4 ms·

Someone hands me a thumb drive and it consists of one byte, 0000 0001. On Monday, the owner of the thumb drive tells me the contents are a counter of how many

by omeze 7y ago

Someone hands me a thumb drive and it consists of one byte, 0000 0001. On Monday, the owner of the thumb drive tells me the contents are a counter of how many apples they had in their kitchen on new year. On Tuesday, the owner tells me the contents are a boolean for whether they took a vacation in 2014. What is the entropy of the data?

fossuser 7y ago
https://www.lesswrong.com/posts/7X2j8HAkWdmMoS8PE/disputing-definitions https://www.lesswrong.com/posts/7X2j8HAkWdmMoS8PE/disputing-... You can deduce that the order of some data implies it's likely information. A person can also have a small amount of data be information even if it would be hard to deduce otherwise. Seems like arguing about this is just arguing about the definition of a word.
- frenchyatwork 7y ago
  Unfortunately, the reason why it seems like arguing about the definition of a word is that there are there are more concepts at play here than we have words to describe them. When people talk about information as in information theory, what they're talking about is devoid of actual semantic contact, but when normal people people talk about information, that normally implies some sort of semantic content.
  - yters 7y ago
    Information theory addresses semantics through mutual information. 'Meaning' entails reference, and when A refers to B they have mutual information. Mutual information is how Shannon measures the carrying capacity of a channel.
    - westoncb 7y ago
      This is still just talking about a property of information that resembles one aspect of what's meant by "meaning". This is all still purely structural and independent of semantic content. The example given above about alternate interpretations of the same bit string on a USB drive gets at the problem. Semantics is about the process of interpreting data into something "meaningful" to some particular system (e.g. a human). It's essentially an importing process: converting data to a format that some system knows how to operate on (i.e. "understands). To keep it brief, meaning (or "information" in the non-jargon sense) is about a relation between some data and a system which that data operates on (or the system operates on the data, depending on how you look at it). "information" from information theory is an intrinsic property of data[0]: the information content doesn't change based on what's interpreting it. (At least that's my understanding.) [0] I'm using "data" here to refer to something like a serialization of a system into a form where information-theoretical analysis can be applied to it (e.g. like a bit string).
      - yters 7y ago
        Like I mentioned previously, there are objective sufficient criteria to identify meaning. Even if the interpretive context is subjective. So, we can guarantee true positives, even if we cannot guarantee true negatives.
        westoncb 7y ago
        > there are objective sufficient criteria to identify meaning You have just shifted the problem by now using a very specific, narrow definition of "meaning" rather than doing so with "information". The objective sufficient criteria to identify meaning only work because this definition of meaning refers to an intrinsic property of a system; however, the common usage of "meaning" involves a relation between multiple systems.
        yters 7y ago
        Not sure what you are saying. Informative content has both intrinsic (randomness deficiency, entropy, kolmogorov sufficient statistic) and relational properties (also randomness deficiency, mutual information, algorithmic mutual information) that can be objectively and quantitatively measured.
        westoncb 7y ago
        Intrinsic/relational may not be the best way of getting at it after all. It would likely regress into an equally involved question on defining "system". So let me try another approach. We need to distinguish between information and the "semantic content" of information. Here are the definitions I have in mind: - Data: an arbitrary string - Information: a numeric property of some data: INFO(d) - Mutual information: a numeric property of two pieces of data: MI(d1, d2) - Semantic content: the particular "effect on a system"[0] that results when the system internalizes the data: SC(s, d) ('s' denotes the system.) My understanding is that you're claiming MI == SC. There are certain ways we could fiddle with my usage of 'system' above so that it can just be 'data,' too (e.g. representing the system as a string, as with Turing Machines that operate on string reps of Turing Machines). In which case we can at least say that MI and CS both have domains like (data X data), so there is some similarity there. But, even just the fact that MI is going to evaluate to a single numeric quantity, while SC evaluates to an "effect on a system" (i.e. following a state transition), implies that they're referring to different things. I think it's also clear that it would be possible for MI(d1, d2) to evaluate to the same number as MI(d3, d4), even though SC(d1, d2) !== SC(d3, d4), since all the informational relations could be equivalent (between (d1, d2) and (d3, d4)), and yet the particular state transitions followed could differ. [0] We can make this more precise. For instance, in the language of automata theory it would be something like following a particular state transition. Edit: I should clarify one implicit thing here: this hinges on an assumption that my definition of "semantic content" would be a satisfactory match to common usage of the phrase for most people. Happy to hear an alternate that also captures common usage, and I can expand on my choice of def.
yters 7y ago
A better example is the difference between the contents of a normal English book vs a randomly generated string of letters. The former has a lot of structural properties the latter lacks, plus shares a large amount of mutual information with a source independent from the physical properties of the book.
- ganzuul 7y ago
  When we cast our randomness as a string we impose a lot of structure. Proportionally, getting to English from there might involve a lot less new structure than in the previous step. Often I wonder if I'm just spouting gibberish, saying things like this, but it does make sense to me.
  - yters 7y ago
    No, the information cost grows exponentially going from random letters to English sentences. We can measure this with crossentropy to get a probability of generating a well formed English sentence from a uniform distribution over the letters. To generate just a few hundred words we end up needing more trials than are available in the lifespan of our universe.
mattkrause 7y ago
8 bits, by definition. The crux of information theory is that you can analyze a message without know what it means. Mutual information would help you quantify how much this changes your mental model of the owner's life. Maybe you already knew about the vacation (you went too?). It would let you quantify how much this change in definitions affected you.
- goatlover 7y ago
  What makes those bits zeros and ones? What makes the thumb drive an information storage device? Why do we need to know the mental model of the owner's life to infer the information stored? For information to be objective, it needs to be understood as mind-independent feature of the world. If you have to reference human minds to make it meaningful, then it's not objective. It's a cultural construct.
  - mattkrause 7y ago
    I took the byte and thumb drive bit literally, but if it were some exotic ternary system, it can hold about ~12.7 bits. If they're decimal digits instead, that sequence has a capacity of 10 bans or 26.6 bits. The name "information theory" seems to throw people; imagine it were called coding theory instead and it might seem clearer that we can talk about the "capacity" of different codes and how different representations interact without worrying too much about what they mean. It's similar to algebra, in that we can make statement about `x + y = y + x` or `n + 0 = n` without worrying about whether x is measured in apples, lightyears, or anything at all.
    - goatlover 7y ago
      Right, but if a future alien archaeologist were to dig up a thumb drive (preserved in a vault) after humans went extinct, what would indicate that it held bits?
      - mattkrause 7y ago
        A 'bit' in information theory is just a unit of measurement, corresponding to the information conveyed by one yes-or-no question (where both answers were previously equally likely). Since it's a fairly natural unit, your alien archaeologist might even use it. However, if they had a weird taboo against binary, they could define a similar concept in other ways. In fact, we humans have too. Bletchley Park used "bans" or hartleys, which is the information carried by a single base-10 digit (without any a priori information). Nats use a base-e representation instead, which ties into statistical physics nicely. There's nothing stopping you from inventing a weird informational unit where 1 corresponds to "the answer to a seven way question were two of the answers are each twice as likely as the rest." So, suppose your alien digs up my thumbdrive. It examines it carefully and notices that each MOSFET is always in one of two states. That suffice to describe its information-carrying capacity in whatever system they use: bits, hartleys, nats, zorblaxen. A similar analysis of their own media would let them say "This thing is so sophisticated, it can hold the work of 10,000 scribes" or "This junk can barely contain a moment of our 16D videos." If your drive were packed with different types of media, they might even discover that some parts of the disk (e.g., ascii text) DON'T use all of the available capacity. None of this, of course, tells them what the data means. For that, you'd need to relate these data to some external source.

