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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.

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    $\begingroup$ 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. $\endgroup$
    – bdeonovic
    Jun 22, 2021 at 19:15
  • $\begingroup$ @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. $\endgroup$
    – knrumsey
    Jun 22, 2021 at 20:09
  • $\begingroup$ @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? $\endgroup$
    – knrumsey
    Jun 22, 2021 at 21:47
  • $\begingroup$ 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. $\endgroup$
    – Xi'an
    Jun 23, 2021 at 5:32

1 Answer 1

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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 $\gamma$.

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