# On the Bayesian setup in inference

I've been trying to get into the chapter 4 in Lehmann's Theory of point estimation, but I can't seem to understand his presentation of the Bayesian setup. He starts of by the introduction below and after a few examples of uses of Bayesian estimators he outlines the idea (after dots in my photo). I don't know what he means by: $EL(\Theta,d)$.

In my opinion there should be two expectations there since we want to find d to minimize (1.1), I can't see how minimizing the one above being sufficient. I've tried with "Law of total expectation" and Fubini but nothing has been satisfactory. I have a similar problem with a theorem which comes right after the second paragraph.

• What is "Lehmanns book"? Can you make the figure larger, so it's easier to read? May 16 '16 at 16:37
• The reference to posterior risk at the end makes it clear that the expression $EL(\Theta,d)$ is the expected loss of state $\Theta$ with decision $d$.
– whuber
May 16 '16 at 17:29
• @whuber well the bayes risk (1.1) is $E_{\Lambda}E_{\theta} L[\theta,\delta]$, then why is this minimize given $EL(\Theta,d)$ is? I suppose they mean this is $E_{\Lambda}L(\Theta,d)$ btw May 16 '16 at 17:55
• Yes; Lehmann took pains to emphasize that $\Theta$ is governed by the distribution $\Lambda$, so this is the distribution involved in the expectation. The question he is answering is, how small can you make the risk by means of your choice of $d$? You might be over-interpreting this situation, which involves nothing more complicated than that idea.
– whuber
May 16 '16 at 18:07
• "Overlooked" in what sense? Lehmann explicitly takes expectations, both in the prior and posterior cases he describes.
– whuber
May 16 '16 at 19:30

This is Fubini's theorem in action: when minimising in $\delta$ $$\mathbb{E_{\Lambda}} \{\mathbb{E}_{\theta}[L(\theta,\delta)]\}=\int_\Theta\int_\mathcal{X} L(\theta,\delta(x))\text{d}P_\theta(x)\text{d}\Lambda(\theta)=\int_\mathcal{X} \int_\Theta L(\theta,\delta(x))\text{d}\Lambda_x(\theta)\text{d}P(x)$$where $\Lambda_x$ denotes the posterior distribution of $\theta$ conditional on $x$, one minimises in $d$ for each value of $x$ the posterior expected loss $$\int_\Theta L(\theta,d)\text{d}\Lambda_x(\theta)$$and set $$\delta(x)=\arg\min_d \int_\Theta L(\theta,d)\text{d}\Lambda_x(\theta)$$assuming all quantities are finite.
• So $\Lambda$ is prior after the first equality? The subscripts confuses me, I tought there expressionen for prior and post where almost the same(of the form in your answer), and that one just changed the distribution of $\Lambda$ in post. May 17 '16 at 4:34
• $\Lambda$ denotes the prior and $\lambda_x$ the posterior, while $P_\theta$ denotes the sampling distribution and $P$ the marginal. May 17 '16 at 5:03