The idea is that the coin before handling has some probability of heads, but each time it is picked up that probability could change (maybe it is bent by the statistician's mighty thumb). The slope gives the amount of change in this probability each flip: for example, a coin that starts fair and which has a slope of 0.01 has a probability of heads of 0.51 (0.50 + 0.01) on its first flip, 0.52 on its second, and so forth. If the act of flipping has absolutely no effect on the probability of heads, slope can be set to be zero, though using stats::dbinom() for this particular edge case should be faster.
dcoin_linear( nheads, nflips, preflip_prob = 0.5, slope = 0.1, log = FALSE, outside_bounds_is_NA = FALSE )
Number of heads
Total number of flips (heads and tails)
Probability of heads before the coin is handled
How much the probability changes each time the coin is flipped
If TRUE return log transformed probabilities.
If TRUE, if any probability of heads is outside the bounds of probability, the function returns NA. Otherwise, it sets the value to the nearer bound.
The likelihood of the data (or log likelihood if log=TRUE)
Of course, if all we have is the total number of heads and total number of flips, we do not know if it was HTT, THT, or TTH. For the particular case of a slope set to exactly zero the order does not matter, but in the general case it will. For example, if the probability of heads increases with each flip, HTT is less likely than TTH even though each has one heads out of three flips. The current code looks at all possibilities exhaustively, but more efficient ways to calculate this undoubtedly exist. Pull requests are welcome. It also means this may be slow as the number of flips increases.
For some slopes and preflip probabilities,the probabilities of heads on a given flip may be outside the 0 to 1 bounds. By default, if this happens the function returns NA. If outside_bounds_is_NA is FALSE, it moves the probabilities to the nearer bound.