'll do it on the blackboard, even though you've calculated around for a while and you get this thing that at first looks ugly, if you look at it long enough, you can see that it makes sense. It totally makes sense. It's interpretable. This is the predicted uncertainty, or at the I-th level. This is the observed uncertainty at the I-th level. And that gives you a prediction error regarding the uncertainty, and not just about the quantity at that level, also about the uncertainty. So this is one main feature here. It's a general structure. It's always the same. It's interpretable. It's computationally extremely efficient. And a great feature when we fit this to actual data from different brains, different people, is that we have parameters that can differ from subject to subject. So when a neuroscientist, a psychologist says subject, they mean a person doing an experiment, so from subject to subject, so from person to person, and can be individually estimated from experimental data. And this enables comparison of parameter estimates between subjects, so between persons, and of evolving beliefs on states within the same person. Because it gives us a measure of the free energy, it provides a basis for model selection, because the free energy is an approximation to the log model evidence, as we saw. AV is always greater than or equal to A. So we can compare different learning models with different hierarchical depths, for example. And we can compare so-called decision models. We'll see what decision models are in just a few minutes. So here's an experiment we published a few years back. It's a very, very simple experiment. It's almost the simplest experiment you can think of. The idea is this. We have a person, a subject sitting here, and hearing a tone. The tone is either low or high. This tone is referred to as a cue, with a duration of 300 milliseconds. And this tone will be followed by a picture of either a house or a face. Now, why are we doing that?  So when we do that in the MRI scanner, it is easy for us to tell whether this person was looking at a house or at a face. We know the kind of activation that we'll see in the brain from looking at houses and from looking at faces, and they're different. So that's why we chose to use houses and faces. And of course, these cues, the high and low tones, are predictive of whether there will be a house or a face. So perhaps a high tone will allow you to predict that it's more likely to see a face. And then conversely, because we constructed it that way, if the high tone is more likely to result in a face, then the low tone is more likely to result in a house. I mean, very simply, that we have probability of house given high tone is 0.8. Probability given house of a house given a low tone is 0.2. And then conversely, probability of a face given a high tone is also 0.2. And probability of a face given a low tone is 0.8. Yes. They know that. And they know that there are going to be these probabilistic associations. That's what we call them, probabilistic associations between houses and the different cues. And the reverse associations between faces and the cues. And now the twist is that this will change in the course of the experiment. So at first, we have this. And then we switch to something like 0.3, 0.7, 0.7, 0.3. And then we switch, and so on. We could actually check that. We didn't do that. That was not the question in that experiment. I would bet that will work, yes, but we didn't do it. Yes? Wanted to keep it simple here. So also at the modeling level, this allows you to assume that with every outcome, people learn about both kinds of contingencies. Because we're fully transparent about how the experiment works. We don't hide anything from them. So from every outcome, they can learn about both kinds of contingencies. So we only have to model one learning trajectory. But we also do other experiments. And we have done other experiments where this was not the case. So where they have to learn about these associations separately, it depends on the experiment. I think I have colleagues who didn't tell people whether there was this coupling between the probabilities. And in this situation, we told them how it was here. So really, it's the choice of the experiment. And of course, there's a whole zoo of experiments like that. And people do small variations and see how the effects change in response to small variations. And one interesting question is always, do sub-populations of subjects. So if you take people who are somehow special, people with schizophrenia, for instance, and you subject them to two different paradigms, do the way they adapt to the change in the paradigm, does that vary in a characteristic way compared to how healthy controls react to such a change in the paradigm? Then you can publish a paper saying people with schizophrenia are more or less able to deal with this kind of change or this kind of difference in the paradigm. No, they have to learn them. That's the point of the experiment. These numbers are hidden from the people. They don't know these numbers. They have to learn them. They just know there is an