Multiple importance sampling revisited: breaking the bounds
 Mateu Sbert^{1, 2}Email authorView ORCID ID profile,
 Vlastimil Havran^{3} and
 Laszlo SzirmayKalos^{4}
https://doi.org/10.1186/s1363401805312
© The Author(s) 2018
Received: 11 July 2017
Accepted: 23 January 2018
Published: 27 February 2018
Abstract
We revisit the multiple importance sampling (MIS) estimator and investigate the bound on the efficiency improvement over balance heuristic estimator with equal count of samples established in Veach’s thesis. We revise the proof for this and come to the conclusion that there is no such bound and henceforth it makes sense to look for new estimators that improve on balance heuristic estimator with equal count of samples. Next, we examine a recently introduced nonbalance heuristic MIS estimator that is provably better than balance heuristic with equal count of samples, and we improve it both in variance and efficiency. We then obtain an equally provably better onesample balance heuristic estimator, and finally, we introduce a heuristic for the count of samples that can be used when the individual techniques are biased. All in all, we present three new sampling strategies to improve on both variance and efficiency on the balance heuristic using nonequal count of samples.
Our scheme requires the previous knowledge of several quantities, but those can be obtained in an adaptive way. The results also show that by a careful examination of the variance and properties of the estimators, even better estimators could be discovered in the future. We present examples that support our theoretical findings.
Keywords
AMS Subject Classification
1 Introduction
The multiple importance sampling (MIS) estimator [1, 2], and in particular balance heuristic, which is equivalent to the Monte Carlo estimator with a mixture of probability density functions (pdfs), has been used for many years with a big success, being a reliable and robust estimator that allows an easy and straightforward combination of different sampling techniques. MIS with balance heuristic estimator has been almost exclusively used with equal count of samples for each technique, mainly following the recommendations based on certain quasioptimality theorems in Veach’s thesis. Specifically, these theorems stated that a balance heuristic estimator could not be worse than n times any other MIS estimator with equal count of samples where n is the number of combined methods. In papers addressing the variance of MIS, strategies were assumed to have equal number of samples [3, 4], and the combination of MIS with jittered sampling was studied in [5].
MIS is often used in rendering applications of computer graphics [6] where the reflected illumination is the product of the intensity in the illumination direction that can be represented by an environment map, and the probability of the reflection from a surface. As it is not feasible to sample with this product of two functions, sampling mimics either directions from where reflection is likely or directions of high intensity. The first approach is called light source sampling and the second approach is called BRDF sampling after the name of the bidirectional reflection distribution function (BRDF).
Lu et al. [7] propose an improvement to balance heuristic estimators for environment map illumination by using a Taylor’s second order approximation of the variance around the equal weights 1/2 to obtain the counts of samples according to the BRDF and the environment map, which is accurate only if the optimal sample numbers are not too far from equal sampling. Heuristics have also been used that prefer certain sampling methods based on the local properties, for example, BRDF sampling is advantageous on highly specular surfaces and light source sampling on diffuse surfaces. Pajot et al. proposed a framework, called representativity, to develop such heuristic [8]. Havran and Sbert [9, 10] introduce a heuristic for unbiased techniques based on assigning a count of samples proportional to the inverse of the variance of each technique that in some cases seemed to put into question the quasioptimality rules of equal sampling. More recently, based on the analysis of the variance of MIS estimators, Sbert et al. [11] discovered a MIS nonbalance heuristic estimator that is provably better than balance heuristic with equal count of samples. The introduction of costs into those schemes allowed an even bigger increase in efficiency. We improve on their work by clarifying the quasioptimality rules in Veach’s thesis by introducing new balance heuristic estimators that are provably better than balance heuristic with equal count of samples, and by introducing a new heuristic valid when the individual estimators are biased. We believe that our work can inspire new provably better estimators or at least sound heuristics for heuristically better estimators.
