I have a dataset of 1 dimensional 20points as prior information, so assuming prior distribution to be Gaussian distribution we can easily find its variance and mean. Now we will use this prior finding the posterior, further some extra event occurred and due to that we will be getting 5 more data points, So now how should we update the prior to get a new posterior?
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$\begingroup$ Updating a Bayesian posterior based on additional directly relevant information is possible. But your question lacks the detail that would make a useful and specific discussion of that possible. Can you please be more specific? // What prior? What kind of 'data points'? (Success/Failure? Normally distributed test scores?) $\endgroup$– BruceETCommented Jun 27, 2020 at 21:32
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1$\begingroup$ I have a dataset of 1 dimensional 20points as prior information, so assuming prior distribution to be Gaussian distribution so we easily find its variance and mean. Now we will use this prior finding the posterior, further some extra event occurred and due to that we will be getting 5 more data points, So now how should we update the prior to get a new posterior? $\endgroup$– Kabir KCommented Jun 27, 2020 at 21:42
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$\begingroup$ Suggest you edit Question appropriately. Preferably with a numerical example. Not everyone reads Comments. $\endgroup$– BruceETCommented Jun 27, 2020 at 21:44
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1 Answer
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You don’t use your current data to choose the prior.
What one usually does is:
- choosing a likelihood for the data, given the parameters: the main machinery of your model
$$ p(D|\theta) $$
- choosing a prior for the parameters based on the knowledge available before data collection (or weakly informative)
$$ p(\theta) $$
- calculating a posterior for the parameters using the observations and Bayes theorem:
$$ p(\theta|D)=\frac{p(D|\theta)p(\theta)}{p(D)} $$
If you have estimated the posterior for your parameters using a dataset $D$ and you obtain a new dataset D$^{\prime}$, you can “update” the posterior of the parameters using the “old” posterior as the prior for the new model fitting,
$$ p(\theta|D^{\prime})=\frac{p(D^{\prime}|\theta)p(\theta|D)}{p(D^{\prime})} $$
Basically, this is what happens when you learn the posterior even with a single dataset $D$. The exchangeability, in that case, allows you to avoid to consider the order of the observations. So, you may consider the case when previous knowledge is available. If your prior is based on that knowledge, it means that the distribution of $\theta$ is conditioned by that knowledge. Let's say $P(\theta|all\,I\,know\,about\,the\,process)$. For simplicity, we write it as $P(\theta)$. In the new posterior, I wrote the conditioning explicitly to show you the dependence from the previous observations $D$.