Conditional posterior probability density (steps) I'm trying to understand how to condition a probabilistic posterior distribution.
Consider the following probability density:
$$
p(\alpha, \beta | y) = \prod_{i=1}^n (\alpha+\beta t_i)^{y_i}e^{-(\alpha+\beta t_i)}
$$
From this, can I say the following:
$$
p(\alpha|\beta, y) \propto \prod_{i=1}^n (\alpha+\beta t_i)^{y_i} e^{-\alpha}
$$
 A: My understanding from your equation (1) is that you mean $\mathbf{y} = (y_1, \cdots, y_n)$ is your sample, $\alpha$ and $\beta$ are parameters, and the density of the sample or likelihood is the Poisson regression:
$$p(\mathbf{y}|\alpha, \beta) \propto \prod_{i=1}^n (\alpha + \beta t_i)^{y_i} e^{-(\alpha + \beta t_i)}$$
To answer the question, conditioning on a random variable, like $\beta$, means that you know or fix $\beta$ and you want to know the density $p(\alpha |\beta, \mathbf{y})$. Assuming you have a non-informative prior on $\alpha$ and $\beta$, then the posterior has the same format as the likelihood function 
$$p(\alpha, \beta |\mathbf{y}) \propto \prod_{i=1}^n (\alpha + \beta t_i)^{y_i} e^{-(\alpha + \beta t_i)}$$
and the conditional density satisfies
$$p(\alpha|\beta, \mathbf{y}) \propto p(\alpha, \beta |\mathbf{y}) \propto\prod_{i=1}^n (\alpha + \beta t_i)^{y_i}e^{-\alpha}=e^{-n\alpha}\prod_{i=1}^n (\alpha + \beta t_i)^{y_i}$$
where the $\propto$ symbols are understood in terms of $\alpha$ only, given that $\beta$ and $Y$ are fixed. 