association between these tones and the outcomes and that these associations are symmetric. And this was done for 320 trials and 64 so-called null events. So we need this for the MRI analysis so that we have a baseline where we just present a tone, but then the outcome is nothing. So we have neither a house nor a face. Now, the thing is this. In the HDF, as I introduced it, we had continuous quantities performing gas in random walks like this. Like we have X2 here and X3. So that's the standard HDF thing. But now the outcome we have on each trial is binary. So outcome coding, we have two kinds of outcomes. We have a high tone followed by a face. We code that. And likewise, because it's the same kind of outcome because the probabilities are coupled, a low tone followed by a house is coded as an outcome of one. Followed by a house as well as a low tone followed by a face is coded as zero. These are the two kinds of outcomes you can have on each trial. And basically, the probability here, we look at just this part here, the probability of getting an outcome of one, this corresponds to, probability of, and we're going to call this X1. X1 equals one is what? Here, who can tell me? The probability of getting an outcome equal to one is, I heard the right answer somewhere around here, 0.2. So we code the outcome where the high tone is followed by a face as one. High tone is given. Probability of a face is 0.2. So there's already our answer. But we can also check whether we're right. So we can look at the other 0.2. And this is the following a low tone. So we're correct. And the probability, so this is also of X1 equals 0, is 0.8. This is our X1. And now the twist is this, this X1. And X1 is always a state of the external world. This is an objective fact. This is how the experiment turns out. It's out there. It's not in my head. And this is a binary state of the world. The outcome of the experiment on each trial is either 1 or 0. It's either one of these or one of these. And there's a clear probability for getting a 1 and for getting a 0. And this is what the subject has to learn. Now how do we get such binary outcomes into our modeling? Since we assume Gaussian random walks and the Gaussian random walk by definition extends to the whole real line, it's unbounded, we cannot model changing probabilities because probabilities are confined to the interval from 0 to 1. But there's a very simple trick. We can map the real line onto the unit interval by using a logistic sigmoid transformation. So the logistic sigmoid function, useful in many situations. And its graph looks like an s. That's why it's called a sigmoid. Here's 1. Here's 0. This is the x-axis. Here we have 1 half. And this maps the whole line here to the unit interval. So we can take a state x2 whose sigmoid is the probability of getting an x1 of 1. And that's what you get here. So the probability of x1 equals 1 is the sigmoid of x2. The probability of x1 equals 0 is 1 minus the sigmoid of x2 because as you'll immediately believe me, if this is the y-axis, then this is the graph of y equals s of x. And this is the graph of y equals 1 minus s. You could try that, but I'm not immediately convinced by the idea because what we want to have is basically a probability on the unit interval. So this is a probability of getting x1 equals 1. So it's between 0 and 1. And as we know from the way we're conducting the experiment, this can change. It can sometimes be here. It can sometimes be here. It can sometimes be here. Now with a step function, you would just get a jump from 0 to 1, right? Yes, you can do a proposal just to parse your proposal.  So it would just be if you're mapping x2 onto x1, then you can do it. But I don't think in this situation it would make much sense. So that's what we do. We model this changing probability of getting an outcome of 1 as a change in x2. And we take this model here, and we apply everything we learned this week to this model. So we basically write it down. Yes, cumulative distribution of the Gaussian. Ah, yes, that's another sigmoid. You could do that. But as you should see, using the logistic sigmoid gives you quite straightforward update equations. And substantially, yes, substantially, it does make much difference what sigmoid you use. You can also use the hyperbolic tangent. I'm using the sigmoid property of the sigmoid. So in theory, in theory, any, you can use the arcastangents. You can use the cumulative Gaussian. Anything that has this that you can use in a sensible way or in a continuous and monotonic way to map the real line onto the unit interval will, in principle, work. In practice, you want to have a function that gives you straightforward update equations. So you could just learn about the probability of how the interface is appearing. So there is, of course, some background to this that I didn't mention. So we wanted to dissociate pure association learning from reward learning. And we had another condition where people didn't just get houses and faces, but got monetary rewards or not. We wanted to see whether the same mechanisms in the brain were in play when prediction errors