Estimate Posterior with: Binomial Likelihood with 1/θ Prior

Question

I am trying to estimate the posterior distribution using Bayes theorem. The following information is given:

\begin{align} \newcommand{\prior}{{\rm prior}} \newcommand{\likelihood }{{\rm likelihood }} \newcommand{\posterior}{{\rm posterior}} \posterior&∝\likelihood \times \prior \\ \likelihood(x|θ)&∝ θ^k\times(1-θ)^{n-k} \\ \prior(θ)&∝1/θ \end{align}

Here I interested in estimating the posterior distribution of θ, with Binomial likelihood(k successes of the n trials).

By multiplying the likelihood and prior I get the following:

\begin{align} \posterior&∝θ^k\times(1-θ)^{n-k}\times1/θ \quad = \quad θ^k\times(1-θ)^{n-k}\timesθ^{-1} \\ \posterior&∝θ^{k-1}\times(1-θ)^{n-k} \end{align}

Question: The posterior distribution looks like Beta distribution: $θ^{\alpha-1}(1-θ)^{\beta-1}$. It is quite clear that $\alpha=k$ , but I am struggling to figure out how to transform power $n-k$ to look like $\beta_{new}={\rm something} - 1$. Could anyone point me to the right direction?

Answer 1

Maybe I'm missing something, but isn't the prior, $p(\theta) \propto \theta^{-1}$ merely a $\mbox{beta}(a=0,b=1)$ distribution? (If $a=0$ is uncomfortable, then set $a=\epsilon$ for small $\epsilon$.) Therefore the usual beta-Bernoulli update rule should apply. That is, with a Bernoulli likelihood function (which this is, for $N$ independent trials), and a conjugate beta prior (for which the present problem is a special case), the posterior is again a beta distribution: $\mbox{beta}(a+k,b+N-k)$. Here we start with $a=\epsilon$ and $b=1$, so the posterior is $\mbox{beta}(\epsilon+k,1+N-k)$.

P.S. For a derivation of the general beta-Bernoulli update rule, see p. 132 of DBDA2E.