"What is the expectation of embryonic death for rhythm method users? Our first assumption was that only half of the embryos are viable. I take it that this value holds for populations using no contraception and not distinguishing between HF and non-HF periods (or using contraceptive techniques that do not distinguish between HF and non-HF periods)."
Assuming the claim that about 50% of all embryos are viable, this much makes sense.
"What is not known is what proportion of embryos are conceived during the HF period as opposed to outside the HF period. Since it is reasonable to assume that only a minority of embryos are conceived outside the HF period, let us make a broad estimate that between 1/10 and 1/3 are so conceived."
Technically speaking, conception can only take place when both an ovum and a sperm are present. It cannot take place prior to ovulation. What he means by this sloppy shorthand is that either sperm deposited during the pre-HF period (possibly as old as 5 days) may fertilize an ovum or a sperm could fertilize and "old" ovum during the post-HF period.
"Then, by our third assumption—that is, that the chance of the viability is twice as high for an embryo conceived during the HF period as for an embryo conceived outside of the HF period, we can calculate that the chance of viability outside the HF period ranges roughly from one in four to one in three."
"[Footnote i.] By the probability calculus, the probability of viability (p) equals the conditional probability of viability given that the conception occurred outside the HF period (q) times the probability that the conception occurred outside the HF period (r) plus the conditional probability of viability given that the conception occurred during the HF period (2q) times the probability that the conception occurred during the HF period (1-r). Hence, p = qr + 2q(1-r). We set p = 1/2 by assumption one and let r ∈ [1/10, 1/3]. Hence q ∈ [10/38, 3/10] ≈ [1/4, 1/3].
Now we must ask if Bovens third assumption, about HF and fringe conceptions having different embryo survival rates, is right. According to this article ("Timing of conception and the risk of spontaneous abortion among pregnancies occurring during the use of natural family planning", AM J OBSTET GYNECOL 1995; 172:1567-72.) in the American Journal of Obstetric Gynecology, it may not be. (Fedora Tip: UnSpace) According to MedlinePlus, "It is estimated that up to 50% of all fertilized eggs die and are lost (aborted) spontaneously, usually before the woman knows she is pregnant. Among known pregnancies, the rate of spontaneous abortion is approximately 10% and usually occurs between the 7th and 12th weeks of pregnancy." So this study does not directly address Bovens’ claims. However, Bovens cites no research whatsoever.
The study found that "the spontaneous abortion rate was similar for 361 optimally timed conceptions (9.1%) and 507 non-optimally timed conceptions (10.9%)" and concluded, "These differences were not statistically significant (relative risk l.19, 95% confidence interval 0.79 to 1.80)." Thus until evidence supports the hypothesis that there is a statistically signficant difference, we can treat the relative risk as being effectively 1.0. However, women with a history of preganacy loss were more than twice as likely to miscarry when then conceived at non-optimal times, i.e. at the fringes of their fertile periods. Thankfully, as I have already mentioned, this unfortunate occurrence can be reduced in likelihood by modifying NFP’s rules. According to the study, if conception was optimally timed, women prone to spontanteous abortions had them about as often as women without such a history. It is reasonable to hypothesize that if intercourse timing does not, for most women, affect the rate that known pregnancies spontaneously abort, then it does not affect the rate of "silent" spontaneous abortions. This is called a testable hypothesis, Mr. Bovens, and it is the cornerstone of the scientific method. If anyone can point to research relating intercourse timing and pre-implantation embryo death, I’d much abliged.
The next section of Bovens’ article explains the mathematical consequences of Bovens’ assumptions. Since the most important assumption, that NFP leads to more spontaneous abortions than other methods of birth control (or even randomly-timed intercourse), has no support, I’ll skip that part. I will, however, point out that Bovens misrepresented some statements by Randy Alcorn.