A good overview of the subject and is very easy to read. In the argument section, I found his method of assessment very interesting which I will attempt to brief here.
Theoretical Commitment and Maximization of Explanatory Breadth - A Framework
"Given that atheistic beliefs are logically consistent, we can consider their non-trivial logical closure; and given that theistic beliefs are logically consistent, we can consider their non-trivial logical closure. (A set of sentences S is logically closed if and only if every sentence that is entailed by sentences that belong to S is itself a member of S. Said differently: a set of sentences is logically closed if and only if every logical consequence of members of S is itself a member of S. The logical closure of a set of sentences S is the set of sentences that is arrived at by adding sentences that are logical consequences of sentences already in S until there are no more sentences to be added. A set of sentences S has a non-trivial logical closure if and only if not every sentence belongs to the logical closure of S. Said differently: A set of sentences has a non-trivial logical closure if and only if S is logically consistent.) I shall call the logical closure of a set of logically consistent atheistic beliefs an atheistic big picture; and I shall call the logical closure of a set of theistic beliefs a theistic big picture. Big pictures divide into two parts: that which is common to a range of competing big pictures, which I call data, and that which is distinctive of particular big pictures, which I call worldview. So, for a given atheistic big picture and theistic big picture, what they agree on is data, and what they disagree on is, respectively, atheistic worldview and theistic worldview."(6.1)
To summarize and simplify (from what I have understood):
1. Assumption: Logical statements within Athiestic belief and Thiestic belief are logically consistent in themselves
2. The logical closure of a set of sentences is the set of all sentences that can be derived or logically inferred from the original set. Here the original set is meant to refer to the principal statements: ie. there is no god who has created the universe, or there is a god who created the universe.
3. The logical closure of a set of logically consistent atheistic beliefs is called an "atheistic big picture." Here we refer to the complete set of statements derived from the logically consistent original beliefs
4. The logical closure of a set of theistic beliefs is called a "theistic big picture." This refers to the complete set of statements that can be derived from logically consistent theistic beliefs.
5. Big pictures consist of two parts: data and worldview. Where data refers to aspects that are common between big pictures. and Worldview refers to aspects that are distinct to each big picture.
.....
"As just noted, the assessment of arguments for the superior theoretical virtue of one kind of best consistent big picture over another requires consideration of all relevant theoretical virtues. But what are relevant theoretical virtues in the assessment of best consistent atheistic, agnostic and theistic big pictures? While this is a controversial philosophical question, I think that the answer is straightforward: when we assess the comparative theoretical virtues of best consistent big pictures, we are interested in the trade-offs that they make between minimisation of theoretical commitments and maximisation of explanatory breadth and depth.....Theoretical commitments come in various different kinds: ontological, ideological, and nomological." (6.4)
Here ontological: commitment to a being or entity.
Ideological: commitments to ideas.
Nomological: commitment to principles.
Given the above: The Critique
6.8 Best theistic big pictures are logically inconsistent.
1 God exists.
2 God is omnipotent.
3 God is perfectly good.
4 There are no limits to what an omnipotent being can do.
5 A good being eliminates evil as far as it can.
6 Evil exists.
Though this is logically inconsistent, there are many theists (not Abrahamic) that do not call their god omnipotent - the whole argument collapses.
Theoretical Commitments
"In my view, the most plausible way to proceed is to argue for the following two claims: (a) best atheistic big pictures have fewer theoretical commitments than best theistic big pictures; and (b) there is no data that theistic big pictures explain better than atheistic big pictures. From these two claims, given the account of theoretical virtue in §6.4, it follows that best atheistic big pictures are superior to best theistic big pictures."
To argue that atheisic big pictures have less theoretical commitments, you must argue the following:
1. that best atheistic big pictures are less ontologically committing than best theistic big pictures;
2. that best atheistic big pictures are less ideologically committing than best theistic big pictures;
3. that best atheistic big pictures are less nomologically committing than best theistic big pictures.
Argument: "Setting gods aside, best atheistic big pictures and best theistic big pictures agree in their ontological, ideological, and nomological commitments....But best theistic big pictures—unlike best atheistic big pictures—require more: their commitment to gods". That is to say that excluding god, both the theist and athiest can agree that reality is a manifold with a distribution of fields over it. Additionally, regarding various other branches or science of philosophy, both atheistic and theistic viewpoints can agree on the objects, ideas, and principles that are important for understanding these different areas of knowledge.
Explanatory Parity
Theists claim, their worldview explains the following better:
(a) our part of natural reality is fine-tuned for our existence
Argument: Claims two things about fine-tuning:
"On the one hand, it may be that, at every prior point in causal reality, it is already fixed that—absent subsequent intervention—our universe is fine-tuned. On the other hand, it may be that there is some point in causal reality prior to which is it not fixed that—absent subsequent intervention—our universe is fine-tuned, but after which it is fixed that—absent subsequent intervention—our universe is fine-tuned."
Simply: Either the universe was always fine-tuned or it became fine-tuned. The atheist may say that in the first case, it was always fine-tuned by necessity (it could not have been any other way), and in the second, if it suddenly became fine-tuned, it is a matter of chance. The theist attributed both cases to God.
Hence: "Since all of these explanations are available in equal measure to atheists and theists, there is nothing in considerations about fine-tuning that supports best theistic big pictures over best atheistic big pictures."
(b) there are irreducibly complex biological entities
Argument: I will not go in-depth on this, because its pretty self-evident:
"we should not think that considerations about irreducible complexity give best theistic big pictures an explanatory advantage over best atheistic big pictures; rather, we should suppose that best atheistic big pictures and best theistic big pictures both accept that biological evolution by way of natural selection provides a complete account of the development of life on earth."
(c) we are capable of logical, mathematical, and statistical reasoning; we have modal knowledge; we have moral knowledge; we are moved by the dictates of conscience; we are capable of appreciating beauty; we are (sometimes) conscious; we have a capacity for humor; and we have a capacity for love.
Argument: "When we consider the full range of relevant data, it is overwhelmingly plausible that those capacities and their limitations are fully explained by billions of years of biological evolution and millions of year of cultural development. But, if that’s right, then best atheistic big pictures and best theistic big pictures tell exactly the same story about the capacities of human beings, and there is no explanatory advantage that accrues on either side as a result of consideration of this data."
(d) there are reliable reports of direct experience of the divine; there are scriptures that record important truths about the divine; there are reliable reports of divine intervention in human history.
Argument: "We have scriptures belonging to all of the religions of the earth, each of which is alleged to support exactly one of those religions. Everyone—no matter which big picture they espouse—supposes that almost all of the reports of miracles and religious experiences, and almost all of the scriptures, do not give support to the religions to which they are claimed to give support; everyone explains away almost all of the reports of miracles and religious experience and almost all of the scriptures. But, in that case, atheists are among those uniquely well-placed to give a uniform account of all reports of miracles and religious experiences and all scriptures; atheists explain them all away in exactly the manner that, for example, theists explain away everything except the data they take to support their own theistic religion.
Since atheists have a uniform account, and since it must be that there is an enormous amount of special pleading among those who do not have a uniform account, it is reasonable to accept that there is no explanatory advantage that accrues to those theists who advert to considerations about religious experience, scripture and miracles: best atheistic big pictures cannot do worse than best theistic big pictures in managing these considerations."
Summation
Given this specific limitation, "best atheistic big pictures have fewer theoretical commitments than best theistic big pictures, but there is nowhere that best theistic big pictures provide better explanations than best atheistic big pictures. So, best atheistic big pictures are more theoretically virtuous than best theistic big pictures, and so should be preferred to them."
Therefore, the atheist big picture is preferable.