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Day 38: Algorithmic complexity

I’ve had an interest in Kolmogorov complexity for a long time, ever since I was first introduced to the minimum description length principle through the work of Peter Grünwald and Jorma Rissanen. The idea is deceptively simple. The inherent complexity of any given string is the length of the shortest program that is guaranteed to produce that string and then stop. So for instance, a string like this

abbabbabbabbabbabbabbabbabbabb"

can be produced with a very short R program:

str_dup("abb",10)
## [1] "abbabbabbabbabbabbabbabbabbabb"

whereas a random-looking string like

gjoxfxyrqbferjumszjuyfqdgkajwi

is not as easy to compress, and is hence more complex. However, compressability is a tricky thing. Firstly, even if you know what programming language is being used to do the work, the Kolmogorov complexity of a sequence is uncomputable. Secondly, the choice of programming language does matter, up to a point. For any string \(s\) the Kolmogorov complexity of that sequence \(K_a(s)\) in programming language \(a\) will not be the same the complexity with respect to language \(b\), though for any pair of languages there is a finite upper bound on how large the discrepancy can be: intuitively, if you can write a finite length of the compiler that translates from one language to the other, then \(|K_a(s)-K_b(s)|\) cannot possibly be larger than the length of the compiler! Thirdly, if we try to use a general purpose programming language to do the compression we can get weird and counterintuitive results. To my mind, this is a pretty random looking sequence:

gjoxfxyrqbferjumszjuyfqdgkajwimpmevrucskvquonuamtsmwlgbcinrkxhliqgmtcwivjimxwkuylskitfsdgdgbqwuulkvprjhzqfdmypztjldasclqzmmetlnffpocaqyponznrpggsletcwpoilnenbhfhxluwkbisiqvwkjxqtpxhexnwetsyoskcyhpcviugfngenodgsyctyviqyyigeinyngbkwjdjqkrrollhpxdkfldlytympmcgmjynihusebtqebcjehegemuanwjbdiedffdzinrcdbyrcmmjzevbkdfvsgmcjzqrikzwyvugtzhkvcjlepzzeojrhmejsmbukzhwcxmcivhpvbssmloyffkiwemlorzgguvczddynejrijdreyxysjuayzjtusmmisvlnrdikcyvwybjncutbmxbhnpgljydbzljebrpzpbemalgysemrlyskdgwlmrtewypneawdayhekllnwborrrmzkwtnwmasseqoiqvsjdkyutyzpaihdbjieqkyqifczjotwoaxujcbvvrditxsgqhyekgcvnrkvtjyqapkbnzsoguqvqmmjlvydtzzjkycyjnoskdiqbwfodiqgqmywjivlicbrrxhyfufawryrvjkocfptwjubqjpxfjrtnvnnkjdhhuudnpnklaxcnvplouezmyxtrqblnyssmdulladktapxsgwiplbsotbshhvcbjophfjmvdrmxodlelfyluewfeeoswsmvazqlatfzxstmweqhlbebyinqgnujwazthrjyiajkjyjoyxgsjotvwafjhssxvukobwynydgmmlbqsjytbzfagsggagrabkmqsbkzvfmagltziyzxmuurpiabcaidpaaevthefbbdbbnusbalessxizzbxvshnlpyghttymwifxbklkrtxvceqwpgwemajldqhgjxztzolzdbapsudezkemoizndlqfduqheg

However, if I use R as the compressing language, there is a very short program that produces it:

set.seed(1);cat(sample(letters,1000,T),sep="")
## gjoxfxyrqbferjumszjuyfqdgkajwimpmevrucskvquonuamtsmwlgbcinrkxhliqgmtcwivjimxwkuylskitfsdgdgbqwuulkvprjhzqfdmypztjldasclqzmmetlnffpocaqyponznrpggsletcwpoilnenbhfhxluwkbisiqvwkjxqtpxhexnwetsyoskcyhpcviugfngenodgsyctyviqyyigeinyngbkwjdjqkrrollhpxdkfldlytympmcgmjynihusebtqebcjehegemuanwjbdiedffdzinrcdbyrcmmjzevbkdfvsgmcjzqrikzwyvugtzhkvcjlepzzeojrhmejsmbukzhwcxmcivhpvbssmloyffkiwemlorzgguvczddynejrijdreyxysjuayzjtusmmisvlnrdikcyvwybjncutbmxbhnpgljydbzljebrpzpbemalgysemrlyskdgwlmrtewypneawdayhekllnwborrrmzkwtnwmasseqoiqvsjdkyutyzpaihdbjieqkyqifczjotwoaxujcbvvrditxsgqhyekgcvnrkvtjyqapkbnzsoguqvqmmjlvydtzzjkycyjnoskdiqbwfodiqgqmywjivlicbrrxhyfufawryrvjkocfptwjubqjpxfjrtnvnnkjdhhuudnpnklaxcnvplouezmyxtrqblnyssmdulladktapxsgwiplbsotbshhvcbjophfjmvdrmxodlelfyluewfeeoswsmvazqlatfzxstmweqhlbebyinqgnujwazthrjyiajkjyjoyxgsjotvwafjhssxvukobwynydgmmlbqsjytbzfagsggagrabkmqsbkzvfmagltziyzxmuurpiabcaidpaaevthefbbdbbnusbalessxizzbxvshnlpyghttymwifxbklkrtxvceqwpgwemajldqhgjxztzolzdbapsudezkemoizndlqfduqheg

All I’ve done here is draw 1000 “random” samples with replacement from the alphabet, and I’ve fixed the seed for the pseudorandom number generator, so that the output is deterministic. To my mind, this is a stupid way to define regularity in a sequence – by construction, the Mersenne Twister is designed to generate deterministic sequences that mimic randomness to the greatest possible degree. Calling the gjoxfxyr... sequence “structured” seems absurd.

