So I am a good example of someone who holds empirical estimations of truth above any other form. Just to get my assumptions out of the way, I would endorse the implications of Cox's postulates. Cox's postulates are sufficient to arrive at all of the sciences and are based on Aristotelean logic.
While I have both Cox's book on the topic and Jaynes book on it, they are not physically available to me, so I am going to engage in heresy and use Wikipedia's description so that I can disclose my starting position. May the gods of science forgive me for using a Wiki.
Cox's construction of probability theory is around assessing the truth of logical statements using information. It is usually a form of inductive reasoning but can be deductive in the binary case. Two of the three postulates are not considered controversial, but one of them may be the most contentious axiom in probability theory.
The primary non-controversial axiom in Wikipedia is, unfortunately, described as consistency with common sense. A more formal look at this would be that any rule created to assess the plausibility of a statement contains no contradiction and that items with equivalent truth tables are the same. The other non-controversial one is that if there are two ways to assess the plausibility of a statement for a given set of information, then each method must arrive at the same result. Now, this may not imply strict equality, for example, if something has a 1/3rd chance of being true and 2/3rds chance of being false then it is equivalent to 1:2 odds the first one is true.
The controversial axiom is that the plausibility of a statement can be measured by or stated as a real number for a given information set. The controversy comes from the fact that Frequentist statistics cannot do this. The strongest statement that can be made with a Frequentist measure is that if a null hypothesis is true, then there is some value less than a real number p, which is in the open interval of (0,1), that is a statement of the worst-case probability of the observation being due to chance alone.
Of course, any reasoning, except for special cases, must be incomplete using Cox's postulates. Do note that neither Cox nor Kolmogorov have a unique axiomatic structure from which to construct a theory of probability. As I said, I am accepting Cox's postulates as sufficient to construct all of science on.
From Cox's postulates, you can derive both Bayes theorem, but also, Bayesian analysis over a parameter or a model space. This permits a clear solution to questions of belief.
For example, let us imagine that I am working to determine the validity of the idea of acceleration due to gravity. I can construct models where gravity doesn't exist, where the acceleration is a constant and where it is a non-constant. I can collect data, maybe poorly, and assess my relative beliefs. Perhaps I can do a very good job. As I gather more and more data, then I can get good beliefs. I can do this with anything that I can state a model and collect data.
Humans were hopelessly impoverished century after century until the coming of science and humans mostly lived short, brutish lives. People endorse empiricism because it produces usable results both to endorse concepts and to reject ideas.
For the idea of a god, however, it is not useful. It is similar to Sagan's comment from one of his instructors where he asked students why people believed the world was flat. Their response was because it looks flat, so he asked them what a round world would look like.
A world with a god looks exactly like this one. A world without a god looks exactly like this one. No amount of data can strictly exclude either case.
Therefore, the question is without importance. Not only can it not be solved, even if it could be solved the solution wouldn't matter at least not to the level that it would change my brand of toothpaste.
Now, this is not to say that individuals cannot be intrigued or driven by the problem. I know someone who is driven by knitting. If you like something, then there generally is no harm and probably quite some good for people to explore it. It is irrelevant to me. It simply cannot impact my life as there is no added content to my life or existence by either answer.
Let me give you a more concrete example. There does not exist a Frequentist solution to the estimator for the parameter, R, in the equation x(t+1)=Rx(t)+e(t+1), where x(t,t+1) are real numbers and R is an unknown real number greater than one and e is a random variable centered on zero with a finite and positive variance. While there may be some mathematical value in exploring why no solution can exist in Frequentist statistics, in fact, there is a great deal of value, it still results in there being no Frequentist way to solve this problem. It is the ancillary knowledge created in the understanding of why that is very important.
Questions about God are valuable regardless of whether a god exists, but treating any answer to the question as having any value or even attempting to strengthen or assert one point of view is without any value. If it cannot be solved then, ignoring the hobby value or the indirect value of testing the boundaries of knowledge, it is a waste of time to work on something that is non-existent.
I believe that listening to music or going on a hike is a valuable thing. I won’t find truth, but I will enjoy it. Maybe if I did psychological research, I would even care about why my mind enjoys music, being in the woods, or simply walking around. I have no doubt this is well studied.
It is possible that philosophers have misdefined truth and are conflating it with things that people persistently enjoy. That observation of regularity is the goal of the realm of probability and statistics. The beautiful thing about observing regularities with probability and statistics is it is possible to resolve, though not always, which patterns are real and which are idiosyncratic.
For me, it is sufficient to assert that truth need not exist. It is enough to define what does and does not constitute a regularity in nature operationally without requiring an actual existence. For example, I don’t believe that gravitation exists. I do not believe, for example, that a rock obeys the law of gravity because it cannot disobey. Instead, I believe that the fabric of existence, space-time if you would prefer, is warped and the effect we call gravity is a human solution to talking about the observed regular effect.
I also do not believe that I can disobey the laws of gravity. There is no agency that allows for a decision on which natural laws to follow and which to ignore. I also do not believe this is not up for debate. While I do spend time reading the philosophers, it has never been to find truth, but to find a way to think about things from a different perspective and to avoid well-known mental traps.
People, such as myself, use empiricism to discover truth because the alternatives cannot provide assurances that they lead anywhere useful.