Putting our results in a wider perspective, we should consider adaptive variance reduction methods that learn some properties of the integrand, e.g., the location of its peak, and refine the sampling strategy onthefly by analyzing the samples that have been already generated. In case of mixing different pdfs with which sampling is straightforward, the weights of the individual methods are the target of the adaptation. Adaptive methods can be imagined as a sequence of Monte Carlo quadrature steps and parameter estimation steps. Care should be exercised to implement adaptive methods since making the sampling pdf dependent on the generated samples might make the estimator biased, which can be attacked by freezing the adaptation after certain samples letting the approach be consistent. Another challenge is the control of the adaptation with randomly sampled, i.e., noisy data. To handle this, adaptive MCMC approaches often choose an artificial but robust criterion, like the acceptance rate, instead of the variance of the estimation [12, 13]. In computer graphics, this technique has been used to control the mutation of MCMC methods [14, 15] and to distribute different numbers of samples among the different techniques [16].
Owen and Zhou [17] survey the principles of adaptively mixing different sampling strategies, including defensive importance sampling and the simultaneous application of control variates and importance sampling. They also investigate MIS and point out that there is a room for improvement. Douc et al. [18] investigated the population Monte Carlo method and derived sufficient convergence conditions for adaptive mixtures, and also iterative formulae for the optimal weights [19]. As population Monte Carlo explores the sampling space with a Markov stochastic process, it uses the information of only the current samples directly. Cornuet et al. [20] improve this in their adaptive multiple importance sampling algorithm and present optimal exploitation of previous samples, which automatically stabilizes the process. Marin et al. [21] argue that the Markov property is important to be able to prove the consistency of the estimator, and the stability of the process can be achieved by increasing the sample size in each iteration.
In their recent work, Martino, Elvira et al. proposed adaptive population importance sampling [22] and gradient adaptive population importance sampling [23] that adapt the mean and the covariance of proposal Gaussian pdfs, and reported significant improvements with respect to MIS that keeps the individual proposal pdfs fixed. We note that in computer graphics the not continuous integrands may pose problems to such gradientbased methods. Elvira et al. also examined a discrete weighting scheme [24] where only a subset of available sampling strategies is selected with the aim of reducing the computational complexity of the sampling process.
The rest of the paper is organized as follows: In Section 2, we review the basic formulae for the variance of MIS. In Section 3, we discuss Theorem 9.5 of Veach’s thesis and present a new proof for a slightly modified theorem. In Section 4, we present a multisample balance heuristic estimator that is provably better than multisample balance heuristic with equal count of samples. In Section 5, we obtain a provably better onesample balance heuristic estimator, and in Section 6, we justify a new heuristic for the count of samples that does not require the individual techniques be unbiased. In Section 7, we discuss the results, and in Section 8, summarize our conclusions. Some proofs are given in Appendices A and B, and numeric 1D examples are presented in Appendix C.
2 MIS variance analysis
Naming convention for the MIS estimators in this paper. We will drop the superindex 1 from primary estimators when not strictly necessary
F  Generic multisample MIS estimator 
\({\mathcal {F}}\)  Generic onesample MIS estimator 
\({\mathcal {F}}^{1}\)  Generic onesample MIS primary estimator 
\(\hat {F}\)  Generic multisample MIS balance heuristic 
estimator  
\(\hat {\mathcal {F}}\)  Generic onesample MIS balance heuristic 
estimator  
F_{eq}, \({\mathcal {F}}_{{\text {eq}}}\), \({\mathcal {F}}^{1}_{{\text {eq}}}\), \(\hat {F}_{{\text {eq}}}\), \(\hat {\mathcal {F}}_{{\text {eq}}}\)  Equal count of samples estimators 
2.1 General multisample MIS estimator
There are two different components of a MIS estimator that determine the variance of the estimates: weighting scheme w_{ i } and the number of samples N_{ i } allocated to different methods. Weighting functions may or may not depend on the number of samples. Additionally, the number of samples can be predetermined, called the multisample model, or selected also randomly, which leads to the onesample model.
2.2 Balance heuristic estimator
where α_{ k }=N_{ k }/N.