In the real literature on algorithmic complexity, these “exceptions” are ignored, and for very good reason. There are results proving that the set of sequences that any language can possibly assign short descriptions to is tiny relative to the full space of possible sequences (I can’t remember the specifics now, but I vaguely recall it being a set of measure 0 - I think I read something that Gregory Chaitin wrote about this at some point).

For finite sequences, especially short sequences, it starts to matter an awful lot more what machine you use to do the compression. Rather than rely on a universal Turing machine, practical versions of the minimum description length principle rely on specific Turing machines (for a particularly elegant example, see Wallace & Dowe 1999)

In statistics, this idea is of considerable interest due to results such as the Kraft inequality which establishes a correspondence between codelength functions and probability mass functions; meaning that any statistical model can be interpreted as a Turing machine, and the probability that model assigns to the data has an interpretation as a codelength. It also means that you can interpret a Bayesian model with prior \(P(\theta)\) defined over a parameters space as a universal code that summarises a collection of likelihood 1 functions \(P(x|\theta)\), though it is worth noting that Jorma Rissanen in particular expressly disavows this interpretation. In any case, there are a number of very interesting papers examining the relationships between minimum description length and probabilistic reasoning.

In cognitive science, a lot of the interest stems from the idea of thinking about “efficient description” as one of the jobs that the mind needs to accomplish. Nick Chater, for instance, has written a number of interesting papers about the simplicity principle, and if you’re ever bored and want to read some of Tom Griffiths’ early work you can find gems like this paper connecting subjective and algorithmic complexity.

All of which is by way of background.

A while back I ran into a Behavior Research Methods paper by Gauvrit et al (2016) which aims to provide a representation of the inherent algorithmic complexity of very short sequences. Better yet, it introducess an R package acss, that catalogs the complexity of every possible sequence of length 12 or less, written in an alphabet containing no more than 9 characters. Obviously, given everything I’ve said above, this is impossible to do in a fully general way, using a universal Turing machine. However, if you restrict the class of Turing machines in some sensible fashion, it is absolutely possible do to this. The method that they used is described here but the gist of it is that they “randomly” select Turing machines from a large family (of machines with \(n\) states and \(m\) symbols) and then calculate the probability \(D_{n,m}(s)\) that a machine sampled in this fashion produces string \(s\). The value of \(-\log_2(D_{n,m}(s))\) is an estimate of the codelength (in bits) of the compressed sequence.

To see it in action, I’ll first define a function that takes a string like "sssssss" and then computes the complexity of the substrings "s","ss","sss", etc

acss_sequence <- function(str) {
  ac <- str %>%
    str_split("") %>% 
    unlist %>% 
    accumulate(paste0) %>% 
    acss %>% 
    subset(select = "K.9")
  
  tibble(
      Group = ac %>% rownames %>% last,
      String = ac %>% rownames,
      Length = ac %>% length %>% seq_len,
      Complexity = ac %>% as.vector
    )
}

And calling it…

acss_sequence("ssssssss")
## # A tibble: 8 x 4
##   Group    String   Length Complexity
##   <chr>    <chr>     <int>      <dbl>
## 1 ssssssss s             1       4.96
## 2 ssssssss ss            2       7.93
## 3 ssssssss sss           3      11.7 
## 4 ssssssss ssss          4      15.0 
## 5 ssssssss sssss         5      18.2 
## 6 ssssssss ssssss        6      20.9 
## 7 ssssssss sssssss       7      23.1 
## 8 ssssssss ssssssss      8      24.9

Not surprisingly, complexity increases as a sequence becomes longer, even if it’s the the same symbol being added. However, when I add different symbols at each step, the complexity rises faster

acss_sequence("abcdefgh")
## # A tibble: 8 x 4
##   Group    String   Length Complexity
##   <chr>    <chr>     <int>      <dbl>
## 1 abcdefgh a             1       4.96
## 2 abcdefgh ab            2       7.93
## 3 abcdefgh abc           3      11.9 
## 4 abcdefgh abcd          4      16.3 
## 5 abcdefgh abcde         5      20.7 
## 6 abcdefgh abcdef        6      25.1 
## 7 abcdefgh abcdefg       7      29.6 
## 8 abcdefgh abcdefgh      8      34.1

More generally, here’s how complexity rises when I try out different possibilities:

  • aaaaaaaaaaaa: Adding the same symbol over and over
  • abcdefghiabc: Adding new symbols every time
  • abcdefaaaaaa: Repeating at two character motif
  • abcabcabcabc: Repeating a three-character motif
  • abbabbabbabb: Repeating a three-character motif that only uses two characters
  • abcdefaaaaaa: Adding the same symbol after initially distinct symbols
sequences <- c(
  "aaaaaaaaaaaa", "abcdefghiabc", "abababababab",
  "abcabcabcabc", "abbabbabbabb", "abcdefaaaaaa"
)

data <- sequences %>%
  map_df(function(x){x %>%  acss_sequence})

pic <- data %>% 
  ggplot(aes(x = Length, y = Complexity, 
            group = Group, colour = Group )) +
  geom_line(lwd = 1.5) + 
  geom_point(size = 3, pch = 21, fill = "white", 
             stroke = 1.5) +
  theme_bw() +
  scale_x_continuous(breaks = c(3,6,9,12))

plot(pic)

Neat. I made a rainbow 🌈.

