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Meaning
Prior
Count
Likelihood
Prior*Lik
Posterior
This simulation is based on Xu and Tenenbaum (2007). For an explanation of the theory and a step-by-step demo with notes, click Introduction. Otherwise, you can click "Be the learner" or "Be the teacher" to jump right in to the actual simulation. "Show taxonomy" allows you to see the hierarchical relationship between all the animals in our context to the left.

This simulation is based on Word Learning as Bayesian Inference, a 2007 paper by Fei Xu and Joshua Tenenbaum. This paper argues that Bayesian inference provides a successful framework for modeling how the meanings of words are learned (by children acquiring language or by adults who happen to be hearing a new word for the first time).

Let's take an example. Suppose you heard someone utter the word "ghoti" to refer to the following entity:

Given just this information, there are several options for what this word could mean, including "animal", "vertebrate", "fish", "clownfish", and "this specific clownfish." Although people do learn some words from direct instruction (e.g. by being taught them in vocabulary classes or looking them up in dictionaries), most word meanings are acquired unconsciously. So how does this unconscious acquisition process happen when there are so many alternatives to consider?

Xu and Tenenbaum's answer is that this process combines prior knowledge and beliefs about likely word meanings with the evidence the learner observes. For example, one type of prior belief that people seem to have is called a "basic-level bias." That is, all else being equal, people will assume that a new word refers to some "basic" level concept (such as "clownfish") rather than an overly specific "subordinate" level concept (such as "this specific clownfish") or an exceedingly abstract "superordinate" level concept (such as "vertebrate").

The available evidence is also important. Suppose that you now hear the word "ghoti" a second time, still referring to the exact same entity:

The word still could mean any of the five concepts from before ("animal", "vertebrate", "fish", "clownfish", or "this specific clownfish.") However, it's a bit suspicious that both times that you've heard this word have been referring to the same specific clownfish, while you've never heard it refer to anything else. Therefore, you're now probably more likely to think "ghoti" might be the name of this specific fish, given this suspicious coincidence.

However, suppose you now hear the word a third time and that now it refers to this tuna:

Now you know for sure that "ghoti" cannot mean "clownfish" or "this specific clownfish" because those meanings would contradict the available evidence. The remaining options are "fish", "vertebrate", and "animal". Even though all these are possible, it is a suspicious coincidence that you've only heard the word refer to fish and not to other vertebrates or other animals. Therefore, at this point, you'll tentatively conclude that "ghoti" means "fish."

Bayesian inference unites the two types of reasoning discussed above. Our proposed meaning of the word is our hypothesis, and the evidence we have is our observations about the words usage. Using Bayes' Theorem, we can then write:

p(ghoti = "fish" | evidence) ∝ p(ghoti = "fish") * p(evidence | ghoti = "fish")

The prior term, p(ghoti = "fish"), captures any prior beliefs the learner might have about word meanings. The likelihood term, p(evidence | ghoti = "fish"), captures our intuitions about how plausible our observations would be under a given meaning for "ghoti." Together they are used to compute the posterior probability p(ghoti = "fish" | evidence), which represents how probable it is that "ghoti" means "fish" in the face of our evidence. (Recall that the symbol α means "is proportional to." It is not a ghoti, even though it looks like one.)

For the simulations here, we will build from the basic level bias to assume that the prior probability of a word meaning is determined entirely by that meanings position in the taxonomic tree, with basic level concepts ("clownfish", "horse", "spider", "tortoise") having the highest prior probability, immediately subordinate ("clownfish 1", "wolf 7", "raven 4") and immediately superordinate ("fish", "mammal", "mollusc") categories having lower prio probabilities, and more distantly superordinate categories ("vertebrate", "animal") having even lower prior probabilities. To see the taxonomic structure of all animal terms considered, click here, and to see the specific prior probabilities we use, click here.

To compute the likelihood, we can use the basic probability theorem that p(A and B | C) = p(A | C) * p(B | C) (assuming A and B are independent). In our case, the likelihood of a sequence of observations would be something like p(["ghoti" is used to refer to clownfish 5] and ["ghoti is used to refer to clownfish 5"] and ["ghoti is used to refer to tuna 1"] | "ghoti" means "fish"), and using this theorem we can rewrite it as p("ghoti" is used to refer to clownfish 5 | "ghoti" means "fish")*p("ghoti" is used to refer to clownfish 5 | ghoti means fish)*p("ghoti" is used to refer to tuna 1 | ghoti means fish).

These separate probabilities can now be computed very simply. Consider p("ghoti" is used to refer to clownfish 5 | ghoti means fish). Supposing that ghoti means fish (which is what a conditional probability does), then we should be equally likely to hear "ghoti" used to refer to any of the fish in our world. Since there are 9 fish in the scenario, this means that p("ghoti" is used to refer to clownfish 5 | ghoti means fish) = 1/9. The same argument gives the same value for p("ghoti" is used to refer to tuna 1 | ghoti means fish), giving us the value of (1/9)^3 = 1/729 as our final likelihood.

To figure out which meaning the learner settles on, we simply iterate over all five meanings, find the posterior probability of each one, and choose the one with the highest posterior probability.

Mouse over any word or number to see which animal or group of animals it represents.

Animal
Vertebrate
Mammal
Wolf 1234567
Rhino 12345
Horse 123456789101112
Bird
Roadrunner 12
Owl 123456789
Goldfinch 123456
Raven 1 2 3 4 5
Fish
Clownfish 12345
Shark 123
Tuna 1
Reptile
Snake 123456789
Tortoise 12345
Amphibian
Frog 1234567
Invertebrate
Arthropod
Spider 12345678
Ant 123
Beetle 1234
Grasshopper 123
Mollusc
Octopus 123
Clam 123
Clownfish
Clownfish 1
Fish
Vertebrate
Animal
Clownfish 1
Clownfish
Fish
Vertebrate
Animal
Fish
Vertebrate
Animal
Fish
Vertebrate
Animal
Fish
Vertebrate
Animal
The unknown word is: Ghoti
Possibilities:
Clownfish 1
Clownfish
Fish
Vertebrate
Animal
Priors:
0.01
0.69
0.2
0.05
0.05
Likelihoods:
1.0
0.200
0.111
0.013
0.010
Prior * likelihood:
0.100
0.120
0.022
0.001
0.001