Likelihood and the Maximum Likelihood Estimate · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Picking the bias the data likes best

Computing · Machine Learning · ages 18-20
Name ______________________   Date ____________

  1. Independent flips have the same head chance p. One observed sequence has four heads and two tails. Which expression is its likelihood?

    • p^2 × (1-p)^4
    • p^4 × (1-p)^2
    • p^4 + (1-p)^2

  2. When comparing candidate coin biases by likelihood, what varies?

    • The bias varies; the observed flips stay fixed.
    • The observed flips vary; the bias stays fixed.
    • Both the bias and the observed flips stay fixed.

  3. A maximum likelihood estimate chooses a parameter value with the highest likelihood for the observed data.

    Circle one:   True   False

  4. Four independent flips with a common head chance p all land tails. Which p maximizes the likelihood of this sequence?

    • p = 1.0
    • p = 0.0
    • p = 0.5

  5. Ten independent flips with a common head chance produce seven heads. What is the maximum likelihood estimate of that head chance? Enter a decimal.

    Answer: ______________

  6. Under which error model does maximizing a line's likelihood give the ordinary least-squares fit?

    • Errors are independent and uniform with mean zero and a common fixed variance.
    • Errors are independent and normal with mean zero and different fixed variances.
    • Errors are independent and normal with mean zero and a common fixed variance.

  7. The parameter with the highest likelihood is guaranteed to be the true parameter.

    Circle one:   True   False

  8. Five independent flips share a head chance p. For three heads, the count likelihood is 10 times the likelihood of any one such sequence. How do their maximizing p values compare?

    • They have the same maximizing p.
    • The count likelihood has a maximizing p ten times as large.
    • The sequence likelihood has a maximizing p ten times as large.
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Answer key

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Picking the bias the data likes best W1-mt_crcBnSqkTE-s1

  1. p^4 × (1-p)^2 · Four heads contribute four p factors. Two tails contribute two (1-p) factors. Multiply these factors for independent flips.
  2. The bias varies; the observed flips stay fixed. · Likelihood compares parameter values using the same observed data.
  3. True · The estimate is selected by comparing likelihood scores for the same observed data.
  4. p = 0.0 · There are no heads, so h/n = 0/4 = 0. The likelihood (1-p)^4 is largest at p = 0.
  5. 0.7 · Divide the head count by the number of flips: 7/10 = 0.7.
  6. Errors are independent and normal with mean zero and a common fixed variance. · Independent, mean-zero normal errors with a common fixed variance make the likelihood maximum coincide with the minimum sum of squared residuals.
  7. False · The maximum identifies the best score within the chosen model, not certainty about the true parameter.
  8. They have the same maximizing p. · Multiplying every candidate's likelihood by the same positive constant preserves which candidate scores highest.
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