MIT15_097S12_lec15

Coin flip example part 2 returning to the coin ip

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Unformatted text preview: od model, maximum likelihood is guaranteed to find the correct distribution, as m goes to infinity. In proving consistency, we do not get finite sample guar­ antees like with statistical learning theory; and data are always finite. Coin Flip Example Part 2. Returning to the coin flip example, equation (2), the log-likelihood is R(θ) = mH log θ + (m − mH ) log(1 − θ). We can maximize this by differentiating and setting to zero, and doing a few lines of algebra: dR(θ) ˆ dθ θ ˆ mH (1 − θML ) ˆ mH − θML mH ˆ θML 0 = m H m − mH − θ 1−θ ˆ = (m − mH )θML ˆ ˆ = mθML − θML mH mH = . m = ˆ θ (5) (It turns out not to be difficult to verify that this is indeed a maximum). In this case, the maximum likelihood estimate is exactly what we intuitively thought we should do: estimate θ as the observed proportion of Heads. 4 2.2 Maximum a p osteriori (MAP) estimation The MAP estimate is a pointwise estimate with a Bayesian flavor. Rather than finding θ tha...
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