As interesting as it might be to play around with different sequences to get a good feel for what this “random Turing machine” approach has to say about the complexity of short strings, I’m running out of time and I have laundry to attend to. So instead I’ll finish with the thought that, because this

set.seed(1); cat(sample(emo::jis[[3]],1000,T))

is a very short program in R, the implication is that R believes that this

## <U+0001F64D><U+0001F3FD><U+200D><U+2642> <U+0001F483><U+0001F3FE> <U+0001F91A><U+0001F3FC> <U+0001F195> <U+0001F9B8><U+200D><U+2640><U+FE0F> 2<U+20E3> <U+0001F1E9><U+0001F1F4> <U+0001F954> <U+0001F418> <U+0001F468><U+0001F3FD><U+200D><U+0001F3EB> <U+0001F9B8><U+0001F3FB><U+200D><U+2642> <U+0001F469><U+200D><U+0001F9B0> <U+0001F30F> <U+0001F9D7> <U+26F3> <U+0001F93D><U+0001F3FD><U+200D><U+2642> <U+0001F6E3><U+FE0F> <U+0001F1F9><U+0001F1F9> <U+0001F9D6><U+0001F3FE><U+200D><U+2640> <U+265F><U+FE0F> <U+0001F1E7><U+0001F1F2> <U+0001F9B9><U+0001F3FD><U+200D><U+2640> <U+0001F337> <U+0001F575><U+FE0F><U+200D><U+2642> <U+0001F64D><U+200D><U+2640> <U+0001F9D7><U+200D><U+2640> <U+0001F643> <U+0001F9D6><U+0001F3FD><U+200D><U+2642><U+FE0F> <U+264E> <U+0001F486><U+0001F3FC><U+200D><U+2642> <U+0001F6B5><U+0001F3FE><U+200D><U+2640> <U+0001F495> <U+0001F93C><U+200D><U+2640><U+FE0F> <U+0001F468><U+0001F3FE><U+200D><U+0001F9B3> <U+26CF><U+FE0F> <U+0001F32E> <U+0001F4BF> <U+0001F468><U+200D><U+0001F680> <U+0001F6A2> <U+0001F3C2><U+0001F3FF> <U+0001F4CC> <U+0001F577> <U+0001F3BC> <U+0001F596><U+0001F3FE> <U+0001F469><U+200D><U+0001F466> <U+0001F4DF> <U+0001F973> <U+0001F6B5><U+0001F3FB><U+200D><U+2642> <U+0001F550> <U+0001F3DE> <U+0001F6B5><U+0001F3FC><U+200D><U+2642> <U+0001F51B> <U+0001F3CA> <U+0001F9DC><U+0001F3FB><U+200D><U+2640> <U+0001F469><U+0001F3FF><U+200D><U+2696> <U+0001F469><U+200D><U+0001F3A8> <U+0001F647><U+0001F3FE><U+200D><U+2642> <U+0001F939><U+0001F3FF><U+200D><U+2640><U+FE0F> <U+0001F336> <U+0001F464> <U+0001F22F> <U+0001F646><U+0001F3FD><U+200D><U+2640><U+FE0F> <U+0001F3CB><U+FE0F><U+200D><U+2642> <U+0001F937><U+0001F3FD><U+200D><U+2642><U+FE0F> <U+0001F33A> <U+0001F9DD><U+0001F3FE><U+200D><U+2642> <U+0001F6B5><U+0001F3FE><U+200D><U+2642><U+FE0F> <U+0001F3C9> <U+0001F469><U+0001F3FF><U+200D><U+0001F3ED> <U+23EE><U+FE0F> <U+0001F486><U+200D><U+2642><U+FE0F> <U+0001F6BF> <U+0001F487><U+0001F3FD> <U+0001F937><U+0001F3FF><U+200D><U+2642> <U+0001F6B5><U+200D><U+2642><U+FE0F> <U+2049> <U+2638> <U+0001F9D7><U+0001F3FB><U+200D><U+2642> <U+265F><U+FE0F> <U+0001F1F0><U+0001F1EA> <U+0001F6A3><U+0001F3FF><U+200D><U+2642> <U+0001F690> <U+0001F9D8><U+0001F3FE><U+200D><U+2642><U+FE0F> <U+0001F926><U+0001F3FF><U+200D><U+2642> <U+0001F387> <U+0001F9B8><U+0001F3FB><U+200D><U+2640> <U+0001F69E> <U+0001F46E><U+0001F3FD><U+200D><U+2640> <U+0001F9DC><U+0001F3FD><U+200D><U+2640><U+FE0F> <U+0001F477><U+200D><U+2642> <U+0001F9DB><U+0001F3FB><U+200D><U+2642> <U+0001F468><U+0001F3FF><U+200D><U+0001F393> <U+0001F41F> <U+0001F53D> <U+0001F3AD> <U+0001F4F8> <U+26F9><U+0001F3FE><U+200D><U+2640><U+FE0F> <U+0001F3C2><U+0001F3FB> <U+0001F4EB> <U+0001F4A6> <U+0001F341> <U+0001F487><U+0001F3FE><U+200D><U+2640><U+FE0F> <U+0001F64E> <U+0001F1F9><U+0001F1FC> <U+0001F998> <U+0001F9B9><U+0001F3FF><U+200D><U+2640> <U+0001F575><U+200D><U+2640><U+FE0F> <U+0001F6B5><U+0001F3FD><U+200D><U+2642><U+FE0F> <U+0001F535> <U+2764> <U+0001F1F3><U+0001F1FA> <U+0001F55B> <U+0001F6B6><U+0001F3FB><U+200D><U+2642> <U+0001F6A3><U+200D><U+2642><U+FE0F> <U+0001F477><U+0001F3FD><U+200D><U+2640> <U+0001F615> <U+0001F69B> <U+0001F468><U+0001F3FD><U+200D><U+2708> <U+0001F3CA><U+0001F3FF><U+200D><U+2640> <U+0001F432> <U+0001F1F9><U+0001F1F7> <U+0001F93D><U+200D><U+2642><U+FE0F> <U+0001F938><U+0001F3FB> <U+0001F471><U+0001F3FD><U+200D><U+2640> <U+2604><U+FE0F> <U+26F9><U+0001F3FB><U+200D><U+2640> <U+0001F939><U+0001F3FB> <U+0001F9B8><U+0001F3FE><U+200D><U+2642><U+FE0F> <U+0001F9DA><U+0001F3FD><U+200D><U+2640><U+FE0F> <U+0001F441><U+200D><U+0001F5E8><U+FE0F> <U+0001F44B><U+0001F3FD> <U+0001F469><U+0001F3FE><U+200D><U+0001F373> <U+0001F476><U+0001F3FF> <U+0001F421> <U+0001F1E6><U+0001F1F1> <U+0001F498> <U+0001F44C><U+0001F3FB> <U+0001F468><U+200D><U+0001F468><U+200D><U+0001F467> <U+0001F1F8><U+0001F1F0> <U+0001F93E><U+200D><U+2640> <U+0001F379> <U+0001F49C> <U+0001F9DB><U+200D><U+2642><U+FE0F> <U+0001F9DD><U+0001F3FE><U+200D><U+2642> <U+23F3> <U+26F9><U+FE0F><U+200D><U+2640><U+FE0F> <U+0001F468><U+0001F3FB><U+200D><U+0001F9B0> <U+0001F325> <U+0001F469><U+200D><U+2708> <U+262F><U+FE0F> <U+0001F97E> <U+0001F590> <U+0001F926><U+0001F3FF><U+200D><U+2640> <U+26F9><U+FE0F><U+200D><U+2640> <U+0001F93D><U+0001F3FC><U+200D><U+2640> <U+0001F469><U+0001F3FD><U+200D><U+0001F9B1> <U+0001F468><U+200D><U+0001F467><U+200D><U+0001F467> <U+0001F468><U+0001F3FD><U+200D><U+0001F373> <U+0001F64E><U+0001F3FE><U+200D><U+2640> <U+0001F9B9><U+0001F3FE><U+200D><U+2640> <U+0001F645><U+0001F3FC><U+200D><U+2640><U+FE0F> ® <U+0001F3CA><U+0001F3FF><U+200D><U+2640> <U+0001F9F5> <U+0001F4F4> <U+0001F3CC><U+0001F3FE> <U+0001F469><U+0001F3FE><U+200D><U+0001F3EB> <U+0001F937><U+0001F3FC><U+200D><U+2640> <U+0001F6A2> <U+0001F486><U+0001F3FB> <U+0001F400> <U+0001F9F9> <U+2196><U+FE0F> <U+0001F9D7><U+0001F3FE><U+200D><U+2642><U+FE0F> <U+0001F9D6><U+0001F3FF><U+200D><U+2640> ™<U+FE0F> <U+0001F990> <U+0001F31A> <U+0001F4A8> <U+0001F520> <U+0001F646><U+0001F3FD><U+200D><U+2640><U+FE0F> <U+0001F470><U+0001F3FE> <U+2714> <U+0001F93E><U+0001F3FD> <U+23F8> <U+0001F935><U+0001F3FD> <U+0001F389> <U+0001F6E9><U+FE0F> <U+0001F1E9><U+0001F1EF> <U+0001F447><U+0001F3FE> <U+0001F68C> <U+0001F9D7><U+0001F3FF><U+200D><U+2640> <U+0001F469><U+0001F3FF><U+200D><U+0001F3A8> <U+0001F1E6><U+0001F1E9> <U+0001F645><U+0001F3FF><U+200D><U+2642> <U+0001F442><U+0001F3FD> <U+0001F469><U+0001F3FC><U+200D><U+0001F680> <U+0001F9F9> <U+0001F647><U+0001F3FB><U+200D><U+2640> <U+0001F515> <U+0001F64D><U+0001F3FB><U+200D><U+2640> <U+0001F9D9><U+200D><U+2640><U+FE0F> <U+0001F939><U+0001F3FB><U+200D><U+2640> <U+0001F64D><U+0001F3FE><U+200D><U+2640><U+FE0F> <U+0001F469><U+0001F3FE><U+200D><U+0001F9B1> <U+0001F939><U+0001F3FF><U+200D><U+2640><U+FE0F> <U+0001F44D><U+0001F3FC> <U+0001F575><U+200D><U+2640><U+FE0F> <U+0001F9DD><U+0001F3FB><U+200D><U+2642> <U+0001F6E4><U+FE0F> <U+0001