were about rewards or when they were simply about associations. So we had to have two things that are associated with each other or associated with a reward. So we wanted to know specifically whether the dopaminergic midbrain becomes active only in learning about reward or in learning about association also. And the answer was the second, the dopaminergic midbrain, cares about both rewards and simple associations. So in some sense, it cares about getting it right. And getting it right is a reward in itself. So you don't need an external monetary or whatever. But if you did the experiment in animals, you would use something like food or drink as a reward. So the dopamine system cares simply also about learning associations. So yes, how you can model that exactly in the same way. Yes, absolutely. The modeling would be even simpler because you wouldn't have this strange kind of mapping here and so on. But there's always sort of a neuroscientific background to this and the neuroscientific question that we are trying to answer through modeling. So the modeling helps the neuroscience. And it's not just done for the sake of the modeling itself. Further questions? Yes, what is the modeling doing? So basically, this is a model, one of several possible models. And we compared many of them for how a subject processes what is going on during the experiment. So what idea does the subject have of how the world works? This is one possible way external reality could produce the sensations you have as a subject while you're doing this. And we do this for several models. So to be strictly exact, no. We are modeling the world here. We're modeling the outside world. So you can see this is x. And we're actually consistent with everything we've done before. I said x is a special case of theta. So that's outside in the world. Once we do inversion, once we do inference, then we're working inside the subject's brain. So once we derive the update equations, as we do here again, so this is what is going on inside the subject. But this is inference. Whenever we're dealing with the muses and the pies, we're dealing with beliefs about the x's. x is the outside world. u and pi is the belief inside my head about x. So this is inside the subject, but the model here, this is a picture of a probabilistic process outside in the world. Because we don't know the exact process as it really is, we have to use sort of simpler ideas of how these things work in general. So this is a quite generic model of how information that has a form of time series can be processed. So the generic HDF update equations, the ones we just saw before, they apply at all the levels where we have continuous states. x2 is continuous. x3 is continuous. x2 and x3 live on the whole real line. But x1 is binary. It lives in the set 0 or 1. So we need to derive update equations for x2 on the basis of the observation on x1. Because we know the outcome. We see whether the outcome was 1 or 0. So this is x1. We don't have to infer x1. We see x1 direct. But we have to infer x2. And our inference on x2 is the form of a belief of a probability distribution with two sufficient statistics. And these are mu2 and pi2, or it's inverse sigma2. So we need a way to get from x1 to the belief on x2. So we need a way for every new observation of x1 to update our mu2 and our sigma2. And again, it's precision weighted prediction. We show on this slide to get out the exact nature of the precision weighting. You can do a Taylor approximation. And you can see that it's all made up of precision terms. And again, it's the same structure. It's prediction plus weighted prediction error. So in the delta 1, delta 1 is minus prediction. Prediction is mu1 half. Yes. Because in the model comparison, three layers were optimal. We had the most model evidence for that number of layers. So whenever we fit the model, we calculate the model evidence, which is basically the free energy that we have. And then we compare that for many models in the same data set. So we use an HDF with two levels, with three levels, with four levels, and we compare that. And we see we have the most evidence for three levels. And then we compare with all kinds of other models. So we also applied for a Scholar-Waffner learning to this. We applied reinforcement learning models, Sutton K1, and so on. What is evidence? P of y given F. And this is integrated over. That's the model evidence. When we had base theorem, we had the posterior given the data and the model is the likelihood divided by the evidence. So in base theorem, here we've got the evidence. The deeper reason why we use that is if you compare two models, now just looking at two models, what you want to look at is the ratio between these two. So the base factor, this is called the base factor, this is the evidence of model one divided by the evidence of model two. Now what is a sensible question to ask when I tell you this is the base factor? Question immediately springs to mind when I tell you this is the base factor.  