2.2.1 Balance heuristic with equal count of samples
2.3 A provably better nonbalance heuristic estimator
Sbert et al. showed [11] that taking N_{ i }∝σ_{i,eq} guarantees that we improve on balance heuristic with equal sampling by using estimator Eq. 19, up to the statistical error in estimating the σ_{i,eq} values.
3 Breaking the bounds for the relative acceleration
In this section, we investigate the bounds for the improvement on the variance of balance heuristic estimator with equal count of samples.
where F is any multiple importance sampling estimator using the same total number of samples N. Veach interpreted Theorem 9.5 as a proof of quasioptimality of balance heuristic with equal count of samples, saying “According to this result, changing the N_{ i } can improve the variance by at most a factor of n, plus a small additive term. In contrast, a poor choice of the w_{ i } can increase variance by an arbitrary amount. Thus, the sample allocation is not as important as choosing a good combination strategy.”
But this inequality is not valid when the weights w_{ i }(x) depend on the number of samples N_{ i }, see Appendix A for a proof. Just to give a single counter example, let us consider the case when zero variance estimator is possible by properly setting the number of samples, making V[F] zero, but the equal count of samples estimator will not have zero variance. However, we show that Theorem 9.5 can be generalized to such cases as well, but it requires the full reconsideration of the original proof.
The interpretation by Veach of Theorem 9.5 is based on the assumption that additive term μ^{2}(n−1)/N is small if the total number of samples, N, gets larger. However, denominator N is implicitly included in the other terms of Eq. 23 as well, thus the considered additive term is, in fact, not negligible. As a result, the selection of N_{ i } sample numbers or weights α_{ i } can make a significant difference in the variance, which is worth examining and opens possibilities to find better estimators.
3.1 Onesample balance heuristic
Theorem 1
Proof
□
When for all i, α_{ i }=1/n, Eq. 29 becomes an equality.Observing that α_{max}≤1, the following corollary is immediate.
Corollary 1
Equations 29 and 32 do not imply that the improvement with respect to the equal count of samples is limited by a factor of n, since μ can be large in comparison with the variances. So, it is worth trying to obtain α_{ i } values that can reduce the variance.
Finally, note that we do not state that Eq. 23 is invalid, but that its proof by Veach in his thesis [2] is wrong and also his interpretation is not correct. On the other hand, we have proved a formally identical relationship, Eq. 34, for the onesample MIS estimator instead of the multisample estimator.
4 A provably better balanceheuristic multisample estimator
In Section 1, we reviewed two strategies for the allocation of samples to individual methods, balance heuristic with equal count of samples (Section 2.2.1) and nonbalance heuristic with sample numbers proportional to the variance (Section 2.3). The variance of the latter is better than that of the equal count estimator. This section shows that we can do it even better if the sample allocation strategy of Section 2.3 is applied to balance heuristic.
Estimator \(\hat {F}\) is now the balance heuristic estimator defined in Eq. 12 with \(N_{i} \propto {\sigma _{i,{\text {eq}}}}/{\sqrt {c_{i}}}\).
4.1 Discussion
Thus, the relative difference between variances will be constant disregarding N. This is a similar situation to that encountered in Section 3.
5 A provably better onesample balance heuristic estimator
Observe that, as μ^{2} is a constant, minimizing \(V[{\mathcal {F}}^{1}]\) means minimizing the first term of the last equality in Eq. 41. When the weights {w_{ i }(x)} are independent of the coefficients {α_{ i }}, we get the following theorem:
Theorem 2
Proof
The proof fully given in Appendix B follows from applying the Lagrange multipliers method to the target function:\(\Lambda (\alpha _{i}, \lambda)=\sum _{i=1}^{n} \frac {1}{\alpha _{i}} M_{i}^{2}  \mu ^{2} + \lambda \left (\sum _{k=1}^{n} \alpha _{k} 1\right)\). □
using α_{ i }∝M_{i,eq} weights is guaranteed to have less variance than the onesample balance heuristic estimator with equal count of samples. If we take into account the costs, we have the following theorem:
Theorem 3
is more efficient than the onesample balance heuristic with equal count of sampling.