by omeze 7y ago

fossuser 7y ago

https://www.lesswrong.com/posts/7X2j8HAkWdmMoS8PE/disputing-definitions https://www.lesswrong.com/posts/7X2j8HAkWdmMoS8PE/disputing-... You can deduce that the order of some data implies it's likely information. A person can also have a small amount of data be information even if it would be hard to deduce otherwise. Seems like arguing about this is just arguing about the definition of a word.

frenchyatwork 7y ago
Unfortunately, the reason why it seems like arguing about the definition of a word is that there are there are more concepts at play here than we have words to describe them. When people talk about information as in information theory, what they're talking about is devoid of actual semantic contact, but when normal people people talk about information, that normally implies some sort of semantic content.
- yters 7y ago
  Information theory addresses semantics through mutual information. 'Meaning' entails reference, and when A refers to B they have mutual information. Mutual information is how Shannon measures the carrying capacity of a channel.
  - westoncb 7y ago
    This is still just talking about a property of information that resembles one aspect of what's meant by "meaning". This is all still purely structural and independent of semantic content. The example given above about alternate interpretations of the same bit string on a USB drive gets at the problem. Semantics is about the process of interpreting data into something "meaningful" to some particular system (e.g. a human). It's essentially an importing process: converting data to a format that some system knows how to operate on (i.e. "understands). To keep it brief, meaning (or "information" in the non-jargon sense) is about a relation between some data and a system which that data operates on (or the system operates on the data, depending on how you look at it). "information" from information theory is an intrinsic property of data[0]: the information content doesn't change based on what's interpreting it. (At least that's my understanding.) [0] I'm using "data" here to refer to something like a serialization of a system into a form where information-theoretical analysis can be applied to it (e.g. like a bit string).
    - yters 7y ago
      Like I mentioned previously, there are objective sufficient criteria to identify meaning. Even if the interpretive context is subjective. So, we can guarantee true positives, even if we cannot guarantee true negatives.
      - westoncb 7y ago
        > there are objective sufficient criteria to identify meaning You have just shifted the problem by now using a very specific, narrow definition of "meaning" rather than doing so with "information". The objective sufficient criteria to identify meaning only work because this definition of meaning refers to an intrinsic property of a system; however, the common usage of "meaning" involves a relation between multiple systems.
        yters 7y ago
        Not sure what you are saying. Informative content has both intrinsic (randomness deficiency, entropy, kolmogorov sufficient statistic) and relational properties (also randomness deficiency, mutual information, algorithmic mutual information) that can be objectively and quantitatively measured.
        westoncb 7y ago
        Intrinsic/relational may not be the best way of getting at it after all. It would likely regress into an equally involved question on defining "system". So let me try another approach. We need to distinguish between information and the "semantic content" of information. Here are the definitions I have in mind: - Data: an arbitrary string - Information: a numeric property of some data: INFO(d) - Mutual information: a numeric property of two pieces of data: MI(d1, d2) - Semantic content: the particular "effect on a system"[0] that results when the system internalizes the data: SC(s, d) ('s' denotes the system.) My understanding is that you're claiming MI == SC. There are certain ways we could fiddle with my usage of 'system' above so that it can just be 'data,' too (e.g. representing the system as a string, as with Turing Machines that operate on string reps of Turing Machines). In which case we can at least say that MI and CS both have domains like (data X data), so there is some similarity there. But, even just the fact that MI is going to evaluate to a single numeric quantity, while SC evaluates to an "effect on a system" (i.e. following a state transition), implies that they're referring to different things. I think it's also clear that it would be possible for MI(d1, d2) to evaluate to the same number as MI(d3, d4), even though SC(d1, d2) !== SC(d3, d4), since all the informational relations could be equivalent (between (d1, d2) and (d3, d4)), and yet the particular state transitions followed could differ. [0] We can make this more precise. For instance, in the language of automata theory it would be something like following a particular state transition. Edit: I should clarify one implicit thing here: this hinges on an assumption that my definition of "semantic content" would be a satisfactory match to common usage of the phrase for most people. Happy to hear an alternate that also captures common usage, and I can expand on my choice of def.

