There is no evidence for God

In #religion

On Alex O'Connor's podcast, he discusses with Joe Schmid, Atheist Slogans You Should Stop Using (https://creators.spotify.com/pod/profile/alex-oconnor4/episodes/Atheist-Slogans-You-Should-Stop-Using---Joe-Schmid-e3eqc52). I am not that far into the episode (only two points in) and already I disagree with much of what they are saying, so I think I will try to write responses to each. I've already written a response to the first one well before the podcast aired, which makes me a prophet!

  • Extraordinary Claims Require Extraordinary Evidence
  • There Is No Evidence for God
  • Who Created God?
  • Science Flies You to the Moon, Religion Flies You into Buildings
  • Your Location Determines Your Religion
  • Claims Are Not Evidence
  • You Can’t Prove a Negative
  • What Can Be Asserted Without Evidence Can Be Dismissed Without Evidence
  • Faith Is Belief Without Evidence
  • Religion Makes Good People Do Bad Things
  • Absence of Evidence Is Evidence of Absence
  • Theism Is Unfalsifiable
  • Science Books Would Come Back, Religions Would Not
  • Evolution Disproves God

The definition of evidence that Joe states is, "Evidence is information that raises the probability of my hypothesis". Alex gives an example, attempting to find a corner case, where he says that under the alien abductions hypothesis you would predict certain government failures, so seeing government failures is evidence of alien abductions. Joe says that this is just weak evidence, but evidence nonetheless. In this way, saying "there is no evidence for God" is hyperbole -- there is just bad evidence, or weak evidence. While I agree that saying "there is no evidence of God" might be in most cases hyperbolic, it may not be true in all cases. Mathematically, the way one can translate Alex and Joe's perspectives on evidence is with the following two statements:

  1. "Evidence is information that raises the probability of my hypothesis":
    $$P(H_1|E)>P(H_1)$$
  2. Anything that is expected on my hypothesis is evidence of the hypothesis, even if only weak evidence:
    $$P(E|H_1) > 0.5$$
    is a necessary condition for \(E\) to be considered evidence.

Unfortunately these two conditions are not guaranteed to be true at the same time, depending on the alternative hypotheses considered. If we take the simple case of a single other alternative, \(H_2\), compared to our hypothesis, \(H_1\). We start with Bayes' Theorem, and apply the statement (1) inequality above

$$ \begin{aligned} P(H_1|E) & = \frac{P(E|H_1)P(H_1)}{P(E|H_1)P(H_1) + P(E|H_2)P(H_2)} \\ \frac{P(H_1|E)}{P(H_1)} & = \frac{P(E|H_1)}{P(E|H_1)P(H_1) + P(E|H_2)P(H_2)} >1 \\ \end{aligned} $$

Thus we have, defining the prior \(P(H_1)\equiv h\),

$$ \begin{aligned} P(E|H_1) &> P(E|H_1)P(H_1) + P(E|H_2)P(H_2) \\ P(E|H_1) &> P(E|H_1)\cdot h + P(E|H_2)\cdot (1-h) \\ P(E|H_1)\cdot (1-h) &> P(E|H_2)\cdot (1-h) \\ P(E|H_1) &> P(E|H_2) \end{aligned} $$

This last relationship is a direct consequence of Statement (1) above and is not always consistent with Statement (2). All we need is any alternative hypothesis which predicts the evidence as more likely under that hypothesis, even by a tiny amount. The evidence, \(E\), would then push the posterior of \(H_2\) up more than that for \(H_1\) and the posterior for \(H_1\) will go down given this evidence.

An easy example one can do is calculate the probability of holding one of two decks of cards, one which is typical (\(H_1\)) and the other has an extra ace (\(H_2\)). Regardless of the prior, drawing an ace (\(A\)) will raise the probability of the second deck (\(H_2\)) and reduce the probability of the first (\(H_1\)).

$$ \begin{aligned} P(H_1|A) &= \frac{\frac{4}{52} \cdot h}{\frac{4}{52} \cdot h + \frac{5}{52} \cdot (1-h)} \\ \frac{P(H_1|A)}{P(H_1)} &= \frac{\frac{4}{52} \cdot h/h}{\frac{4}{52} \cdot h + \frac{5}{52} \cdot (1-h)} & \frac{\frac{4}{52} }{\frac{4}{52} \cdot h + \frac{5}{52} \cdot (1-h)}\\ &=\frac{4}{4\cdot h+5\cdot (1-h)} \end{aligned} $$

This final term ranges from 4/5 up to 1 (where \(H_2\) is impossible), so if there is any non-zero prior for \(H_2\) the "evidence" of an ace reduces the probability of \(H_1\). You can see this even if there are 50 aces in the first deck, as long as the second one has more than that.

It's also the case that even with the simplistic case Alex presents, the scenario is not just one data point. His case is that under the alien abductions hypothesis you would predict certain government failures, so seeing government failures is evidence of alien abductions. However, you'd expect certain types of failures. There may be cases of failures you'd expect but don't actually observe. Personally, I don't think governments are great at keeping secrets, so the alien secret would likely get out in some form and we aren't seeing that. In the case of two data points, the analysis might go something like this, where \(a_1\) is the likelihood of \(A\) given \(H_1\), \(b_1\) is the likelihood of \(B\) given \(H_1\), \(h_1\) is the prior of \(H_1\), etc...

$$ \begin{aligned} \frac{P(H_1|A,B)}{P(H_1)} &\le 1 \text{ (for no evidence)} \\ &= \frac{\left(\frac{a_1 b_1h_1}{a_1 b_1h_1 + a_2 b_2(1-h_1)}\right)}{h_1}\\ a_1 b_1&\le a_1 b_1h_1+ a_2 b_2(1-h_1) \\ (a_1 b_1-a_2 b_2)\underbrace{(h_1-1)}_{\le 0} &\ge 0 \\ &\text{thus} \\ a_2 b_2 &\ge a_1 b_1 \end{aligned} $$

Consider two pieces of data. The first, \(A\), strongly supports the primary hypothesis (\(a_1 \sim 1\)) but is only moderately likely under the alternative (\(a_2 \sim 0.5\), like a coin flip). However, if the second piece of data, \(B\), is something you'd strongly expect under \(H_1\) but don't observe (\(b_1 \ll 1\)), then even though \(B\) is equally likely under the alternative (\(b_2 \sim 0.5\)), the combined data would still not be considered evidence for \(H_1\).

So Alex O'Connor's example would in fact not be evidence for alien abductions, not even weak evidence. Because the alternatives (e.g. many examples of normal people mucking up governments) end up scaling down the posterior probability for the alien abduction hypothesis.