1What is the core idea behind 'rational analysis' in cognitive science?
It treats people as approximately optimal machines that adapt to their environment and tasks in the best possible way.
2In rational analysis, how is learning typically modeled as an optimal process?
Learning is modeled via Bayesian update, using Bayes' rule to optimally revise beliefs based on evidence.
3What kind of models are based on probability theory and are widely used to describe learning, reasoning, and memory in cognitive science?
Probabilistic cognitive models, with Bayesian models being a very important subclass.
4According to rational theory, behavior is assumed to implement optimal relations between _____ and cognitive processes.
environment structure
5In the 'auxiliary fronting' example, what is the naive rule a child might infer to form a question like "Is the man smiling?"
Move the verb to the beginning of the sentence.
6Given the declarative "The man who is tall is smiling," what ungrammatical question does the 'closest-verb' rule produce?
*Is the man who tall is smiling?
7Given the declarative "The man who is tall is smiling," what grammatical question does the 'structurally highest' rule produce?
Is the man who is tall smiling?
8The 'poverty of the stimulus' argument points out that children converge on the structurally-based rule for question formation without hearing direct _____ that the linear rule is wrong.
evidence
9What is the classical conclusion from generative grammar regarding the 'poverty of the stimulus'?
Children are endowed with a universal grammar that privileges hierarchical structure over linear order.
10How does unbounded recursion (e.g., "I know that he knows that she knows...") present a 'poverty of the stimulus' problem?
Children generalize to potentially unbounded recursion despite rarely encountering deeply recursive sentences in their input.
11In formal language theory, a language is defined as a set of _____ over some alphabet.
strings
12Given an alphabet {\$a,b\$}, what is the language \$L_1 = \{a^n b^m \mid n, m e 1\}\$?
The set of all strings with one or more 'a's followed by one or more 'b's.
13For a base set \$X = \{\text{abc}\}\$, the _____ is the set of all finite concatenations, including the empty string \$\epsilon\$.
Kleene star (\$X^*\$)
14The formal language \$L_0 = a^n N\$ can be used to model what natural language construction?
A sequence of any number of adjectives ('a') followed by a noun ('N').
15The formal language \$L = \{a^n b^n \mid n \in \mathbb{N}\}\$ is often used to model what type of grammatical construction?
Center embedding, where the number of opening elements (e.g., nouns) must match closing elements (e.g., verbs).
16What is the general formula for conditional probability, \$P(A|B)\$?
\$P(A|B) = \frac{P(A \cap B)}{P(B)}\$, for \$P(B) > 0\$.
17A _____ is a function from a sample space of possible events to a set of real numbers.
random variable
18What type of function is used to describe the probability distribution of a discrete random variable?
A probability mass function (PMF).
19What type of function is used to describe the probability distribution of a continuous random variable?
A probability density function (PDF).
20A continuous random variable \$X\$ that follows a normal distribution is denoted by what notation?
\$X \sim N(\mu, \sigma^2)\$
21What is the probability mass function for a geometric distribution, \$P(X=k)\$?
\$P(X=k) = \alpha(1-\alpha)^{k-1}\$, for \$k=1,2,3,\dots\$
22A geometric distribution is typically interpreted as the probability that the first success in a series of independent trials occurs on the _____-th trial.
k
23What is the formula for Bayes' rule, expressing \$P(H|E)\$?
\$P(H|E) = \frac{P(E|H)P(H)}{P(E)}\$
24In Bayes' rule, the term \$P(H|E)\$ is known as the _____ probability.
posterior
25In Bayes' rule, the term \$P(H)\$ is known as the _____ probability.
prior
26In Bayes' rule, the term \$P(E|H)\$ is known as the _____.
likelihood
27The term \$P(E)\$ in Bayes' rule is the marginal likelihood, which acts as a _____.
normalizing constant
28How is the evidence term \$P(E)\$ calculated as a sum over all possible hypotheses \$H_i\$?
\$P(E) = \sum_i P(E|H_i)P(H_i)\$
29What is the proportional form of Bayes' rule, often used for comparing hypotheses?
\$P(H|E) \propto P(E|H)P(H)\$
30The 'reverse-engineering' approach in Bayesian cognitive modeling works backward from observed behavior to infer the _____ and _____ that make that behavior optimal.
priors and likelihoods
31What is the central claim of Fodor's Language of Thought (LOT) hypothesis?
Humans possess basic conceptual primitives and logical operations that combine compositionally to form complex concepts.
32In the Bayesian LOT framework, how is learning modeled?
Learning is modeled as applying Bayes' rule over a hypothesis space defined by the Language of Thought.
33In the LOT for strings, what is the `plus(S, c)` or `\$S \oplus c\$` operation?
It adds character 'c' to the end of string 'S'.
34In the LOT for strings, what does the `rest(S)` operation return?
It returns all characters of string 'S' except for the first one.
35The LOT program \$F_0 = \text{if}(\text{flip}(1/3), \epsilon, F_0 \oplus a)\$ generates what formal language?
The set of all positive-length sequences of 'a', i.e., \$\{a, aa, aaa, ...\}\$.
36The LOT program \$F_0 = \text{if}(\text{flip}(1/3), \epsilon, F_0 \oplus a)\$ assigns probabilities to strings that follow what named distribution?
A geometric distribution.
37In the LOT framework, how is the prior probability \$P(H)\$ of a hypothesis (a program) calculated?
It's approximated as \$p^m\$, where \$m\$ is the number of primitives used and \$p\$ is the prior probability of a single primitive.
38Assigning lower prior probability to hypotheses that require more primitives implements a _____ penalty.
complexity
39The complexity penalty in the LOT prior is analogous to what principle from information theory?
The Minimum Description Length (MDL) principle.
40In Bayesian learning, what happens to a hypothesis if it assigns a likelihood of 0 to an observed piece of data?
Its posterior probability becomes 0, and it is effectively eliminated from consideration.
41In a learning model, _____ is the proportion of strings the model correctly accepts out of all true language strings.
recall
42In a learning model, _____ is the proportion of strings the model classifies as 'in the language' that truly belong to it.
precision
43The _____ score is the harmonic mean of precision and recall.
F1
44The Bayesian LOT model learns languages like \$a^n\$ and \$a^n b^n\$ well but struggles with which computationally harder language involving three symbols?
\$a^n b^n c^n\$
45The Bayesian model's difficulty in learning a language with a fixed, small number of embeddings (e.g., 'up to 4') suggests that fully _____ rules may be simpler to encode in LOT.
recursive
46How does the Bayesian LOT framework model the developmental trajectory of number word learning in children?
Posterior probability mass shifts sequentially from one-knower, to two-knower, and eventually to a general counting principle (CP-knower) hypothesis.
47In the number learning model, grasping a general, recursive counting principle corresponds to the _____ hypothesis gaining the most posterior probability.
CP-knower
48In the 'wudsy' experiments, what was the learning task for human participants?
To infer the meaning (the concept) of the novel word 'wudsy' from labeled examples of objects.
49In the LOT for concepts model, how are complex concepts like "blue and circle" represented?
They are represented as Boolean formulas (e.g., 'blue \$\land\$ circle') over primitive predicates for properties like color and shape.
50According to Marr's levels of analysis, Bayesian models of cognition are typically considered to be explanations at the _____ level.
computational