1. N-Gram Model Formulas Word sequences Chain rule of probability
Bigram approximation
N-gram approximation

2. Estimating Probabilities
N-gram conditional probabilities can be estimated from raw text based on the relative frequency of word sequences.
To have a consistent probabilistic model, append a unique start (<s>) and end (</s>) symbol to every sentence and treat these as additional words.
Bigram:
N-gram:

3. Perplexity Measure of how well a model “fits” the test data.
Uses the probability that the model assigns to the test corpus.
Normalizes for the number of words in the test corpus and takes the inverse.
Measures the weighted average branching factor in predicting the next word (lower is better).

4. Laplace (Add-One) Smoothing
“Hallucinate” additional training data in which each possible N-gram occurs exactly once and adjust estimates accordingly.
where V is the total number of possible (N1)-grams (i.e. the vocabulary size for a bigram model).
Bigram:
N-gram:
Tends to reassign too much mass to unseen events, so can be adjusted to add 0<<1 (normalized by V instead of V).

5. Interpolation
Linearly combine estimates of N-gram models of increasing order.
Interpolated Trigram Model:
Where:
Learn proper values for i by training to (approximately) maximize the likelihood of an independent development (a.k.a. tuning) corpus.

6. Formal Definition of an HMM
A set of N +2 states S={s0,s1,s2, … sN, sF}
Distinguished start state: s0
Distinguished final state: sF
A set of M possible observations V={v1,v2…vM}
A state transition probability distribution A={aij}
Observation probability distribution for each state j B={bj(k)}
Total parameter set λ={A,B}

7. Forward Probabilities
Let t(j) be the probability of being in state j after seeing the first t observations (by summing over all initial paths leading to j).

8. Computing the Forward Probabilities
Initialization
Recursion
Termination

9. Viterbi Scores
Recursively compute the probability of the most likely subsequence of states that accounts for the first t observations and ends in state sj.
Also record “backpointers” that subsequently allow backtracing the most probable state sequence.
btt(j) stores the state at time t-1 that maximizes the probability that system was in state sj at time t (given the observed sequence).

10. Computing the Viterbi Scores
Initialization
Recursion
Termination
Analogous to Forward algorithm except take max instead of sum

11. Computing the Viterbi Backpointers
Initialization
Recursion
Termination
Final state in the most probable state sequence. Follow
backpointers to initial state to construct full sequence.

12. Supervised Parameter Estimation
Estimate state transition probabilities based on tag bigram and unigram statistics in the labeled data.
Estimate the observation probabilities based on tag/word co-occurrence statistics in the labeled data.
Use appropriate smoothing if training data is sparse.

13. Simple Artificial Neuron Model (Linear Threshold Unit)
Model network as a graph with cells as nodes and synaptic connections as weighted edges from node i to node j, wji
Model net input to cell as
Cell output is: 1 3 2 5 4 6
w12
w13
w14
w15
w16 oj 1
(Tj is threshold for unit j) Tj
netj

14. Perceptron Learning Rule
Update weights by:
where η is the “learning rate”
tj is the teacher specified output for unit j.
Equivalent to rules:
If output is correct do nothing.
If output is high, lower weights on active inputs
If output is low, increase weights on active inputs
Also adjust threshold to compensate:

15. Perceptron Learning Algorithm
Iteratively update weights until convergence.
Each execution of the outer loop is typically called an epoch.
Initialize weights to random values
Until outputs of all training examples are correct
For each training pair, E, do:
Compute current output oj for E given its inputs
Compare current output to target value, tj , for E
Update synaptic weights and threshold using learning rule

16. Context Free Grammars (CFG)
N a set of non-terminal symbols (or variables)
 a set of terminal symbols (disjoint from N)
R a set of productions or rules of the form A→, where A is a non-terminal and  is a string of symbols from ( N)*
S, a designated non-terminal called the start symbol

17. Estimating Production Probabilities
Set of production rules can be taken directly from the set of rewrites in the treebank.
Parameters can be directly estimated from frequency counts in the treebank.