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## Inference from a data set and sequential use of Bayes formula

In the elementary examples shown above, the inference has been done from a single data point . If we have a set of observations (data), indicated by , we just need to insert in the Bayes formula the likelihood , where this expression indicates a multi-dimensional joint pdf.

Note that we could think of inferring on the basis of each newly observed datum . After the one observation:

 (45)

and after the second:
 (46) (47)

We have written Eq. (47) in a way that the dependence between observables can be accommodated. From the product rule in Tab. 1, we can rewrite Eq. (47) as
 (48)

Comparing this equation with (47) we see that the sequential inference gives exactly the same result of a single inference that properly takes into account all available information. This is an important result of the Bayesian approach.

The extension to many variables is straightforward, obtaining

 (49)

Furthermore, when the are independent, we get for the likelihood
 (50) (51)

that is, the combined likelihood is given by the product of the individual likelihoods.

Next: Multidimensional case Up: Inferring numerical values of Previous: Poisson model
Giulio D'Agostini 2003-05-13