Supposing a parameter of interest is assumed to follow an exponential distribution with rate $\lambda$. Two competing prior hypotheses are formed such that the first prior is given by $\text{Ga}(\alpha, \beta)$ and the second prior is given by $\text{Ga}(\alpha+1, \beta)$. Furthermore, data is collected on the parameter such that $x_{1}, x_{2}, \dots, x_{n}$ represent $n$ observations of recorded data.
The individual responsible for forming these priors suggests a degree of belief such that $p(\lambda \sim \text{Ga}(\alpha, \beta)) = 0.75$ and $p(\lambda \sim \text{Ga}(\alpha+1, \beta)) = 0.25$.
From the above:
- How would a Bayes factor for the two prior distributions be formed?
Can this be achieved by calculating the ratio between the two prior distributions?
i.e. $$\text{Bayes} factor = \frac{p(x) \sim \text{Ga}(\alpha, \beta)}{p(x) \sim \text{Ga}(\alpha+1, \beta)} = \frac{\frac{\beta^{\alpha}}{\Gamma(\alpha)}x^{\alpha-1}e^{-\beta x}}{\frac{\beta^{\alpha+1}}{\Gamma(\alpha+1)}x^{\alpha}e^{-\beta x}}$$
If so, given that there are multiple observations ($x_{i}$), how should each observation be inserted into the distribution to give a composite distribution (is it the sum of $p(x) \sim \text{Ga}(\alpha, \beta)$ such that $\sum_{i=1}^{n} p(x) \sim \text{Ga}(\alpha, \beta)$)? In which case the Bayes factor is given by:
$$\frac{\sum_{i=1}^{n} p(x_{i}) \sim \text{Ga}(\alpha, \beta)}{\sum_{i=1}^{n} p(x_{i}) \sim \text{Ga}(\alpha+1, \beta)}$$
- How can an estimate of the observed data be combined with the model probabilities to achieve a posterior ratio of model probabilities?
Can this be achieved by getting the product of the prior and likelihood distributions?
$$p(\lambda|{\bf x}) = p(x|\lambda)p(\lambda)$$
If so, assuming there is only a collection of observations and no suggested likelihood, how can the correct likelihood be determined?
This a generic homework-like template question, as opposed to a specific exercise hence the absence of specific parameter and data values.