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Edit: A Bayesian model that exhibits overfitting (#10)
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yousuketakada committed Apr 8, 2018
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Expand Up @@ -1971,12 +1971,12 @@ \subsubsection*{#1}
whereas the precision~$\alpha$ of the parameters~$\mathbf{w}$ in the prior~(3.52) is very small
(i.e., the conditional distribution of $t$ given $\mathbf{w}$ is narrow whereas
the prior over $\mathbf{w}$ is broad), leading to insufficient \emph{regularization}
(see also Section~3.1.4).
Then, the posterior~$p(\mathbf{w}|\bm{\mathsf{t}})$ given the data set~$\bm{\mathsf{t}}$ is
(see Section~3.1.4).
Then, the posterior~$p(\mathbf{w}|\bm{\mathsf{t}})$ given the data set~$\bm{\mathsf{t}}$ will be
sharply peaked around the maximum likelihood estimate~$\mathbf{w}_{\text{ML}}$ and
the predictive~$p(t|\bm{\mathsf{t}})$ is also sharply peaked
(well approximated by the likelihood conditioned on $\mathbf{w}_{\text{ML}}$)
so that the assumed model reduces to the least squares method,
the predictive~$p(t|\bm{\mathsf{t}})$ be also sharply peaked
(well approximated by the likelihood conditioned on $\mathbf{w}_{\text{ML}}$).
Stated differently, the assumed model reduces to the least squares method,
which is known to suffer from overfitting (see Section~1.1).

Of course, we can extend the model by incorporating hyperpriors over $\beta$ and $\alpha$,
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