It looks more like a quotient, doesn't it? But why are we calling it the base factor? So the answer to the question is it is what lets you, the factor that you need to multiply your prior odds with to get your posterior odds. Yes. So yes, that's an important point. We have two different models. For instance, and that's where we started, we have an HDF with four levels and an HDF with three levels. So m1 is three levels, m2 is four levels. For instance, and we fit the same data and we calculate the model evidence here. We can only calculate it approximately, but the variational free energy is a good approximation to that. Other approximations are the AIC or the BIC, things that you see in model scoring. And then there's a whole zoo of these measures. But if you have the variational free energy available, then it's a very good approximation to the, it's one of the better approximations to the model evidence. So let's say you and me both have an idea how probable model one and model two are. And we both write this down. Now we collect some data wide and this changes how probable we think two models are. So this is my belief in the probability in the relative probability of these models. Before I see the data y and this is my two beliefs after seeing data y. And now the thing is, this is called the prior odds. So the ratio of the probability of two events is called the odds. Anybody familiar with betting will know that. When you go to the betting shop, they give you the odds. So they say the odds of this horse winning are seven to one. And the longer the odds are, then the more unlikely it is that horse to win. Now these are the posterior odds. And what gets us from the prior odds to the posterior odds is the base fact. So this is the y. And now the thing is, this here is a objective. The prior odds may be different for you and for me. But the factor that takes me and you from the prior odds to the posterior odds is the same for us both. And this is a measure of how good in the light of the data y is model one relative to model two. If in the light of data y, they are both equally good, then this ratio will be one. And both of us will not change our opinion. So the posterior odds will be equal to the prior odds. However, if applied to these data y, the model m1 is superior to the model m2, then both of our beliefs will change in the favor of model one. So this is the model evidence. That's why it's because it allows us to compare models. That's why it's such an important point. We can only calculate it approximately in all interesting cases. But it's fundamental for model comparison. And on the basis of this, do we decide how many levels to include in the modeling? Yes. So do we have any sort of predictions of how the model evidence will change if we add a layer? No, I don't know. Yes. But it's always, as we saw, it's always a trade-off with the accuracy. The complexity will usually rise by adding new layers. But the accuracy may also rise. And if the accuracy rises more than the complexity, then the more, no. Because you've got the complexity term in there, which safeguards against overfitting. So we always compare how much does an additional layer, or in general additional parameters, how much do they affect my complexity and my accuracy? And it depends on the balance of this effect, whether your new model will be superior. So if the introduction of new parameters increases the accuracy by more than it increases the complexity, then the introduction of your new parameters will be justified. So if the introduction of the new parameters increases the complexity more than the accuracy, then you can say if adding another layer adds more to the complexity than to the accuracy, then you shouldn't. So that's just the logic. So always when comparing different models, you will prefer the ones with more evidence. And evidence is accuracy minus complexity. So I calculate the evidence for two layers, for three layers, for four layers. I see a maximum at three layers, so I take three layers. Theoretically, I could go up to, so you think it could go down for four layers and then up again for five layers. So there are principled reasons not to expect that, you know, because adding a layer will only help you if it increases your accuracy. So if you, if adding a fourth layer doesn't help you, then adding a fifth layer won't suddenly start helping you. I mean, this is a quite straightforward situation. Not all situations may be so straightforward. So you cannot always predict what will happen to complexity and accuracy when you tweak the model at this end as opposed to that. Sorry, I didn't get that. So the fetas here are the states and parameters of the model. So theta is a collection of all that we have here. The x's are parts of theta. The kappas, the omegas, the fetas, and this sort of thing. So because this is a time series model, what gets updated is, again, in the inversion process. And in the inversion process, the parameters kappa and omega and theta here are not updated because they are assumed to be constant here, which is a modeling assumption. However, the sufficient statistics describing our belief on x1, x2, x3, they are updated step by step. That's what we have the update equations for. So