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I'm currently reading a theoretical economics paper, where they used an example I don't quite understand. I hope you guys can help me out🙂

The following excerpt is the example I don't quite understand, how do I get the probabilities at the end? Could someone maybe tell me how to get there?

An investor is deciding whether to invest in an entrepreneur’s new startup based on the entrepreneur’s past history of successes and failures. The entrepreneur’s first two startups failed, and the last three succeeded. The investor’s problem is to predict the probability of success of the sixth startup. The investor’s prior is that the startup’s probability of success, θ, is uniformly distributed on [0, 1]. Assume that, in the absence of persuasion, the investor would adopt the default view that the same success probability governs all of the entrepreneur’s startups. Also assume for the purpose of the example that this is the true model. The persuader wants the investor to invest, and thus wishes to propose models that maximize the investor’s posterior expectation of θ. The receiver will entertain models suggesting that the entrepreneur’s success probability was redrawn from the uniform distribution on [0, 1] at some point, so that only the most recent startups are relevant for estimating θ. Assuming these are the only models the receiver will entertain, the persuader will propose the model that the entrepreneur’s last three startups are relevant, but the first two are not. Under the default model that the success probability is constant over time, the receiver predicts the success probability of the next startup to be 57 percent. Under the persuader’s proposed model, the receiver instead predicts it to be 80 percent.

This example is from the paper "Using Models to Persuade" (Schwartzstein and Sunderam, 2021, AER).

Thanks in advance!

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1 Answer 1

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This is a straightforward application of a standard introductory Bayesian example, namely that a beta prior is conjugate for a Bernoulli sampling distribution. Any introductory Bayesian statistics textbook will explain this in detail.

In brief, the idea is that if your prior distribution on a parameter is $\theta\sim B(\alpha,\beta)$, and you observe $n$ Bernoulli trials with success probability $\theta$ and $x$ successes (and $n-x$ failures), then your posterior distribution on $\theta$ is $B(\alpha+x,\beta+n-x)$. (The fact that the posterior is still a Beta is what makes this "conjugate".)

In the present case, your prior is uniform on $[0,1]$, which is just a special case of a Beta, namely $B(1,1)$.

  • In the "default" model, we have observed $3$ successes and $2$ failures, so the posterior is $B(1+3,1+2)=B(4,3)$. The expectation of this Beta distribution is $\frac{4}{4+3}\approx 57\%$.
  • In the "proposed" model, we discard the two failures and only consider the $3$ successes. Now the posterior is a $B(1+3,1)=B(4,1)$, with an expectation of $\frac{4}{4+1}=80\%$.
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  • $\begingroup$ Thanks a lot for your answer. This is the first time dealing with conjugate distributions for me, this really helped me a lot! $\endgroup$
    – futur3boy
    Apr 24, 2021 at 14:03

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