# Questions about stochastic time tracking **URL:** \ **Category:** Akrasia\ **Created:** [April 9, 2020, 12:45pm UTC](https://forum.beeminder.com/t/questions-about-stochastic-time-tracking/6559 "2020-04-09T12:45:35Z")\ **Posts on this page:** 20\ **Page:** 2
**Author:** ![zedmango](https://avatars.discourse-cdn.com/v4/letter/z/c67d28/32.png) [@zedmango](https://forum.beeminder.com/u/zedmango)\ **Post date:** [April 11, 2020, 11:04am UTC](https://forum.beeminder.com/t/questions-about-stochastic-time-tracking/6559/21 "2020-04-11T11:04:03Z")
What do you have against Poisson processes @narthur? ☠ 🇫🇷 🐟 ---
**Author:** ![zedmango](https://avatars.discourse-cdn.com/v4/letter/z/c67d28/32.png) [@zedmango](https://forum.beeminder.com/u/zedmango)\ **Post date:** [April 11, 2020, 11:06am UTC](https://forum.beeminder.com/t/questions-about-stochastic-time-tracking/6559/22 "2020-04-11T11:06:10Z")
With your latest proposal, say you’ve gone 80 minutes without a ping - you’re guaranteed one in the next 10 minutes, so working at that time is worth more than normal. With Poisson you avoid that problem, because it’s memoryless! ---
**Author:** ![dreev](https://sea1.discourse-cdn.com/flex019/user_avatar/forum.beeminder.com/dreev/32/4311_2.png) [@dreev](https://forum.beeminder.com/u/dreev)\ **Post date:** [April 11, 2020, 5:21pm UTC](https://forum.beeminder.com/t/questions-about-stochastic-time-tracking/6559/23 "2020-04-11T17:21:31Z")
@zedmango’s exactly right, as was your first instinct to implement it by having a 10% probability of a ping every 5 minutes (other than the small loophole where you can slack off in the 5 minute window following each ping). The [official implementation](https://forum.beeminder.com/t/official-reference-implementation-of-the-tagtime-universal-ping-schedule/4282) is equivalent to that same algorithm, just doing that literally every second instead of every 5 minutes. ---
**Author:** ![tomyum](https://avatars.discourse-cdn.com/v4/letter/t/cdc98d/32.png) [@tomyum](https://forum.beeminder.com/u/tomyum)\ **Post date:** [April 11, 2020, 8:44pm UTC](https://forum.beeminder.com/t/questions-about-stochastic-time-tracking/6559/24 "2020-04-11T20:44:30Z")
I think narthur’s idea can still work, but only if we properly weight the measurements in the final estimate. We sample the next ping interval uniformly between 0 minutes and 90 minutes. When determining the average rate for a label, we multiply each datapoint by (90-T)/45 where T is the number of minutes in that interval. (I think this is the correct weighting coefficient, but I might be wrong.) This leads to an unbiased estimate of the true time spent on each label, no matter how the user changes their behavior as a function of the elapsed time in the interval. Intuitively: True, if 89 minutes have elapsed, I can be sure that there will be a ping in one minute. However, this ping will count very little to the final estimate, so it’ I’m curious if (a) what I wrote is actually correct (i.e. can we really get unbiased estimates like that? And is it sufficient to ensure non-exploitability?) (b) would this actually lower the variance of the estimate? Or rather, in what situations would it do so? Effectively, we could still get something like a drought, if we get a bunch of intervals close to 90 minutes in a row, which all count very little. © Are there any smart ways to adaptively sample / importance-sample the intervals that both ensure unbiasedness / non-exploitability, but perhaps substantially reduce the variance compared to Poisson sampling? What do you guys think? ---
**Author:** ![zedmango](https://avatars.discourse-cdn.com/v4/letter/z/c67d28/32.png) [@zedmango](https://forum.beeminder.com/u/zedmango)\ **Post date:** [April 11, 2020, 11:06pm UTC](https://forum.beeminder.com/t/questions-about-stochastic-time-tracking/6559/25 "2020-04-11T23:06:13Z")
So one could be 100 times more efficient by only working in the 10 microseconds surrounding each possible sample time: like work 11:59:59.995 - 12:00:00.005, relax for 990 microseconds, then work 12:00:00.995 - 12:00:01.005, and so forth. Now we just have to find something that we can do in 0.99 seconds. ---