yters 7y ago

A better example is the difference between the contents of a normal English book vs a randomly generated string of letters. The former has a lot of structural properties the latter lacks, plus shares a large amount of mutual information with a source independent from the physical properties of the book.

ganzuul 7y ago
When we cast our randomness as a string we impose a lot of structure. Proportionally, getting to English from there might involve a lot less new structure than in the previous step. Often I wonder if I'm just spouting gibberish, saying things like this, but it does make sense to me.
- yters 7y ago
  No, the information cost grows exponentially going from random letters to English sentences. We can measure this with crossentropy to get a probability of generating a well formed English sentence from a uniform distribution over the letters. To generate just a few hundred words we end up needing more trials than are available in the lifespan of our universe.

mattkrause 7y ago

8 bits, by definition. The crux of information theory is that you can analyze a message without know what it means. Mutual information would help you quantify how much this changes your mental model of the owner's life. Maybe you already knew about the vacation (you went too?). It would let you quantify how much this change in definitions affected you.

goatlover 7y ago
What makes those bits zeros and ones? What makes the thumb drive an information storage device? Why do we need to know the mental model of the owner's life to infer the information stored? For information to be objective, it needs to be understood as mind-independent feature of the world. If you have to reference human minds to make it meaningful, then it's not objective. It's a cultural construct.
- mattkrause 7y ago
  I took the byte and thumb drive bit literally, but if it were some exotic ternary system, it can hold about ~12.7 bits. If they're decimal digits instead, that sequence has a capacity of 10 bans or 26.6 bits. The name "information theory" seems to throw people; imagine it were called coding theory instead and it might seem clearer that we can talk about the "capacity" of different codes and how different representations interact without worrying too much about what they mean. It's similar to algebra, in that we can make statement about `x + y = y + x` or `n + 0 = n` without worrying about whether x is measured in apples, lightyears, or anything at all.
  - goatlover 7y ago
    Right, but if a future alien archaeologist were to dig up a thumb drive (preserved in a vault) after humans went extinct, what would indicate that it held bits?
    - mattkrause 7y ago
      A 'bit' in information theory is just a unit of measurement, corresponding to the information conveyed by one yes-or-no question (where both answers were previously equally likely). Since it's a fairly natural unit, your alien archaeologist might even use it. However, if they had a weird taboo against binary, they could define a similar concept in other ways. In fact, we humans have too. Bletchley Park used "bans" or hartleys, which is the information carried by a single base-10 digit (without any a priori information). Nats use a base-e representation instead, which ties into statistical physics nicely. There's nothing stopping you from inventing a weird informational unit where 1 corresponds to "the answer to a seven way question were two of the answers are each twice as likely as the rest." So, suppose your alien digs up my thumbdrive. It examines it carefully and notices that each MOSFET is always in one of two states. That suffice to describe its information-carrying capacity in whatever system they use: bits, hartleys, nats, zorblaxen. A similar analysis of their own media would let them say "This thing is so sophisticated, it can hold the work of 10,000 scribes" or "This junk can barely contain a moment of our 16D videos." If your drive were packed with different types of media, they might even discover that some parts of the disk (e.g., ascii text) DON'T use all of the available capacity. None of this, of course, tells them what the data means. For that, you'd need to relate these data to some external source.