Then you infer an estimate using Bayes' theorem.

Otherwise it is not private, as a reply has pointed out.

Saying "whatever you want" will incur a very large sampling error, especially as the population of those saying whatever increases.

What is needed is a notion of scalable privacy, where as the population of those saying "whatever" increases the privacy strength also increases yet the absolute error remains at worst constant.

https://arxiv.org/abs/1708.01884

Honest question, just too lazy to read the manuscript you link...

" the estimation error quickly increases with the population size due to the underlying truthful distribution distortion. For example, say we are interested in how many vehicles are at a popular stretch of the highway. Say we configure flip1 = 0.85 and flip2 = 0.3. We query 10,000 vehicles asking for their current location and only 100 vehicles are at the particular area we are interested in (i.e., 1% of the population truthfully responds “Yes"). The standard deviation due to the privacy noise will be 21 which is slightly tolerable. However, a query over one million vehicles (now only 0.01% of the population truthfully responds “Yes") will incur a standard deviation of 212. The estimate of the ground truth (100) will incur a large absolute error when the aggregated privatized responses are two or even three standard deviations (i.e., 95% or 99% of the time) away from the expected value, as the mechanism subtracts only the expected value of the noise."

"In this paper, our goal is to achieve the notion of scalable privacy. That is, as the population increases the privacy should strengthen. Additionally, the absolute error should remain at worst constant. For example, suppose we are interested in understanding a link between eating red meat and heart disease. We start by querying a small population of say 100 and ask “Do you eat red meat and have heart disease?". Suppose 85 truthfully respond “Yes". If we know that someone participated in this particular study, we can reasonably infer they eat red meat and have heart disease regardless the answer. Thus, it is difficult to maintain privacy when the majority of the population truthfully responds “Yes".

Querying a larger and diverse population would protect the data owners that eat red meat and have heart disease. Let’s say we query a population of 100,000 and it turns out that 99.9% of the population is vegetarian. In this case, the vegetarians blend with and provide privacy protection of the red meat eaters. However, we must be careful when performing estimation of a minority population to ensure the sampling error does not destroy the underlying estimate."

Thanks.

If the coin comes you heads you answer truthfully. If it comes up tails, you flip the coin again and answer if the yes if the coin is heads and no if the coin is tails. You can then no longer know if anybody is (or is not) a dog.

The probabilities can be adjusted to provide more or less privacy (while making the data less or more useful). For example, if you only answer truthfully 0.1% of the time it would be hard to know anything about anyone, at the cost of knowing the total number of dogs less precisely.

[1] https://arxiv.org/abs/1407.6981

I am not sure how any of that helps in a mass collection of data like OS telemetry.

Which text were you reading that lead you to this conclusion?

> It is likely that some attackers will aim to target specific users by isolating and analyzing reports from that user, or a small group of users that includes them. Even so, some randomly-chosen users need not fear such attacks at all...

For the limitation, the whole section 6.1 explains that this only protects a single question. If you collect more than single question, you must rely on other techniques to protect the privacy.

The text you've quoted is about how a random subset of the population is already immune to the issue of repeated queries, not that subsampling the population helps in any way. If you don't interrupt the quotation mid-sentence, it reads:

> Even so, some randomly-chosen users need not fear such attacks at all: with probability (1/2 f)^h, clients will generate a Permanent randomized response B with all 0s at the positions of set Bloom filter bits. Since these clients are not contributing any useful information to the collection process, targeting them individually by an attacker is counter-productive.

The whole of section 6.1 is not about how it only protects a single question, it is about how one ensures that the single-question protections generalize to larger surveys, concluding that

> This issue, however, can be mostly handled with careful collection design.

But my point is precisely that this technique helps with a single question. As soon as you are doing continuous mass collection you don't really get any privacy protection from this technique, and you have to rely on other techniques (encryption, etc).

But ... [please see reply to omarforgotpwd].

But, as someone mentioned, if the coin comes tails you should answer with another coin flip, not "yes".

https://en.wikipedia.org/wiki/Randomized_response