Return to Question

In Bayesian inference, we have a dataset $x$ and assumed to come from a known parameterized distribution with unknown parameters $\theta$. We then seek to maximize the posterior $P(\theta|x)$ in order to estimate $\theta$.

Here, $x$ is treated as fixed. How can we study the robustness of the MAP estimator, $\theta^* = \text{argmax} P(\theta|x)$? I.e. how is the distribution/variance of $x$ used to make statements about how $\theta^*$ changes under different samplings of $x$? I know $\text{Var}(P(\theta|x))$ tells us how "unsure" we are of $\theta^*$ for example.

More specifically, the influence function is a measure of robustness of an estimator. How can this be applied to the MAP?

In Bayesian inference, we have a dataset $x$ and assumed to come from a known parameterized distribution with unknown parameters $\theta$. We then seek to maximize the posterior $P(\theta|x)$ in order to estimate $\theta$.

Here, $x$ is treated as fixed. How can we study the robustness of the MAP estimator, $\theta^* = \text{argmax} P(\theta|x)$? I.e. how is the distribution/variance of $x$ used to make statements about how $\theta^*$ changes under different samplings of $x$? I know $\text{Var}(P(\theta|x))$ tells us how "unsure" we are of $\theta^*$ for example.

More specifically, the influence function is a measure of robustness of an estimator. Can this be applied to the MAP?

In Bayesian inference, we have a dataset $x$ and assumed to come from a known parameterized distribution with unknown parameters $\theta$. We then seek to maximize the posterior $P(\theta|x)$ in order to estimate $\theta$.

Here, $x$ is treated as fixed. How can we study the robustness of the MAP estimator, $\theta^* = \text{argmax} P(\theta|x)$? I.e. how is the distribution/variance of $x$ used to make statements about how $\theta^*$ changes under different samplings of $x$? I know $\text{Var}(P(\theta|x))$ tells us how "unsure" we are of $\theta^*$ for example.

More specifically, the influence function is a measure of robustness of an estimator. How can this be applied to the posterior?

In Bayesian inference, we have a dataset $x$ and assumed to come from a known parameterized distribution with unknown parameters $\theta$. We then seek to maximize the posterior $P(\theta|x)$ in order to estimate $\theta$.

Here, $x$ is treated as fixed. How can we study the robustness of the MAP estimator, $\theta^* = \text{argmax} P(\theta|x)$? I.e. how is the distribution/variance of $x$ used to make statements about how $\theta^*$ changes under different samplings of $x$? I know $\text{Var}(P(\theta|x))$ tells us how "unsure" we are of $\theta^*$ for example.

More specifically, the influence function is a measure of robustness of an estimator. How can this be applied to the MAP?

In Bayesian inference, we have a dataset $x$ and assumed to come from a known parameterized distribution with unknown parameters $\theta$. We then seek to maximize the posterior $P(\theta|x)$ in order to estimate $\theta$.

Here, $x$ is treated as fixed. How can we study the robustness of the MAP estimator, $\theta^* = \text{argmax} P(\theta|x)$? I.e. how is the distribution/variance of $x$ used to make statements about how $\theta^*$ changes under different samplings of $x$?

  I know the variance of the posterior tells us how "sure" we are of $\theta^*$, but what tools are available?

In Bayesian inference, we have a dataset $x$ and assumed to come from a known parameterized distribution with unknown parameters $\theta$. We then seek to maximize the posterior $P(\theta|x)$ in order to estimate $\theta$.

Here, $x$ is treated as fixed. How can we study the robustness of the MAP estimator, $\theta^* = \text{argmax} P(\theta|x)$? I.e. how is the distribution/variance of $x$ used to make statements about how $\theta^*$ changes under different samplings of $x$? I know $\text{Var}(P(\theta|x))$ tells us how "unsure" we are of $\theta^*$ for example.

More specifically, the influence function is a measure of robustness of an estimator. How can this be applied to the posterior?