F1F0><U+0001F1EE> <U+0001F469><U+0001F3FC><U+200D><U+0001F3A8> <U+0001F3C6> <U+0001F1EB><U+0001F1EF> <U+0001F5D2> <U+0001F64B><U+0001F3FF><U+200D><U+2642><U+FE0F> <U+0001F4AE> <U+0001F1EC><U+0001F1F8> <U+0001F1EC><U+0001F1F9> <U+0001F486><U+0001F3FB><U+200D><U+2642> <U+0001F64D><U+0001F3FD> <U+0001F9D4><U+0001F3FC> <U+0001F926><U+200D><U+2642><U+FE0F> <U+0001F93E><U+0001F3FF><U+200D><U+2640><U+FE0F> <U+0001F535> <U+0001F939><U+0001F3FB> <U+0001F9DD><U+0001F3FD><U+200D><U+2642> <U+0001F9D3><U+0001F3FC> <U+0001F3CC><U+200D><U+2640><U+FE0F> <U+2198><U+FE0F> <U+0001F487><U+0001F3FF> <U+0001F575><U+0001F3FD><U+200D><U+2640> <U+0001F46F> <U+0001F43F><U+FE0F> <U+0001F9D7><U+0001F3FB><U+200D><U+2642> <U+26F0> <U+26F0> <U+0001F918><U+0001F3FE> <U+0001F3C4><U+0001F3FF><U+200D><U+2640> <U+26F9><U+200D><U+2640><U+FE0F> <U+0001F64B><U+0001F3FC><U+200D><U+2642><U+FE0F> <U+270D><U+0001F3FC> <U+0001F199> <U+0001F477><U+0001F3FF> <U+0001F3CC><U+0001F3FB><U+200D><U+2642> <U+0001F9B9><U+0001F3FB><U+200D><U+2640> <U+0001F3C4><U+0001F3FE><U+200D><U+2640><U+FE0F> 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<U+0001F9B9><U+0001F3FE><U+200D><U+2640> <U+0001F448><U+0001F3FF> <U+0001F482><U+0001F3FD><U+200D><U+2642> <U+0001F926><U+0001F3FE><U+200D><U+2642> <U+0001F98A> <U+0001F9DD><U+200D><U+2642> <U+0001F413> <U+0001F938><U+0001F3FF> <U+0001F1E8><U+0001F1E9> <U+2194> <U+0001F483> <U+0001F647><U+200D><U+2642> <U+0001F6E0><U+FE0F> <U+26F9><U+0001F3FF><U+200D><U+2642><U+FE0F> <U+0001F647><U+0001F3FD><U+200D><U+2642> <U+0001F468><U+200D><U+0001F3A8> <U+0001F468><U+0001F3FB><U+200D><U+2696><U+FE0F> <U+26F0> <U+0001F32D> <U+0001F170><U+FE0F> <U+0001F481><U+0001F3FC><U+200D><U+2640> <U+0001F1E7><U+0001F1EB> <U+0001F9B8><U+200D><U+2640> <U+2328> <U+0001F9D9><U+0001F3FE><U+200D><U+2642> <U+0001F63D> <U+0001F6D0> <U+0001F37D> <U+0001F1E8><U+0001F1FC> <U+0001F95F> <U+26B0> <U+0001F6B6><U+0001F3FE><U+200D><U+2640> <U+0001F9D7><U+0001F3FF><U+200D><U+2642> <U+0001F44A><U+0001F3FB> <U+0001F468><U+0001F3FC><U+200D><U+0001F3A4> <U+0001F931><U+0001F3FB> <U+0001F485> <U+0001F302> <U+0001F54E> <U+0001F483><U+0001F3FD> <U+0001F56F><U+FE0F> <U+0001F468><U+0001F3FD><U+200D><U+0001F393> <U+0001F434> <U+0001F6B6><U+0001F3FB><U+200D><U+2642> <U+0001F91D> <U+0001F232> <U+0001F936><U+0001F3FE> <U+0001F3C3><U+0001F3FC><U+200D><U+2640> <U+0001F9C2> <U+0001F94D> <U+0001F469><U+200D><U+2764><U+FE0F><U+200D><U+0001F468> <U+2692> <U+0001F469><U+200D><U+0001F469><U+200D><U+0001F466> <U+0001F93D><U+0001F3FF><U+200D><U+2640><U+FE0F> <U+0001F3CC><U+0001F3FD><U+200D><U+2640> <U+0001F6B6><U+0001F3FF><U+200D><U+2640> <U+0001F575> <U+0001F481><U+0001F3FC><U+200D><U+2642><U+FE0F> <U+0001F64E><U+0001F3FC><U+200D><U+2640> <U+26F8><U+FE0F> <U+0001F004> <U+0001F477><U+0001F3FB><U+200D><U+2642> <U+0001F939><U+0001F3FF><U+200D><U+2642> <U+0001F445> <U+0001F93E><U+0001F3FD><U+200D><U+2642><U+FE0F> <U+0001F9D7><U+200D><U+2640> <U+0001F3C4><U+0001F3FF><U+200D><U+2642> <U+0001F61D> <U+25FE> <U+0001F469><U+200D><U+0001F527> <U+0001F93E><U+200D><U+2640><U+FE0F> <U+0001F4CA> <U+2764><U+FE0F> <U+0001F3C4><U+0001F3FC><U+200D><U+2642><U+FE0F> <U+270B><U+0001F3FB> <U+0001F4DE> <U+0001F471><U+0001F3FE> <U+0001F1F2><U+0001F1F6> <U+0001F6B4><U+0001F3FE><U+200D><U+2640><U+FE0F> <U+0001F1E6><U+0001F1F7> 