# What is the interpretation of this modified Metropolis algorithm?

## Modified Metropolis-Hastings

Consider a model with parameters $$\theta = (\alpha, \gamma)$$ and consider a modified Metropolis-Hastings algorithm which can be summarized (with brevity) as follows.

1. Propose a new value $$\alpha_\star$$
2. Conditional on $$\alpha_\star$$, fix $$\gamma_\star$$ according to some estimation rule (e.g. MLE or MAP)
3. Calculate MH acceptance probability and choose whether or not to accept $$\theta_\star = (\alpha_\star, \gamma_\star)$$.

## An Example

To help clarify, we can consider an extremely simple example. Let $$X_1, X_2, \ldots X_{n}$$ be iid samples from a $$N(\alpha, \gamma)$$ distribution. Let $$\pi(\alpha, \gamma, {\bf X})$$ denote the posterior distribution.

1. Initialize $$\alpha^0$$.
2. Set $$\gamma^0 = \arg\max_{\gamma}\left\{ \pi(\alpha^0, \gamma, {\bf X})\right\}$$
3. Compute $$\pi^0 = \pi(\alpha^0, \gamma^0, {\bf X})$$

For $$t = 1, 2, \ldots T$$

1. Propose $$\alpha_\star$$ (using a proposal, symmetric about $$\alpha^{t-1}$$ for simplicity).
2. Set $$\gamma_\star = \arg\max_{\gamma }\left\{ \pi(\alpha_\star, \gamma, {\bf X})\right\}$$
3. Calculate $$\pi_\star = \pi(\alpha_\star, \gamma_\star, {\bf X})$$
4. Set $$\alpha^t = \alpha_\star, \gamma^t = \gamma_\star$$ and $$\pi^t = \pi_\star$$ with probability $$\max\left(\frac{\pi_\star}{\pi^{t-1}}, 1\right)$$ and set $$\alpha^t = \alpha^{t-1}$$, $$\gamma^t = \gamma^{t-1}$$ and $$\pi^t = \pi^{t-1}$$ otherwise.

## Discussion and Questions

• I have performed a simple simulation study using the simple example described above, and found that this modified approach performed similarly (slightly better, actually) in terms of inference for $$\alpha$$ than the standard Bayesian approach.

• It seems that this approach can be viewed as placing a prior distribution on $$\theta = (\alpha, \gamma)$$ which places all of the prior density on regions of the parameter space where $$\gamma$$ is maximized with respect to $$\alpha$$ (a manifold, I believe).

• I am looking at this for much more complicated problems than the simple example above. In particular, problems where it is very difficult to propose reasonable values of $$\gamma$$ and maintain a computationally tractable Metropolis-Hastings ratio.

• Questions: Is this a valid approach to Bayesian inference? Does this algorithm have a name? Is there a nice way to interpret this approach? Literature discussing this in any way is quite welcome.

• This sounds a bit dubious to me, but I could be wrong. I'm worried here since the distribution from which you are getting $\alpha$ is constantly changing. In the end the $\alpha$ draws you get certainly are not from $\pi(\alpha,\gamma,\mathbf{X})$, nor are they from the marginal $\pi(\alpha,\mathbf{X})$ with $\gamma$ marginalized out. I'm not saying that such an approach can't be useful for something (e.g. the MAP $\hat{\alpha}$ you get with such an approach may be consistent or have other favorable statistical properties, but that doesn't mean you have sampled the posterior of interest. Jun 22, 2021 at 19:15
• @bdeonovic I agree with you completely. This cannot lead to samples from the posterior of interest. I think that the modifications essentially lead to a strange proposal distribution for \theta, which is not being accounted for in the MH acceptance step. Still, I think it can be a reasonable approach from a practical perspective in certain challenging problems, and am curious if this has been studied in other contexts. Jun 22, 2021 at 20:09
• @Xi'an Thank you for this comment. Can you say anything else about this approach in general? Do you know of anybody taking a similar approach? Jun 22, 2021 at 21:47
• The closest reference I can think of is the stochastic EM algorithm of Diebolt & Celeux (1981) or Lavielle & Moulines (1997). The latent variable is simulated and the true parameter is optimised at each iteration. Jun 23, 2021 at 5:32

This Metropolis-Hastings algorithm is exact for the distribution over $$\alpha$$ $$\tilde \pi(\alpha| \mathbf x) \propto \pi(\alpha,\arg\max_\gamma\pi(\alpha,\gamma,\mathbf x),\mathbf x)$$ which is a form of profile posterior. It thus does not target $$\pi(\alpha,\gamma|\mathbf x)$$ and does not bring information about $$\gamma$$. The difference with the average $$\pi(\alpha| \mathbf x) \propto \int \pi(\alpha,\gamma,\mathbf x)\,\text d\gamma$$ means that there is no accounting for the variability in $$\gamma$$. Plus, setting $$\gamma$$ to depend on $$\mathbf x$$ as well as $$\alpha$$ means a potential for overfitting.
To check the difference, I look at a Normal case: $$\pi(\alpha,\gamma|\mathbf x)\propto \alpha^{-n-3}\exp[\{-n(\bar x-\gamma)^2-s^2-\delta\gamma^2-\beta\}/2\alpha^2]$$ where $$\pi(\alpha|\mathbf x)\propto\alpha^{-n-2}\exp[\{-\underbrace{(n^{-1}+\delta^{-1})^{-1}\bar x^2}_{=\dfrac{n\delta\bar x^2}{n+\delta}}-s^2-\beta\}/2\alpha^2]$$ and $$\gamma^\star(\alpha,\bar x)=\frac{n\bar x}{n+\delta}$$ implying $$\tilde\pi(\alpha|\mathbf x)\propto\alpha^{-n-3}\exp[\{-n^2\delta\bar x^2/(n+\delta)^2-s^2-\beta\}/2\alpha^2]$$ which shows the difference between both "posteriors". The pseudo-posterior $$\tilde\pi(\alpha|\mathbf x)$$ is more concentrated near zero, reflecting upon the extra confidence on $$\gamma$$ brought by using the "best" value of $$\gamma$$.