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I'm doing Bayesian analysis using a $\mathrm{Lindley}(\lambda)$ likelihood and a $\mathrm{Gamma}(\alpha,\beta)$ prior.

For $n$ i.i.d. data with $\mathrm{Lindley}(\lambda)$, the likelihood is:

$$f(\mathbf{y} \mid \lambda)=\frac{\lambda^{2n}}{(\lambda+1)^n}\prod(1+y_i)\exp\bigl(-\lambda\sum{y_i} \bigr)$$

The prior is:

$$p(\lambda \mid \alpha,\beta)=\frac{\beta^{\alpha}}{\Gamma(\alpha)}\lambda^{\alpha-1}\exp(-\beta\lambda)$$

Then I got the posterior:

$$\pi(\lambda \mid \mathbf{y},\alpha,\beta)\propto\frac{\lambda^{2n+\alpha-1}}{(\lambda+1)^n}\exp\Bigl\{-\bigl(\sum{y_i}+\beta \bigr)\lambda\Bigr\}$$

I'm wondering if my calculations is correct. If so, does this posterior have a name?

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As usual, it is easier to work with the kernels rather than the densities. Our posterior has kernel $$f(\lambda \mid \boldsymbol y, \alpha, \beta) \propto f(\boldsymbol y \mid \lambda) f(\boldsymbol \lambda \mid \alpha, \beta) \propto \frac{\lambda^{2n}}{(\lambda+1)^n} e^{- n \bar y \lambda} \lambda^{\alpha-1} e^{-\beta \lambda} = \frac{\lambda^{2n+\alpha-1}}{(\lambda+1)^n} e^{-(n \bar y + \beta)\lambda}.$$ So my calculation agrees with yours. This is clearly neither Lindley distributed nor gamma distributed.

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