Hello

  • Cross entropy
    • H(p,q) = D(p||q) + H(p)
      • H(p) is some inherent randomness in p
      • D(p||q) is what we care about. we can try to get D(p||q) by calculating cross entropy.
    • Conclusion: a model is good is that it assign good approximation to the observed data. So we need to find some good q
  • Main points:
    • Example: She loves her. (It’s a correct string, but English is not like this. It should be “She loves herself.”)
    • We need a meaning pair.
    • Two orthogonal dimensions:
      • probability for the strings.
      • Units Prob
        String {anbn|n>=1} P(w1, w2,…, wn)
        Structure A tree structure PCFG
      • L1 = L2: Language 1 is equal to Language 2
        • Weak equivalence
          • Sense of string are the same.
        • Strong equivalence
          • Structure of language 1 and 2 are the same.
          • G1 = G2 iff {x| G1 generates string x} = {x|G2 generates string x} (all and only the same structures)
          • G1 G2
            S->a s s->s a
            s->e s->e
          • G1 and G2 are weak equivalent (they generate the same strings) but not strong equivalent
    • Example: Jon loves mary
    • Questions:
      • How to measure equivalence?
      • binary judgements?
  • EM
    • Question: How to find a good model? Expectation maximization (EM)
    • The structure of model is given, we need to find the parameters for the model.
    • Coin: H H H H T T T T T T
    •  
    • MLE: argmax [p(x|mu)]
      • Solve: 
      • Result: p = k/n
  • HMM <A,B, pi>

Lecture3: Information Theory

  • Today’s class is about:
    • Hypothesis testing
    • collocations
    • Info theory
  • Hypothesis Testing
    • Last lecture covered the methodology.
  • Collocation
    • “religion war”
    • PMI, PPMI
      • PMI = pointwise mutual information
        • PMI = log2(P(x,y)/(P(x)P(y))) = I(x,y)
      • PPMI = positive PMI = max(0, PMI)
        • Example: Hong Kong, the frequency of “Hong” and “Kong” are low, but the frequency for “Hong Kong” is high.
      • Non-statistical measure but very useful: Not a classic hypothesis testing.
      • Note: If the frequency of the word is less than 5, then even if the PMI is high, the bigram is meaningless.
    • RFR
      • rfr(w) = (freq(w in DP)/(#DP))/(freq (w in background corpus)/(#background corpus))
      • DP is an child class in background corpus.
    • Chi-square test
      • Draw the table is important.
    • T-test
      • Non-statistical measure but very useful.
        • Not a classic hypothesis testing.
      • sliding window scan the sentence to find the bigram.
        • found ->1
        • not found ->0
      • A fox saw a new company’s emerge…..
        • A fox ->0
        • fox saw -> 0
        • saw a ->0
        • a new -> 0
        • new company -> 1
        • company emerge -> 0
  • Info theory
    • Entropy: degree of certainty of X; information content of X; pointwise information of X; (ling) surprisal, perplexity
      • X~P, H(X) = E[-log2(p(x))] = -sum(p(x)log2(p(x)))
      • H(X,Y) = H(X) + H(Y) – I(X, Y)
        • H(X, Y) is the cost of communicating between X and Y;
        • H(X) is the cost of communicating X;
        • H(Y) is the cost of communicating Y;
        • I(X, Y) is the saving because X and Y can communicate with each other.
    • Example: 8 horses run, the probability of winning is:
      • horse:    1  .     2   .   3   .    4   .    5   .   6   .     7   .       8
      • Prob:    1/2, 1/4   , 1/8,  1/16, 1/32, 1/64, 1/128, 1/256
      • Code1:   000, 001, 010,  011,   100,  101,     110,      111
        • H(X) = 3;
      • Code2:0, 10, 110,  1110,  111100, 1111101, 111110, 111111
        • Huffman code.
        • H(X) = 2;
    • Structure
      • [W -> encoder] -> X-> [                    ]->    Y  -> [decoder  -> W^]
      •        Source                    channel (P(y|x))
      • Anything before Y is the noisy channel model.
      • P(X,Y) is the joint model, P(X|Y) discriminative model.
      • T^ (estimated T)
        • =argmax_T (P(T|W))
        • = argmax_T(P(W|T)P(T)/P(W))
        • = argmax_T(P(W|T)P(T)) (W can be omitted because its’s just a scaler)
        • = argmax_T(P(w1|t1)P(w2|t2)…p(w.n|t.n) P(t1)P(t2|t1)P(t3|t2,t1)…P(tn|t1,t2, …, t.n-1))
        • = argmax_T(P(w1|t1)P(w2|t2)…p(w.n|t.n) P(t1)P(t2|t1)P(t3|t2)…P(t.n))
      • E^ = argmax P(F|E)P(E)
        • F = French, E = English, translation from English to French.
    • Relative entropy: KL divergence
      • We have an estimated model q and a true distribution p
      • D(p||q) = sum_x(p(x).log(p(x))/ log(q(x)))
      • Properties:
        • D(p||q) >= 0, equal is true when p == q
        • D(p||q) != D(q||p)
        • D(p||q) = E_p[p(x)] – E_p[q(x)]   E_p(q(x)) is the cross entropy
      • Example:
        • Model q: P(X)P(Y) independent
        • truth   p: Q(X, Y) joint
        • D(p||q)
          • = sum_x,y [p(x,y) log2(p(x,y)/p(x)p(y))]
          • = E_x,y [log2(p(x,y)/p(x)p(y))] average mutual info is the expectation of pointwise mutual info
    • Cross entropy: (perplexity)
      • Question: how good is my language model? How do I measure?
      • p(w1, w2, w3, w4) = p(w1) p(w2|w1) … p(w4|w1 w2 w3)
        • p(w4|w1 w2 w3) = p(w4|w*, w2 w3)
        • make it Markov, and combine everything beside the prev and the prev-prev.
      • D(p||m) = sum[p(x.1 … x.n)log(p(x.1 … x.n)/m(x.1 … x.n))]
        • but how do you know the truth?
        • what if n is very large?
      • H(p,m)=H(p) + D(p||m)  = -sum[p(x)log(m(x))]
        • H(p,m) can be an approximation of D(p||m), then:
        • We can sample from p and check H(p,m) by the samples, then:
        • H(L,m) = lim_(n->infinity) [- 1/n sum(p(x.1 … x.n)log(m(x.1 … x.n)))], then:
        • we need p(x.1 … x.n) be a good sample of L.
        • H(L,m) = lim_(n->infinity) [1/n log m(x.1 … x.n)]
        • m(x.1 … x.n) = product(m(x.i| x.i-1))
        • If m(*) = 0 -> H(p,m) = infinity
        • Should associate the cost with something…
          • smoothing (lots of ways for smoothing)
    • Perplexity:
      • 2^(H(x.1 … x.n, m)) i.e. 2^ (cross entropy)
      • effective vocabulary size
      • recover the unit from “bits” to the original one.