**Author:** ![zedmango](https://avatars.discourse-cdn.com/v4/letter/z/c67d28/32.png) [@zedmango](https://forum.beeminder.com/u/zedmango)\ **Post date:** [April 11, 2020, 11:10pm UTC](https://forum.beeminder.com/t/questions-about-stochastic-time-tracking/6559/26 "2020-04-11T23:10:19Z")
> [@tomyum](#): > > When determining the average rate for a label, we multiply each datapoint by (90-T)/45 where T is the number of minutes in that interval Using that coefficient, seems like you’d benefit from working extra hard right after a ping, since in the few minutes after a ping, the ping will be worth about twice as much. More generally, given any weighting coefficient that varies with T, you can exploit it by working harder at values of T where the coefficient is higher. It’s like buying more lottery tickets when the jackpot is higher. So you really need a constant weighting coefficient. ---
**Author:** ![narthur](https://sea1.discourse-cdn.com/flex019/user_avatar/forum.beeminder.com/narthur/32/5113_2.png) [@narthur](https://forum.beeminder.com/u/narthur)\ **Post date:** [April 11, 2020, 11:13pm UTC](https://forum.beeminder.com/t/questions-about-stochastic-time-tracking/6559/27 "2020-04-11T23:13:19Z")
It’s unfortunate because in my ideal world I could choose when to report, so I wouldn’t be interrupted when I was concentrating. But I really don’t think there’s any way to make that work in a way that couldn’t be gamed. ☹ Besides the fact that if I avoid reporting when I’m concentrating on work then that kind of prevents me from tracking work time… 😛 ---
**Author:** ![zedmango](https://avatars.discourse-cdn.com/v4/letter/z/c67d28/32.png) [@zedmango](https://forum.beeminder.com/u/zedmango)\ **Post date:** [April 11, 2020, 11:15pm UTC](https://forum.beeminder.com/t/questions-about-stochastic-time-tracking/6559/28 "2020-04-11T23:15:22Z")
Sure there is - just do it non-stochastically. For instance, work in pomodoros of 25 minutes, then report each one after it’s done. Or, start a stopwatch, work on something as long as you’re concentrated, then once you stop working on it, stop the stopwatch and report. ---
**Author:** ![narthur](https://sea1.discourse-cdn.com/flex019/user_avatar/forum.beeminder.com/narthur/32/5113_2.png) [@narthur](https://forum.beeminder.com/u/narthur)\ **Post date:** [April 11, 2020, 11:46pm UTC](https://forum.beeminder.com/t/questions-about-stochastic-time-tracking/6559/29 "2020-04-11T23:46:42Z")
Truth. I’m already doing pomodoros (was before I started messing with stochastic tracking). Unfortunately my work life got so crazy that a lot of my time is spent in activities for which I can’t commit focused blocks, like meetings. Hopefully that will calm down soon. So I may switch to something more conventional to complement my pomodoros–Toggl, perhaps. ---
**Author:** ![dreev](https://sea1.discourse-cdn.com/flex019/user_avatar/forum.beeminder.com/dreev/32/4311_2.png) [@dreev](https://forum.beeminder.com/u/dreev)\ **Post date:** [April 12, 2020, 12:37am UTC](https://forum.beeminder.com/t/questions-about-stochastic-time-tracking/6559/30 "2020-04-12T00:37:39Z")
> [@narthur](#): > > so I wouldn’t be interrupted when I was concentrating I’ve found that pings while working don’t feel like interruptions. They’re like the opposite, reinforcing that I’m doing what I meant to be doing. They’re not pulling my attention to something else. If that doesn’t work for you because the ping sound itself breaks your concentration, @zedmango’s idea could work. If you can guarantee that you’re totally focused during the entirety of a pomodoro then you can answer any pings from that pomodoro retroactively when it’s done. ---
**Author:** ![zedmango](https://avatars.discourse-cdn.com/v4/letter/z/c67d28/32.png) [@zedmango](https://forum.beeminder.com/u/zedmango)\ **Post date:** [April 12, 2020, 12:38am UTC](https://forum.beeminder.com/t/questions-about-stochastic-time-tracking/6559/31 "2020-04-12T00:38:40Z")
Ooh, or how bout this - keep a pen and paper time log, keep your pings on silent, and then go back and retro-respond to them at the end of each day? ---
**Author:** ![dreev](https://sea1.discourse-cdn.com/flex019/user_avatar/forum.beeminder.com/dreev/32/4311_2.png) [@dreev](https://forum.beeminder.com/u/dreev)\ **Post date:** [April 12, 2020, 12:42am UTC](https://forum.beeminder.com/t/questions-about-stochastic-time-tracking/6559/32 "2020-04-12T00:42:40Z")
> [@zedmango](#): > > keep your pings on silent, and then go back and retro-respond to them at the end of each day? I think that defeats the point of TagTime! If you could reliably do that then you’d have a perfect account of all your time and TagTime’s noisy estimates would be superfluous. ---