Theorem 4
The onesample balance heuristic estimator using the {α_{ i }} values in Eq. 51 is more efficient than the onesample balance heuristic estimator with equal count of sampling when the {α_{ i }} values are decreasing with costs {c_{ i }}.
When the variation of M_{i,eq} values is small relatively to the variation of c_{ i } values, then we can consider that \(\alpha _{i} \propto {M_{i,{\text {eq}}}}/{\sqrt {c_{i}}}\) values are decreasing with c_{ i }, and the efficiency of MIS estimator using \(\alpha _{i} \propto {M_{i,{\text {eq}}}}/{\sqrt {c_{i}}}\) would be better than taking α_{ i } = 1/n.
5.1 Discussion
One might ask why we are interested in the onesample estimator when its variance is always higher than its multisample version due to the additional randomization. In addition, as the costs C_{ T } are the same, as we use on average the same number of samples from each technique for both the onesample and the multisample estimators, we also have \(C_{T} V[\!{F}] \le C_{T} V[\!\mathcal {F}]\), this is, given a onesample estimator, its corresponding multisample counterpart will always be more efficient. However, there might be cases where only the onesample estimator is applicable, e.g., when we do not know a priori the number of samples or we need just a single sample.
6 A new heuristic
In [9, 10] the heuristic α_{ i }∝1/v_{ i } was presented, where \(v_{i}=\int \frac {f^{2}(x)}{p_{i}(x)} {\mathrm {d}}x \mu ^{2}\) is the variance of technique i. This heuristic is only valid when all techniques p_{ i }(x) are unbiased. We will present here a generalized heuristic that is valid even when the techniques are biased and based on the following theorems.
Theorem 5
where \(m_{k}^{2} = \int \frac {f^{2}(x)}{p_{k}(x)} {\mathrm {d}}x\) is the second moment of technique k.
Proof
Observe that we do not require the {p_{ k }} techniques be unbiased. □
which follows from that μ^{2} can be written as \(\sum _{k} \alpha _{k} \mu ^{2}\). Note that from \(V[\!\hat {F}] \le V[\!\hat {\mathcal {F}}]\), the bound in Eqs. 53 and 55 also holds for \(V[\!\hat {F}]\).
That is, whichever combination {α_{ k }} is taken, we cannot make it worse than just using the worst technique.
Let us use Theorem 5 to obtain a heuristic valid for biased techniques.
Theorem 6
Consider a function f(x) and n not necessarily unbiased Monte Carlo integration techniques, {p_{ i }(x)}, with second moments \(\left \{m_{i}^{2}\right \}\), and the sequence of n positive values {α_{ i }}, \(\sum _{i} \alpha _{i}=1\). The rearrangement that minimizes the bound in Eq. 53 happens when the {α_{ i }} are ordered inversely to \(\left \{m_{i}^{2}\right \}\).
Proof
As μ is a constant value only depending on f(x), the rearrangement that minimizes the bound in Eq. 53 is the one that minimizes \(\sum _{k=1}^{n} \alpha _{k} m_{k}^{2}\), and by virtue of the rearrangement inequality [25, page 261][26], it happens when the {α_{ i }} are ordered inversely to \(\left \{m_{i}^{2}\right \}\). □
Theorem 6 justifies the introduction of the new heuristic \(\alpha _{i} \propto {1}/{m_{i}^{2}}\). We can extend this heuristic to take the costs of the techniques into account, in the same way as in [9] and [10] by using \(\alpha _{i} \propto {1}/{\left (c_{i} m_{i}^{2}\right)}\).