After class (M& S book)

  • 6.2.1 MLE: gives the highest probability to the training corpus.
    • C(w1, … ,wn) is the frequency of n-gram w1…wn in the training set.
    • N is the number of training instances.
    • P_MLE(w1, …, wn) = C(w1, … , wn) / N
    • P_MLE(wn | w.1, …, w.n-1) = C(w1, … , wn)/C(w1, … , w.n-1)
  • 6.2.2 Laplace’s law, Lidstone’s law and the Jeffreys-Perks law
    • Laplace’s law
      • original smoothing
      • Lidstone’s law
      • Jeffreys-Perks law
        • Expected likelihood estimation (ELE)
  • 6.2.3 Held Out Estimation
    • Take further test and see how often bigrams that appeared r times in the training text tend to turn up in the future text.
  • 6.2.4 Cross Validation
    • Cross validation:
      • Rather than using some of the training data only for frequency counts and some only for smoothing probability estimates, more efficient schemes are possible where each part of the training data is used both as initial traiining data and as held out data.
      • Deleted Estimation: two-way cross validation:

The test speed of neural network?

Basically, the time spent on testing depends on:

  • the complexity of the neural network
    • For example, the fastest network should be the fully-connected network.
    • CNN should be faster than LSTM because LSTM is sequential (sequential = slow)
    • Currently, there are many ways to compress deep learning model (remove nodes with lighter weight)
  • the complexity of data
    • Try to build parallel models.

Analyze of different deep learning models:

  • RNN
    • CNN
    • LSTM
    • GRU
  • Seq2seq:
    • used for NLP
  • Autoencoder:
    • learn the features and reduce dimension’

Other machine learning models:

  • random forest
    • used for classification

Machine learning thought: