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I'm fitting a model using Bayesian MCMC. The model parameters include a parameter vector $\beta$ which is assumed to reside on a simplex $$S^d=\left\{ \beta=(\beta_1,\beta_2,\cdots,\beta_d);\space \beta_i>0\space\forall i;\space \sum_{i=1}^d \beta_i =1\right\} $$ of low dimension ($d$ typically between 5 and 10). Currently I am updating this parameter by transforming it to $\mathbb{R}^{d-1}$ using the isometric logratio transform $\xi=\text{ilr}(\beta)$, which is given by $$\xi_i=\sqrt{\frac{i}{i+1}}\text{ln}\left(\frac{g(\beta_1,\beta_2,\cdots,\beta_i)}{\beta_{i+1}}\right) $$ for $i=1,2,\cdots,d-1$, where $g$ denotes the geometric mean (for more details see e.g. this article) and using a multivariate normal MCMC proposal on this real space. The reason why I'm using the ILR transform is because it seems easier to define a MCMC proposal (e.g. MVN) on the 'unrestricted' real space $\mathbb{R}^{d-1}$ than on the simplex $S^d$. Anyway, this approach works fine, but there is also the question of what prior to impose on the transformed variable $\xi=\text{ilr}(\beta)$ in the real space $\mathbb{R}^{d-1}$.

I've tried a multivariate normal prior on $\xi$, but I've found that this typically puts most of the mass at the edges of the simplex (for large standard deviations) or puts most of the mass in the centre of the simplex (for small standard deviations). I'm looking for something more uniform on the simplex. If $\beta$ is uniformly (or Dirichlet) distributed on the simplex, what distribution would the transformed variable $\xi$ have in real space? In principle it should be possible to work this out using change of variables, but it looks quite messy, so I wondered if anyone was aware of a reference covering this matter?

I'd also be interested in ideas for proposals defined directly on the simplex - unfortunately I couldn't find much material about this in the literature so, again, if you're aware of a reference covering examples of such proposals then I'd be very interested to know about it.

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    $\begingroup$ If $\beta$ is a physical, meaningful parameter, the one which you have some knowledge about to say that it lies on a simplex, why in particular do want to do the transform anyway? $\endgroup$ – innisfree Mar 14 at 11:26
  • $\begingroup$ @innisfree because it seems easier to define a MCMC proposal (e.g. MVN) on unrestricted real space than on the simplex... but I'd also be interested in ideas for proposals defined directly on the simplex - unfortunately I couldn't find much material about this. $\endgroup$ – S. Catterall Mar 14 at 11:30
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    $\begingroup$ A Dirichlet centred at the current location$$\beta^\prime\sim\mathcal D(\alpha\beta_1,\ldots,\alpha\beta_d)$$ is the Dirichlet version of a Gaussian random walk proposal (but it is not symmetric and hence requires the proposal densities to appear in the Metropolis-Hastings ratio. The quantity $\alpha$ can be calibrated to achieve an acceptance rate between $1/4$ and $1/2$. $\endgroup$ – Xi'an Mar 14 at 11:37
  • $\begingroup$ @Xi'an thanks for the suggestion, that's an interesting idea that I hadn't come across before. Have you ever seen such a proposal being used in published work? $\endgroup$ – S. Catterall Mar 14 at 11:41
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