the update equations are for mu2 and sigma2 or mu2 and pi2 and for mu3 and pi3. That's what gets updated step by step. So here, this is an example of exactly what I'm talking about. So the three parameters, this is a simulation. The three parameters are chosen. So I go and I choose a theta of 0.5 and omega of minus 2.2, a kappa of 1.4. And then I give these x1's, the little green dots to my little model. And this is how mu2 gets updated in the course of time of these 320 trials that we have in the experiment. And this is how mu3 gets updated. And at both levels, also the uncertainty gets updated. So at this level, it's not plotted in this plot. But at this level, we have pi2. At this level, we have pi3. That's also updated. But the parameters theta, omega, and kappa are constant. So in this example, we see an interesting effect. Because what I did when I simulated this was I took the exact same inputs in the first 100 trials as in the last 100 trials. But you can see that the belief updates, and what I plotted here at the first level is the prediction of the next outcome being 1. So this is our little agent making a prediction about the next outcome in the form of telling me the probability of getting an outcome of 1. You can see that in these phases where there's lots of outcomes equal to 1, or I promise to show everything over here, in these phases where there's lots of outcomes equal to 1, the belief that the outcome will be 1 increases. And then there's almost no outcome equal to 1 anymore, but lots of outcomes equal to 0. So the prediction that the outcome will be 0 goes down. And as soon as there's another 1, it ticks back up a little and so on. Because we have this intervening period of volatility here. So here, the probability that the black line here is the objective ground truth probability of getting a 1 or a 0. And we leave it at 0.5. So 1 and 0 are equally likely for the first 100 trials. And quite appropriately, the belief that the next outcome will be a 1 sort of fluctuates around 0.5. Actually, learning is taking place at a somewhat too quick pace here. So you can see this is basically overfitting by our little agent. So our little agent interprets too much into the fluctuations that you have between 1 and 0 because they're entirely random. So ideally, you would just have a flat line around 0.5 because there's actually nothing to learn from these outcomes. They're entirely random, equally probable to get a 1 as to get a 0. But then this changes. The probability of getting a 1 jumps up to 0.8. And our agent learns that this is now more probable. Probability jumps down to 0.1 or 0.2, I don't know. I don't remember. And the belief goes down and up again and down again and up again and down again. And now, because the agent also learns about volatility, its belief in the volatility has increased. This is the volatility level. And now, the same input that the agent got here is interpreted in a different way. Namely, it is seen as much more informative in the sense that because we're in a volatile environment, we have to learn quickly. So even though the agent was already learning a bit too much here, it's now learning even more. So its beliefs are wildly swinging around, but in response to the same inputs here. And this is just, yes, because the belief about the volatility of the environment is increasing. Yes, yes, yes, exactly. You build a memory of how volatile the environment is or has been, and that affects your learning rate. So to put it very slim, simply here you've got a higher learning rate than back here. And this is because the agent has learned that these changes in contingencies are quite likely to occur. And if you keep it stable for a very long while, this belief about the volatility will come down again. It's not long enough here. In the bottom, the black line is the ground truth from which the green inputs are sampled, but the agent doesn't know this. So here I sample my outcomes, 0 and 1, from a Bernoulli distribution with parameter 0.5. And here I sample the outcomes from a Bernoulli distribution with parameter 0.8. And here from a Bernoulli distribution with 0.2, and so on. So this is basically what I'm doing, what I'm using to feed the agent with inputs, with x1s, how I'm generating the x1s. Yes. Everything changes again, and you have to relearn. It's not useless, because here you accurately learn that it's now much more likely to be 1, and here you accurately learn. But as soon as he's learned what he needs to learn, we change the world again. You need to, yes. So let's look at, I will show you some real data. So this is an actual subject. You can see this is actual data. Here the x's indicate that this person missed the trial. So he wasn't quick enough in giving an answer. And you can see this guy learns a lot. So he has a kappa of 4.1. We'll see what that means. But you can see this guy has an enormous learning rate. It's ridiculously high. And because everything's changing all the time, you can see his volatility belief also increasing most of the time. So this is one subject. And this is an inferred learning process for this actual