**Author:** ![zedmango](https://avatars.discourse-cdn.com/v4/letter/z/c67d28/32.png) [@zedmango](https://forum.beeminder.com/u/zedmango)\ **Post date:** [April 12, 2020, 12:50am UTC](https://forum.beeminder.com/t/questions-about-stochastic-time-tracking/6559/33 "2020-04-12T00:50:30Z")
Yeah, but I at least can’t do that reliably, so a combo of TagTime and manual timekeeping might give more accurate results. ---
**Author:** ![zedmango](https://avatars.discourse-cdn.com/v4/letter/z/c67d28/32.png) [@zedmango](https://forum.beeminder.com/u/zedmango)\ **Post date:** [April 12, 2020, 12:51am UTC](https://forum.beeminder.com/t/questions-about-stochastic-time-tracking/6559/34 "2020-04-12T00:51:43Z")
Or just put your phone on silent some of the time (like work hours) and go back and retrofill after - the rest of the time you have the pings audible. ---
**Author:** ![zedmango](https://avatars.discourse-cdn.com/v4/letter/z/c67d28/32.png) [@zedmango](https://forum.beeminder.com/u/zedmango)\ **Post date:** [April 12, 2020, 4:39am UTC](https://forum.beeminder.com/t/questions-about-stochastic-time-tracking/6559/35 "2020-04-12T04:39:37Z")
> [@dreev](#): > > I think @bee’s TagTime overhead time is much less since her ontology is simpler and she I don’t think she often refactors old tags or cleans up possible errors and stuff.) > > @bee’s solution is just to have a [tocks](http://blog.beeminder.com/tocks) goal that’s a little steeper than her tagtime-based hours goal. Can you or @bee elaborate on these points? What’s her tag-tology and how does the tocks goal work? ---
**Author:** ![tomyum](https://avatars.discourse-cdn.com/v4/letter/t/cdc98d/32.png) [@tomyum](https://forum.beeminder.com/u/tomyum)\ **Post date:** [April 12, 2020, 9:40am UTC](https://forum.beeminder.com/t/questions-about-stochastic-time-tracking/6559/36 "2020-04-12T09:40:49Z")
> [@zedmango](#): > > More generally, given any weighting coefficient that varies with T, you can exploit it by working harder at values of T where the coefficient is higher. It’s like buying more lottery tickets when the jackpot is higher. So you really need a constant weighting coefficient. That’s interesting. To be honest, I haven’t fully understood the issue, maybe you can help me. Maybe we use different definitions of “exploiting”? In the method I proposed, it is true that the pings right after a previous ping count more. However, at this time, the probability of the ping occurring in the next millisecond is as low as it gets. As you go progress in the interval, the probability of a ping in the next time instant increases while the weight decreases. These two effects can be set up to cancel each other out, in some sense. They ensure that no behavior policy can look at the time since the last ping in order to lead the system to systematically over- or underestimate the time spent working. Any “behavior policy” has a certain true rate of some activity of interest. This rate will be some number between 0 and 1 and specifies the fraction of time spent doing that activity. I claim that if you have a method that estimates this quantity without bias for any behavioral policy, i.e. `E[estimated_rate] == true_rate`, then it cannot be exploited. Would you agree with that? The fixed-ping schedule can be exploited because a policy that only executes the activity for an epsilon-interval around the measurement point will lead to a biased estimate: the true rate can be arbitrarily low while the measured rate is guaranteed to be 1. (And since there is no stochasticity, no re-weighting will help). Here, we have a measurement that is biased for some policies. The method proposed by narthur + taking the reweighting coefficients into account, should, however, be unbiased. Or is there any behavior policy that results in a skewed measurement? Maybe there are more sensible definitions of “exploitability” other than biasedness which I haven’t considered? ---
**Author:** ![zedmango](https://avatars.discourse-cdn.com/v4/letter/z/c67d28/32.png) [@zedmango](https://forum.beeminder.com/u/zedmango)\ **Post date:** [April 12, 2020, 11:23am UTC](https://forum.beeminder.com/t/questions-about-stochastic-time-tracking/6559/38 "2020-04-12T11:23:02Z")
Let p(T) be the probability of getting a ping where T is the time since the last ping. Let w(T) be the weight of the ping - that is, the score sent to Beeminder. Then if p(T) \* w(T) is not constant, you can exploit the algorithm by working more at values of T with higher p(T) \* w(T). With a Poisson distribution, p(T) is constant, and of course in the TagTime implementation w(T) is constant as well. With a non-Poisson distribution in which p varies only as a function of T, I think what you’re suggesting is to make the weight inversely proportional to p(T) - that is, make w(T) = k/p(T) so that p(T) \* w(T) is constant. While that would indeed prevent you from gaming the algorithm, it’s not clear to me whether your results would average out to accurately reflect the time you spent. ---
