To get more visitors to your website you are considering paying for an ad to be shown 100 times on a popular social media site. According to the social media site, their ads get clicked on 10% of the time. A reasonable generative model in this case is the binomial distribution to simulate how people click on ads. The histogram represents the probability distribution of the number of visitors to your site: Drawing 100’000 samples of a binomial distribution conditioned on 10% click-through rate, for each 100 ads shown, the number of visitors you’d most probably get is situated somewhere between 6 and 12.

The probable number of visitors you’ve got earlier assumes that the ads get clicked on exactly 10% of the time. But, are you sure this is a good assumption? Click-through rate could be as low as 0% or as high as 20%. This uncertainty about the underlying proportion of clicks can be modeled in statistical terms with a uniform probability distribution to reflect the range between 0% and 20%. So here let’s incorporate this other unknown parameter to the model. This really changes the game! Getting no clicks at all becomes way more real if you add the uncertain click-through rate to the model.

Once you actually ran your ad campaign, 13 people clicked and visited your site when the ad was shown a 100 times. This new information can now be used to update the Bayesian model and see how it influences the proportion of clicks you’re likely to get next.

Thanks to this updated parameter, now you can predict how many visitors you would get if you reran the ad campaign. Specifically, you are replacing the prior knowledge you had with the updated posterior (which is now your new prior). It now looks pretty likely that there would be more than 5 visitors following the campaign, with a probability over 97%.

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