This version of the essay has been lightly edited. You can find the original here.
One theme of our work is trying to help populations that many people don't feel are worth helping at all.
We've seen major opportunities to improve the welfare of factory-farmed animals, because so few others are trying to do it. When working on immigration reform, we've seen big debates about how immigration affects wages for people already in the U.S., and much less discussion of how it affects immigrants. Even our interest in global health and development is fairly unusual: Many Americans may agree that charitable dollars go further overseas, but prefer to give domestically because they so strongly prioritize people in their own country compared to people in the rest of the world.
The question "Who deserves empathy and moral concern?" is central for us. We think it's one of the most important questions for effective giving, and generally.
Unfortunately, we don't think we can trust conventional wisdom and intuition on the matter: History has too many cases where entire populations were dismissed, mistreated, and deprived of basic rights for reasons that fit the conventional wisdom of the time but today look indefensible. Instead, we aspire to radical empathy: working hard to extend empathy to everyone it should be extended to, even when it is unusual or seems strange to do so.
To clarify the choice of terminology:
"Radical" is intended as the opposite of "traditional" or "conventional." It doesn't necessarily mean "extreme" or "all-inclusive"; we don't extend empathy to everyone and everything (this would leave us essentially no basis for making decisions about morality). It refers to working hard to make the best choices we can, without anchoring to convention.
"Empathy" is intended to capture the idea that one could imagine oneself in another's position, and recognizes the other as having experiences that are worthy of consideration. It is not intended to refer to literally feeling what another feels, and is therefore distinct from the "empathy" critiqued in Against Empathy (a book that acknowledges the multiple meanings of the term and explicitly focuses on one).
Conventional wisdom and intuition aren't good enough
In The Expanding Circle, Peter Singer discusses how, over the course of history, "The circle of altruism has broadened from the family and tribe to the nation and race ... to all human beings" (and adds that "the process should not stop there").  By today's standards, the earliest cases he describes are striking:
At first [the] insider/ outsider distinction applied even between the citizens of neighboring Greek city-states; thus there is a tombstone of the mid-fifth century B.C. which reads:
This memorial is set over the body of a very good man. Pythion, from Megara, slew seven men and broke off seven spear points in their bodies ... This man, who saved three Athenian regiments ... having brought sorrow to no one among all men who dwell on earth, went down to the underworld felicitated in the eyes of all.
This is quite consistent with the comic way in which Aristophanes treats the starvation of the Greek enemies of the Athenians, starvation which resulted from the devastation the Athenians had themselves inflicted. Plato, however, suggested an advance on this morality: he argued that Greeks should not, in war, enslave other Greeks, lay waste their lands or raze their houses; they should do these things only to non-Greeks. These examples could be multiplied almost indefinitely. The ancient Assyrian kings boastfully recorded in stone how they had tortured their non-Assyrian enemies and covered the valleys and mountains with their corpses. Romans looked on barbarians as beings who could be captured like animals for use as slaves or made to entertain the crowds by killing each other in the Colosseum. In modern times Europeans have stopped treating each other in this way, but less than two hundred years ago some still regarded Africans as outside the bounds of ethics, and therefore a resource which should be harvested and put to useful work. Similarly Australian aborigines were, to many early settlers from England, a kind of pest, to be hunted and killed whenever they proved troublesome. 
The end of the quote transitions to more recent, familiar failures of morality. In recent centuries, extreme racism, sexism, and other forms of bigotry — including slavery — have been practiced explicitly and without apology, and often widely accepted by the most respected people in society.
From today's vantage point, these seem like extraordinarily shameful behaviors, and people who were early to reject them — such as early abolitionists and early feminists — look to have done extraordinary amounts of good. But at the time, looking to conventional wisdom and intuition wouldn't necessarily have helped people avoid the shameful behaviors or seek out the helpful ones.
Today's norms seem superior in some respects. For example, racism is much more rarely explicitly advocated (which is not to say that it is rarely practiced). However, we think today's norms are still fundamentally inadequate for the question of who deserves empathy and moral concern. One sign of this is the discourse in the U.S. around immigrants, which tends to avoid explicit racism but often embraces nationalism — excluding or downplaying the rights and concerns of people who aren't American citizens (and even more so, people who aren't in the U.S. but would like to be).
Intellect vs. emotion
I sometimes hear the sentiment that moral atrocities tend to come from thinking of morality abstractly, losing sight of the basic emotional basis for empathy, and distancing oneself from the people one's actions affect.
I think this is true in some cases, but importantly false in others.
People living peaceful lives are often squeamish about violence, but it seems that this squeamishness can be overcome disturbingly quickly with experience. There are ample examples throughout history where large numbers of "conventional" people casually and even happily practiced direct cruelty and violence to those whose rights they didn't recognize.  Today, watching the casualness with which factory farm workers handle animals (as shown in this gruesome video), I doubt that people would eat much less meat if they had to kill animals themselves. I don't think the key is whether people see and feel the consequences of their actions. More important is whether they recognize those whom their actions affect as fellow persons, meriting moral consideration.
On the flip side, there seems to be at least some precedent for using logical reasoning to reach moral conclusions that look strikingly prescient in retrospect. For example, see Wikipedia on Jeremy Bentham, who is known for basing his morality on the straightforward, quantitative logic of utilitarianism:
He advocated individual and economic freedom, the separation of church and state, freedom of expression, equal rights for women, the right to divorce, and the decriminalising of homosexual acts. [My note: he lived from 1747-1832, well before most of these views were common.] He called for the abolition of slavery, the abolition of the death penalty, and the abolition of physical punishment, including that of children. He has also become known in recent years as an early advocate of animal rights.
Aspiring to radical empathy
Who deserves empathy and moral concern?
To the extent that we get this question wrong, we risk making atrocious choices. If we can get it right to an unusual degree, we might be able to do outsized amounts of good.
Unfortunately, we don't think it is necessarily easy to get it right, and we're far from confident that we are doing so. But here are a few principles we try to follow, in making our best attempt:
Acknowledging our uncertainty. For example, we're quite unsure of where animals should fit into our moral framework. My own reflections and reasoning about philosophy of mind have, so far, seemed to indicate against the idea that e.g. chickens merit moral concern. And my intuitions value humans astronomically more. However, I don't think either my reflections or my intuitions are highly reliable, especially given that many thoughtful people disagree. And if chickens do indeed merit moral concern, the amount and extent of their mistreatment is staggering. With worldview diversification in mind, I don't want us to pass up the potentially considerable opportunities to improve their welfare.
I think the uncertainty we have on this point warrants putting significant resources into farm animal welfare, as well as working to generally avoid language that implies that only humans are morally relevant.
That said, I don't feel uncertain about all of our unusual choices. I'm confident that differences in geography, nationality, and race ought not affect moral concern, and our giving should reflect this.
Being extremely careful about too quickly dismissing "strange" arguments on this topic. Relatively small numbers of people argue that insects, and even some algorithms run on today's computers, merit moral concern. It's easy and intuitive to laugh these viewpoints off, since they seem so strange on their face and have such radical implications. But as argued above, I think we should be highly suspicious of our instincts to dismiss unusual viewpoints on who merits moral concern. And the stakes could certainly be high if these viewpoints turn out to be more reasonable than they appear at first.
So far I remain unconvinced that insects, or any algorithms run on today's computers, are strong candidates for meriting moral concern. But I think it's important to keep an open mind.
Exploring the idea of supporting deeper analysis. Luke Muehlhauser is currently exploring  the current state of research and argumentation on the question of who merits moral concern (which he calls the question of moral patienthood). It's possible that if we identify gaps in the literature and opportunities to become better informed, we'll recommend funding further work. In the near future, work along these lines could affect our priorities within farm animal welfare — for example, it could affect how we prioritize work focused on improving the treatment of fish. Ideally, our views on moral patienthood would be informed by an extensive literature drawing on as much deep reflection, empirical investigation, and principled argumentation as possible.
Not limiting ourselves to the "frontier," because widely recognized problems still do a great deal of damage. In our work, we often find ourselves focusing on unconventional targets for charitable giving, such as farm animal welfare and potential risks from advanced artificial intelligence. This is because we often find that opportunities to do disproportionate amounts of good are in areas that have been, in our view, relatively neglected by others. However, our goal is to do the most good we can, not to seek out and support those causes which are most "radical" in our present society. When we see great opportunities to play a role in addressing harms in more widely-acknowledged areas — for example, in the U.S. criminal justice system — we take them.
For example, according to data from Giving USA, only approximately 4% of US giving in 2015 was focused on international aid. (Reported by Charity Navigator here.) ↩︎
Page 120. ↩︎
Pages 112-113. ↩︎
As a side note, it is often tricky to avoid such language. We generally use the term "persons" when we want to refer to beings that merit moral concern, without pre-judging whether such beings are human and also without causing too much distraction for casual readers. A more precise term is "moral patients." ↩︎