7<U+FE0F><U+20E3> <U+0001F300> <U+0001F366> <U+0001F982> <U+0001F469><U+0001F3FC><U+200D><U+0001F33E> <U+0001F3C4><U+0001F3FB><U+200D><U+2642><U+FE0F> <U+0001F469><U+200D><U+0001F467><U+200D><U+0001F467> <U+0001F1E8><U+0001F1FE> <U+0001F68E> <U+0001F6E9><U+FE0F> <U+0001F6B4><U+0001F3FE><U+200D><U+2642><U+FE0F> <U+0001F46E><U+0001F3FB><U+200D><U+2640><U+FE0F> <U+0001F3BC> <U+0001F3CA> <U+0001F6A3><U+200D><U+2642><U+FE0F> <U+2620> <U+0001F477><U+200D><U+2640> <U+0001F3C4><U+0001F3FE> <U+0001F3B3> <U+0001F61A> <U+0001F48C> <U+0001F171><U+FE0F> <U+0001F30C> <U+0001F64D><U+0001F3FD> <U+2622><U+FE0F> <U+0001F937><U+0001F3FF><U+200D><U+2642><U+FE0F> <U+270D><U+0001F3FC> <U+0001F6A3><U+0001F3FC><U+200D><U+2642> <U+0001F468><U+0001F3FB><U+200D><U+2695> <U+23F0> <U+0001F447><U+0001F3FF> <U+0001F308> <U+0001F468><U+200D><U+2695><U+FE0F> <U+0001F699> <U+0001F481><U+0001F3FB><U+200D><U+2642><U+FE0F> <U+0001F645><U+200D><U+2640><U+FE0F> <U+2694> <U+0001F469><U+0001F3FB><U+200D><U+0001F4BC> <U+0001F468><U+0001F3FB> <U+0001F487><U+0001F3FC><U+200D><U+2642> <U+261D><U+0001F3FD> <U+0001F454> <U+0001F64E><U+0001F3FE> <U+0001F9B8><U+200D><U+2642><U+FE0F> <U+0001F9D6><U+0001F3FB><U+200D><U+2642> <U+0001F6B4><U+0001F3FC><U+200D><U+2640> <U+0001F9EA> <U+0001F46E><U+0001F3FD><U+200D><U+2640><U+FE0F> <U+0001F95F> <U+0001F93D><U+0001F3FC><U+200D><U+2642><U+FE0F> 9<U+FE0F><U+20E3> <U+0001F91E><U+0001F3FE> <U+0001F575><U+0001F3FF><U+200D><U+2642><U+FE0F> <U+0001F3CA><U+0001F3FD><U+200D><U+2642> <U+0001F930><U+0001F3FC> <U+0001F6A3><U+200D><U+2640><U+FE0F> <U+0001F9D9><U+0001F3FF><U+200D><U+2642> <U+0001F1F0><U+0001F1ED> <U+26F9><U+0001F3FF> <U+2663> <U+0001F473><U+200D><U+2640><U+FE0F> <U+262E><U+FE0F> <U+0001F9B8><U+0001F3FB><U+200D><U+2642> <U+0001F469><U+0001F3FF><U+200D><U+0001F9B0> <U+0001F9D4> <U+0001F44E><U+0001F3FF> <U+0001F6F0> <U+23EB> <U+0001F683> <U+0001F6B5><U+0001F3FC><U+200D><U+2642><U+FE0F> <U+0001F4CA> <U+0001F626> <U+0001F1FB><U+0001F1EE> <U+0001F413> <U+0001F3C4><U+0001F3FD><U+200D><U+2640> <U+0001F47E> <U+2744><U+FE0F> <U+0001F9B9> <U+0001F3F4><U+000E0067><U+000E0062><U+000E0073><U+000E0063><U+000E0074><U+000E007F> <U+2139> <U+0001F690> <U+23F2> <U+0001F6B4><U+0001F3FC><U+200D><U+2640><U+FE0F> <U+269B> <U+0001F9D4><U+0001F3FC> <U+0001F415> <U+0001F646><U+200D><U+2640> <U+26F9><U+0001F3FF><U+200D><U+2640> <U+0001F469><U+0001F3FF> <U+0001F469><U+0001F3FE><U+200D><U+0001F9B0> <U+0001F468><U+0001F3FB><U+200D><U+0001F393> <U+0001F1E9><U+0001F1F2> <U+0001F486><U+200D><U+2640> <U+0001F939><U+0001F3FE><U+200D><U+2640><U+FE0F> <U+0001F403> <U+0001F9DB><U+0001F3FD><U+200D><U+2640> <U+0001F939><U+200D><U+2640> <U+0001F4B7> <U+0001F487><U+0001F3FB><U+200D><U+2642> <U+21AA> <U+0001F476><U+0001F3FB> <U+0001F1F2><U+0001F1F8> <U+26C8><U+FE0F> <U+0001F64E><U+0001F3FC><U+200D><U+2642> <U+0001F368> <U+0001F487><U+0001F3FB><U+200D><U+2642><U+FE0F> <U+0001F1EA><U+0001F1F8> <U+0001F486><U+0001F3FE> <U+0001F648> <U+0001F487><U+0001F3FF><U+200D><U+2640><U+FE0F> <U+0001F9D7><U+0001F3FC><U+200D><U+2640> <U+0001F6B6><U+0001F3FC><U+200D><U+2642><U+FE0F> <U+0001F1EF><U+0001F1F4> <U+0001F9D6><U+0001F3FF><U+200D><U+2642><U+FE0F> <U+0001F595><U+0001F3FF> <U+0001F3F3><U+FE0F> <U+25AA><U+FE0F> <U+0001F9DC><U+0001F3FC> <U+0001F68F> <U+0001F487><U+0001F3FC><U+200D><U+2642> <U+0001F918><U+0001F3FC> <U+0001F31D> <U+0001F4C8> <U+264E> <U+0001F9D2><U+0001F3FB> <U+0001F9D9><U+0001F3FC><U+200D><U+2640> <U+0001F3C3><U+0001F3FF><U+200D><U+2642><U+FE0F> <U+0001F64B><U+200D><U+2642> <U+0001F6F8> <U+0001F3EF> <U+0001F17F> <U+0001F6BA> <U+0001F3B4> <U+0001F6C0> <U+0001F91F><U+0001F3FE> <U+0001F469><U+0001F3FB><U+200D><U+0001F373> <U+23E9> <U+0001F1ED><U+0001F1FA> <U+0001F93E><U+0001F3FC><U+200D><U+2642> <U+0001F3C1> <U+0001F482><U+200D><U+2642><U+FE0F> <U+0001F9DB><U+200D><U+2642><U+FE0F> <U+0001F938><U+0001F3FB><U+200D><U+2640> <U+0001F6B4><U+0001F3FF><U+200D><U+2640><U+FE0F> <U+0001F6A3><U+200D><U+2640><U+FE0F> <U+0001F467><U+0001F3FB> <U+0001F990> <U+0001F69B> <U+0001F487><U+0001F3FB><U+200D><U+2640> <U+0001F1EA><U+0001F1F7> <U+0001F31D> <U+0001F468><U+200D><U+2695><U+FE0F> <U+0001F1F5><U+0001F1FC> <U+0001F9D9><U+0001F3FD> <U+0001F9D2><U+0001F3FC> <U+0001F9DF><U+200D><U+2640><U+FE0F> <U+0001F6D1> <U+0001F9DC><U+0001F3FB><U+200D><U+2642><U+FE0F> <U+0001F9DB><U+0001F3FE> <U+0001F637> <U+0001F64D><U+0001F3FC><U+200D><U+2642><U+FE0F> <U+0001F3DC> <U+0001F476><U+0001F3FC> <U+0001F469><U+0001F3FB><U+200D><U+0001F3EB> <U+0001F9D8><U+200D><U+2642><U+FE0F> <U+0001F938><U+0001F3FB><U+200D><U+2640> <U+0001F430> <U+0001F6E5> <U+0001F469><U+0001F3FD><U+200D><U+0001F373> <U+0001F3C4> <U+0001F1F3><U+0001F1EA> <U+0001F58B><U+FE0F> <U+0001F9D9><U+0001F3FD><U+200D><U+2642><U+FE0F> <U+0001F938><U+0001F3FB><U+200D><U+2640><U+FE0F> <U+0001F61C> <U+0001F9DD><U+0001F3FF><U+200D><U+2642> <U+26F9><U+0001F3FC><U+200D><U+2640> <U+0001F31A> <U+0001F1F9><U+0001F1F1> <U+0001F937><U+0001F3FC><U+200D><U+2642> <U+0001F1E9><U+0001F1F4> <U+0001F1F0><U+0001F1F2> 4<U+FE0F><U+20E3> <U+0001F93C> <U+0001F39B><U+FE0F> <U+0001F4C4> <U+0001F36A> <U+0001F450><U+0001F3FB> <U+0001F937><U+200D><U+2642><U+FE0F> <U+0001F603> <U+0001F468><U+0001F3FC><U+200D><U+2696> <U+0001F468><U+0001F3FF><U+200D><U+0001F4BC> <U+0001F615> <U+0001F937><U+0001F3FD><U+200D><U+2640><U+FE0F> <U+0001F46E><U+0001F3FF><U+200D><U+2642><U+FE0F> <U+0001F9B2> <U+0001F466> <U+0001F62F> <U+0001F472> <U+2699><U+FE0F> <U+0001F94F> <U+0001F64E><U+0001F3FB><U+200D><U+2642> <U+0001F469><U+0001F3FF><U+200D><U+0001F9B3> <U+0001F9DA><U+0001F3FB> <U+0001F468><U+0001F3FE><U+200D><U+0001F3EB> <U+0001F469><U+0001F3FC><U+200D><U+0001F393> <U+0001F477><U+0001F3FD><U+200D><U+2640><U+FE0F> <U+0001F469><U+0001F3FB><U+200D><U+0001F33E> <U+0001F468><U+0001F3FC> <U+0001F469><U+200D><U+2764><U+200D><U+0001F468> <U+0001F4DE> <U+0001F9F1> <U+0001F468><U+0001F3FE><U+200D><U+2696><U+FE0F> <U+0001F62C> <U+0001F3CA><U+0001F3FE><U+200D><U+2642><U+FE0F> <U+0001F473><U+0001F3FF><U+200D><U+2642><U+FE0F> <U+0001F6E2> <U+0001F3D9><U+FE0F> <U+2705> <U+0001F926><U+0001F3FF><U+200D><U+2642> <U+0001F1F2><U+0001F1F8> <U+0001F1F8><U+0001F1EF> <U+0001F467> <U+3030><U+FE0F> <U+0001F4CF> <U+2708> <U+0001F646><U+200D><U+2642> <U+0001F93D><U+0001F3FF><U+200D><U+2640><U+FE0F> <U+0001F3C4><U+0001F3FD><U+200D><U+2640> <U+0001F456> <U+0001F3F3><U+200D><U+0001F308> <U+0001F9DB><U+0001F3FC><U+200D><U+2642> <U+0001F64E><U+200D><U+2642><U+FE0F> <U+0001F560> <U+2602><U+FE0F> <U+0001F1E6><U+0001F1FA> <U+0001F3CB><U+200D><U+2640><U+FE0F> <U+0001F519> <U+0001F64B><U+0001F3FF><U+200D><U+2640> <U+0001F9B8><U+0001F3FF><U+200D><U+2642><U+FE0F> <U+0001F538> <U+0001F468><U+0001F3FD><U+200D><U+2695> <U+0001F3C2><U+0001F3FE> <U+0001F3CB><U+FE0F> <U+0001F3CC><U+0001F3FD><U+200D><U+2642><U+FE0F> <U+0001F963> <U+0001F387> <U+0001F202> <U+0001F4C9> <U+0001F469><U+0001F3FD><U+200D><U+0001F4BB> <U+0001F469><U+0001F3FF><U+200D><U+0001F9B0> <U+0001F41F> <U+25C0><U+FE0F> <U+0001F443><U+0001F3FD> <U+0001F64D><U+0001F3FC> <U+269C><U+FE0F> <U+0001F469><U+0001F3FE><U+200D><U+0001F9B3> <U+0001F938><U+0001F3FF><U+200D><U+2642> <U+0001F62A> <U+0001F6B6><U+0001F3FC><U+200D><U+2640><U+FE0F> <U+0001F3CA><U+0001F3FE><U+200D><U+2642><U+FE0F> <U+0001F575><U+FE0F><U+200D><U+2642> <U+0001F993> <U+0001F481><U+0001F3FD><U+200D><U+2640> <U+0001F9DB><U+0001F3FB><U+200D><U+2642> <U+0001F6B6><U+0001F3FF><U+200D><U+2642><U+FE0F> <U+2754> <U+0001F1F9><U+0001F1F1> <U+0001F550> <U+0001F1F9><U+0001F1FF> <U+0001F449><U+0001F3FB> <U+0001F6A3><U+0001F3FE><U+200D><U+2642><U+FE0F> <U+0001F1F1><U+0001F1EE> <U+0001F477><U+0001F3FD><U+200D><U+2640> <U+0001F468><U+0001F3FE><U+200D><U+2696><U+FE0F> <U+0001F925> <U+0001F45B> <U+0001F3E1> <U+0001F4D8> <U+0001F482><U+0001F3FF><U+200D><U+2642> <U+0001F473><U+0001F3FD><U+200D><U+2640><U+FE0F> <U+0001F1F9><U+0001F1E9> <U+0001F9D8><U+0001F3FC><U+200D><U+2642><U+FE0F> <U+0001F469><U+0001F3FE><U+200D><U+0001F9B3> <U+0001F6B5><U+0001F3FC><U+200D><U+2642> <U+0001F446><U+0001F3FD> <U+0001F926><U+0001F3FE><U+200D><U+2640> <U+0001F3F4><U+000E0067><U+000E0062><U+000E0077><U+000E006C><U+000E0073><U+000E007F> <U+0001F469><U+200D><U+0001F469><U+200D><U+0001F466><U+200D><U+0001F466> <U+0001F482><U+0001F3FD> <U+0001F3CA><U+0001F3FB> <U+0001F43A> <U+0001F9B8><U+0001F3FC><U+200D><U+2640><U+FE0F> <U+0001F482><U+0001F3FE> <U+2663><U+FE0F> <U+0001F426> <U+0001F645><U+0001F3FD><U+200D><U+2642> <U+0001F470><U+0001F3FE> <U+0001F64D><U+0001F3FD><U+200D><U+2642>

is a very structured sequence.


  1. Yes, yes, I know that frequentists get annoyed when Bayesians use the term “likelihood” this way, but honestly I think you should be annoyed at Fisher instead. He’s the one who co-opted everyday probabilistic language that dates back to the middle ages to describe a score function \(L(\theta|x)\) that frequentists are expressly forbidden to interpret as a probability. You can hardly blame those of us who are happy to treat \(\theta\) as a random variable for using the word “likelihood” in something closer to its well-understood everyday meaning. And let’s not get started on Neyman’s horrible abuse of the word “confidence”, shall we?

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Danielle Navarro
Associate Professor of Cognitive Science

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