human being. And here you have another guy. And this guy is much slower. And he's so slow he misses giving an answer all the time. But anyway, we have hundreds of answers from him. And this still allows us to fit his belief trajectory. And actually, the learning rate here is much more appropriate. So you can see these fluctuations in the beliefs. But it's not as if it was pointless for them to try to do a good job on the task because they do get more answers right than wrong. So most of the more of their predictions are correct than incorrect. It's not as if you couldn't learn anything here. It's hard. It's not easy. But it's not as if it were pointless. It would be pointless if just randomly answering gave you the same score in the end as trying to learn something. That's not the case here. So you will be far from being correct on every answer, but you will be correct on considerably more than half of your answers as you're doing this task. And also just as a consideration of experimental design, people are lying in the scanner while they're doing this. And if what they have to do is too easy or too boring, they just go to sleep.  So if it gets boring, people go to sleep. So just out of that consideration also, you have to gauge your task so that it is interesting. It has to be somewhat challenging. It shouldn't be too easy, but it shouldn't be pointless either. So and this worked well in this task. So it takes about 25 minutes for people to do this. Yes. Sorry. This is from the behavioral data we recorded while people were in the scanner. We also recorded how their brains reacted to what they were doing. And we have nice pictures of how the brain processes stuff that's going on here and how the brain processes what's going on here. So it's mostly the dopamine-ergic system dealing with this. At least that's what we found in this experiment. And mostly the colonergic system. So these are two different neurotransmitters, two different neuromodulators. So dealing with this is mostly in the midbrain.  It's an old part of your brain. And here you've got mostly stuff going on at the volatility level in the basal forebrain, which is a younger part of the brain. And one is mostly populated by dopamine-ergic neurons. So neurons that release dopamine as a neurotransmitter. And here the activity is mostly in a region where the brain has neurons that release acetylcholine. Yes. No, that's the actual data you get from their behavior. It's their button presses. So they lie in the scanner. And they have a button box. And one button means I predict a face. And the other button means I predict a house. I get the MRI data. I get the MRI signal from their brain. No. So this is just from the behavior. All we have is their button presses. So the button presses are the orange dots. If they press 1, or if their button press corresponds to a 1. So we don't have it on the blackboard anymore. But if the button press corresponds to 1, the orange dot is up here. If the button press corresponds to a 0, the orange dot is down here. By feeding their response, your vision is fit together. So we have one estimate for omega. We have one estimate for kappa. You saw the one before had a kappa estimate of around 4, if I remember correctly. Now this one has a kappa estimate of 1.2. And the theta estimate here in omega is fixed to minus 1. And to minus 4, sorry. And but it's a good question because we need to have a model of, where is this? We need to have a model of how people translate their beliefs into decisions. On top of the learning model, which describes how they update their beliefs, we need a so-called decision model. And as I said, we update the mu2, sigma2s, mu3, sigma3s on each trial. And then we have a decision model that is based in this case. But you can build other decision models. But this little example here is based on the sigmoid of mu2. Why is that? Because on mu2, the belief fluctuates. So here we have 0. Now if we apply a sigmoid to this, we get what we call mu1 hat. So this is the prediction on x1. So when mu2 is up here, the belief that the next outcome will be 1 is high. When mu2 is back here, the belief is low. And this corresponds to these belief trajectories that we just saw. Let's take this one. This is mu2 going around. Here is 0. So whenever it goes high, then the probability that the next outcome is 1 is high. Whenever it goes low, the probability of getting a 1 as an outcome is low. So our decision model is based on the subject's belief here about the probability of the outcome 1. Sigmoid of mu2 is basically a measure of the subject's belief that the next outcome will be 1. And now this is mapped onto the probability of the subject predicting an outcome 1. So we want to allow for the probability, for the possibility, that even when a subject has a belief that the next outcome is 90% probable to be a 1, the probability of predicting that will not be 90% because there may be many reasons for that. The simplest explanation is that people are, in some sense, noisy in their decisions. The decisions are not an accurate reflection of their beliefs always. Another explanation, a