Python for Probability, Statistics, and Machine Learning by José Unpingco

Author:José Unpingco
Language: eng
Format: epub, pdf
ISBN: 9783030185459
Publisher: Springer International Publishing

where we have determined and from the data. Given a new point of interest, , we would certainly compute

as the predicted value for . This is the same as saying that our best prediction for y based on is the above conditional expectation. The variance for this is the following:

Note that we have the covariance above because and are derived from the same data. We can work this out below using our previous notation from Eq. 3.7.0.1,

After plugging all this in, we obtain the following:

where, in practice, we use the plug-in estimate for the .

There is an important consequence for the confidence interval for . We cannot simply use the square root of to form the confidence interval because the model includes the extra noise term. In particular, the parameters were computed using a set of statistics from the data, but now must include different realizations for the noise term for the prediction part. This means we have to compute

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