7 Results
7.1 1D functions

As expected [11] multisample estimators are always better than the corresponding onesample estimators with the same distribution of samplesTable 2
Variances for multisample estimator for the four examples in Appendix C and the environment map example (× 10^{−3}) for balance heuristic using equal count of samples, for the count inversely proportional to the variances of independent estimators [9] and [10], for the provably better, nonbalance heuristic, estimator defined in [11], and for the three new estimators defined in this paper
Ex. 1
Ex. 2
Ex. 3
Ex. 4
Ex. EM
\(\alpha _{k} \propto \frac {1}{n}\)
29.16
4.91
10.68
28.14
4.63
24.11
4.55
2.02
330.85
1.47
α_{ k }∝σ_{k,eq} [11]
29.09
4.90
10.61
27.94
4.61
\(\alpha _{k} \propto \frac {1}{m^{2}_{k}}\)
26.55
4.77
0.37
809.28
3.16
α_{ k }∝σ_{k,eq}
29.90
4.99
9.48
6.61
4.36
α_{ k }∝M_{k,eq}
28.24
4.83
7.05
18.04
3.87
Table 3Variances for onesample balance heuristic for the four examples in Appendix C and the environment map example (× 10^{−3}) for balance heuristic using equal count of samples, for the count inversely proportional to the variances of independent estimators [9] and [10], and for the three new estimators defined in this paper
Ex. 1
Ex. 2
Ex. 3
Ex. 4
Ex. EM
\(\alpha _{k} \propto \frac {1}{n}\)
30.16
5.02
13.35
35.16
5.14
24.22
4.60
2.05
332.97
1.50
\(\alpha _{k} \propto \frac {1}{m^{2}_{k}}\)
27.04
4.85
0.39
863.50
3.46
α_{ k }∝σ_{k,eq}
31.03
5.10
11.67
57.38
4.84
α_{ k }∝M_{k,eq}
29.07
4.92
8.72
22.59
4.28
Table 4Variance times cost (i.e., inefficiency, the smaller the better) for multisample estimator for the four examples in Appendix C and the environment map example (× 10^{−3}) for balance heuristic using equal count of samples, for the count inversely proportional to the variances of independent estimators [9] and[10], for the provably better, nonbalance heuristic, estimator defined in [11], and for the three new estimators defined in this paper
Example 1
Example 2
Example 3
Example 4
Example EM
\(\alpha _{k} \propto \frac {1}{n}\)
102.26
17.24
37.47
98.68
13.41
40.41
9.28
4.03
300.12
1.31
\(\alpha _{k} \propto \frac {\sigma _{k,{\text {eq}}}}{\sqrt {c_{k}}}\) [11]
89.40
15.44
31.80
83.78
11.31
\(\alpha _{k} \propto \frac {1}{c_{k} m^{2}_{k}}\)
43.06
9.82
3.12
534.37
1.68
\(\alpha _{k} \propto \frac {\sigma _{k,{\text {eq}}}}{\sqrt {c_{k}}}\)
81.43
13.54
28.68
91.01
5.88
\(\alpha _{k} \propto \frac {M_{k,{\text {eq}}}}{\sqrt {c_{k}}}\)
79.73
13.08
25.74
31.77
4.92
Table 5Variance times cost (i.e., inefficiency, the smaller the better) for onesample estimator for the four examples in Appendix C and the environment map example (× 10^{−3}) for balance heuristic using equal count of samples, for the count inversely proportional to the variances of independent estimators [9] and[10], and for the three new estimators defined in this paper
Example 1
Example 2
Example 3
Example 4
Example EM
\(\alpha _{k} \propto \frac {1}{n}\)
105.78
17.60
46.82
123.30
14.90
40.89
9.37
4.04
300.48
1.32
\(\alpha _{k} \propto \frac {1}{c_{k} m^{2}_{k}}\)
44.43
9.93
3.18
545.53
2.30
\(\alpha _{k} \propto \frac {\sigma _{k,{\text {eq}}}}{\sqrt {c_{k}}}\)
83.82
13.27
33.36
105.80
6.39
\(\alpha _{k} \propto \frac {M_{k,{\text {eq}}}}{\sqrt {c_{k}}}\)
81.66
13.08
30.11
37.57
5.30
Table 6\(\alpha _{i}^{j}\): α_{ i } values for the jth 1D example and for the environment map example
Example 1