more principled explanation, is people are exploratory. Even when you're certain that one option would be the most rewarding option, you may want to choose another option because you want to learn about your environment. You know that if you go to restaurant A, you will have an enjoyable evening. You know that because you've been there 1,000 times. But now there's a new restaurant. And your expectation of having an enjoyable evening is actually less than because it's a new restaurant who knows and your prior is sort of set around the average new restaurant. But you want to go there because you want to explore. So the probabilities about the outcomes don't always are not entirely predictive of behavior. So it's not 90% probable that people will predict an outcome of 1 when they believe then an outcome of 1 is 90% probable. So why now is the decision by the subject? And this is our model. This is now also a sigmoid model, but it's a different kind of sigmoid here. It's the unit square sigmoid that basically is a sigmoidal map from the unit interval to the unit interval. So we have a decision model. And based on this decision model, we can infer the beliefs of the subject as the subject does the task. Yes, I have the actual decisions of the subject. I have the orange dots. And I know what I put in. I have the orange dots and the green dots. And then we go in search of the best parameters, the parameters that best explain, parameter values that best explain the subject's decisions in light of the input. That's what we do when we fit this model. Yes, we estimate that subject by subject. Each subject has its own value there, because some people are more noisy than others. Some people, very reliably, choose the option they think is more likely. So that would be a high zeta. So you can see for a zeta value of 64, you get an agent who, as soon as the probability of getting a 1 is greater than 0.5, will always predict 1. So that's sort of the reward maximizing behavior. Yes, if you take an economic definition of rationality, where a rational agent is one who maximizes his reward, that would be the most rational. Yes. But I mean, there are many ways to think about this. And there are more kinds of rewards than the immediate reward you get from an outcome. There are also rewards from knowing your environment. If you want to know more about this, there's a nice paper on active inference and epistemic value by first authorist Carl Friston, active inference and epistemic value. And you can see how the actual value of an action has more to it than what economists would call a rational choice, simply based on utility. So there's more to utility in deciding what to do, even in a very vigorous way you can show that. Yes. Yes. I'm happy to do that. This is actually a good slide where to do that. So this is, again, our generative model. So it's x here. Sorry, I'm again on the wrong side. Should do it over here or do it with the mouse. x3, the parameter we have in the model is theta. This is the variance of the Gaussian random walk at the top level. x2, because the equation for x3 at time k is normal around x3 k minus 1 and theta. So that's the top level. That's where the theta comes from, this theta. At the second level, x2k normally distributed around x2 k minus 1. And then we have the exponential of kappa 3k plus omega. Yes. So we fixed omega, because the data only allowed us to estimate, and there's a deep reason for that, which would take too much time to explain here, but the data only allowed us to estimate kappa and theta. These two are estimated. Omega is fixed to minus 4 in this experiment. Yes, exactly. In the decision model, there's also zeta. And zeta is also estimated, subject by subject. And then to make it even slightly more complicated, the initial values for all the trajectories, these are also parameters. And some of them are fixed and some of them are estimated. I think we estimated, I don't know exactly which one of the initial values we estimated in which ones we fixed. So in general, if you have data from 320 trials with binary decisions, the amount of information you have is 320 bits, so 320 ones or zeros. That is not much information. So there are limits to how many parameters you can estimate with an amount of information that's just 320 bits. And for that reason, and also the reason why we fixed omega, there's a deep reason for that. And for the initial values, there's a limit to the number of parameters that you can estimate. And maybe we'll have time to go into how we decide which ones these are. We also have to look at the posterior correlation, for example, of these parameter estimates. Because if they're highly correlated either positively or negatively, that means two parameters are fundamentally explaining the same thing. So all you can do in such a situation is to choose a value for one of them and estimate the other one. But so the interesting parameters that we fixed here were the theta and the kappa and the decision noise theta. So I think we're scheduled to go until 1, right? Now it's 5 past 1. OK, so I wish you a happy weekend and see you next week.