**Author:** ![zedmango](https://avatars.discourse-cdn.com/v4/letter/z/c67d28/32.png) [@zedmango](https://forum.beeminder.com/u/zedmango)\ **Post date:** [April 12, 2020, 11:30am UTC](https://forum.beeminder.com/t/questions-about-stochastic-time-tracking/6559/39 "2020-04-12T11:30:25Z")
Oh, ok, I see what you mean now. If you’re choosing the time between pings uniformly from, say, (0, 90), after T minutes the probability of getting a ping in the next minute should be 1 / (90 - T). So the weight should then be k(90 - T). That way p(T) \* w(T) is a constant and you can’t exploit the algorithm. ---
**Author:** ![tomyum](https://avatars.discourse-cdn.com/v4/letter/t/cdc98d/32.png) [@tomyum](https://forum.beeminder.com/u/tomyum)\ **Post date:** [April 12, 2020, 6:23pm UTC](https://forum.beeminder.com/t/questions-about-stochastic-time-tracking/6559/40 "2020-04-12T18:23:15Z")
> [@zedmango](#): > > With a Poisson distribution, p(T) is constant, and of course in the TagTime implementation w(T) is constant as well. With a non-Poisson distribution in which p varies only as a function of T, I think what you’re suggesting is to make the weight inversely proportional to p(T) - that is, make w(T) = k/p(T) so that p(T) \* w(T) is constant. Yeah I think that’s a good way of putting it! > [@zedmango](#): > > If you’re choosing the time between pings uniformly from, say, (0, 90), after T minutes the probability of getting a ping in the next minute should be 1 / (90 - T). So the weight should then be k(90 - T). Yup, that’s consistent with my calculation. > [@zedmango](#): > > While that would indeed prevent you from gaming the algorithm, it’s not clear to me whether your results would average out to accurately reflect the time you spent. I think it would indeed accurately reflect the time you spent. This is how I’d describe it, a bit informally: Let _U(t)_ be the user’s policy: it specifies the true average rate of engaging in some fixed activity as a function of the time _t_ since the last ping. The domain of _U_ is all nonnegative reals, and the codomain is the interval [0, 1]. The true rate of the activity is now _R_ = ∫_U(t)_ _p(t)_ d_t_ from _t=0_ to infinity where p(t) is the marginal distribution over times-since-last-ping that occur in the process. For narthur’s sampling rule, p(t) looks triangular with highest density at _t_=0 and zero density at _t_=90. For any measured point, its estimated rate _R\_hat_ is the observed user’s action, times a weight _W(t)_ where _t_ is the time since the last ping. If we take the expectation, we get E[_R\_hat_] = ∫ _U(t)_ _W_(_t_) _q(t)_ d_t_ where again _U_ is the policy, _W_ is the weighting function that we want to choose, and _q_ is the distribution over times-since-last-ping in the measured points. In narthur’s sampling rule, _q_ is a uniform distribution from t=0 to 90. In order to make E[_R\_hat_] = _R_, for all policies _U_, we have to make this integral equal to the integral for _R_ above. We achieve that by setting _W_(_t_) = _p(t)_ / _q(t)_, which results in the coefficient W(t) = (90-t)/45 that we determined earlier in the thread. (More generally, W(t) = 2 - 2 \* t / max\_duration). ---
**Author:** ![zedmango](https://avatars.discourse-cdn.com/v4/letter/z/c67d28/32.png) [@zedmango](https://forum.beeminder.com/u/zedmango)\ **Post date:** [April 12, 2020, 11:44pm UTC](https://forum.beeminder.com/t/questions-about-stochastic-time-tracking/6559/41 "2020-04-12T23:44:50Z")
I’ll look over that in more detail in a bit - but you seem to be assuming that the user’s behavior is purely a function of the time since the last ping. If a user acts differently depending on what happened earlier - say, more or less focused on days when there have previously been many short intervals - would that introduce a bias? Can you clarify for me how p(t) works? What’s a marginal distribution? [Previous page](https://forum.beeminder.com/t/questions-about-stochastic-time-tracking/6559.md?page=1) [Next page](https://forum.beeminder.com/t/questions-about-stochastic-time-tracking/6559.md?page=3)