Example 2
Example 3
Example 4
Example EM
\(\alpha _{1}^{1}\)
\(\alpha _{2}^{1}\)
\(\alpha _{3}^{1}\)
\(\alpha _{1}^{2}\)
\(\alpha _{2}^{2}\)
\(\alpha _{3}^{2}\)
\(\alpha _{1}^{3}\)
\(\alpha _{2}^{3}\)
\(\alpha _{3}^{3}\)
\(\alpha _{1}^{4}\)
\(\alpha _{2}^{4}\)
\(\alpha _{3}^{4}\)
α _{1}
α _{2}
\(\alpha = \frac {1}{n}\)
0.33
0.33
0.33
0.33
0.33
0.33
0.33
0.33
0.33
0.33
0.33
0.33
0.5
0.5
0.42
0.48
0.10
0.35
0.21
0.44
0.90
0.10
0.00
0.95
0.04
0.00
0.93
0.07
0.79
0.14
0.06
0.68
0.07
0.25
0.98
0.02
0.00
0.99
0.01
0.00
0.99
0.01
α_{ k }∝σ_{k,eq}[11]
0.33
0.31
0.35
0.34
0.34
0.31
0.37
0.32
0.30
0.37
0.32
0.30
0.52
0.48
\(\alpha _{k} \propto \frac {\sigma _{k,{\text {eq}}}}{\sqrt {c_{k}}} \) [11]
0.51
0.19
0.29
0.52
0.21
0.26
0.55
0.19
0.25
0.55
0.19
0.25
0.70
0.30
\(\alpha _{k} \propto \frac {1}{m_{k}^{2}}\)
0.38
0.39
0.23
0.35
0.29
0.36
0.53
0.46
0.01
0.60
0.36
0.04
0.66
0.34
\(\alpha _{k} \propto \frac {1}{c_{k} m_{k}^{2}}\)
0.52
0.16
0.32
0.69
0.09
0.22
0.87
0.12
0.00
0.56
0.19
0.25
0.90
0.10
α_{ k }∝M_{k,eq}
0.33
0.35
0.30
0.32
0.31
0.35
0.34
0.36
0.28
0.33
0.32
0.34
0.57
0.43
\(\alpha _{k} \propto \frac {M_{k,{\text {eq}}}}{\sqrt {c_{k}}}\)
0.52
0.21
0.25
0.50
0.19
0.30
0.52
0.22
0.24
0.50
0.19
0.29
0.75
0.25

As expected the multisample estimators defined in [11] (Section 2.2.1) are better than balance heuristic with equal count of samples, both in variance and efficiency.

The new multisample estimators defined in Section 4 have similar variance values than balance heuristic with equal number of samples, but have better values for efficiency, both for the onesample and the multisample estimator.

The new multisample estimators defined in Section 4 provide similar variance as the ones in [11] but better efficiency, except in the fourth example where the efficiency is slightly worse.

The optimal distribution of samples obtained in Section 5 performs always better for both the onesample and the multisample estimator cases than the balance heuristic estimator with equal count of samples, in both variance and efficiency. This improvement is notable, especially when we consider that example 4 is particularly well suited for the equal count of sampling.

The multisample estimators using the optimal distribution of samples obtained in Section 5 perform better than the multisample estimators defined in [11], in both variance and efficiency.

The heuristically based estimators defined in [9] and in Section 6 give the best results for the first three examples, but are by far the worst for the fourth example.

The heuristically based estimators for unbiased techniques defined in [9] and [10] give better results than the one defined in Section 6 for possibly biased techniques, except for the third example, where the new heuristic performs much better.

We obtain in one case an acceleration of more than 30 times over the variance of the balance heuristic estimator with equal count of samples.
7.2 Adaptive results for 1D functions
7.3 2D analytical environment map example with LafortunePhong BRDF model
where ρ_{ d } is the diffuse albedo, ρ_{ s } is the specular albedo, m is the shininess, and θ is the angle between incident direction ω_{ i } and the surface normal.
and p_{2}(x)=x is the environment map.
We used typical values, m = 5, ρ_{ d } = ρ_{ s } = 0.5, and compared for equal cost and for the empirically obtained sampling costs, c_{1} = 1,c_{2} = 4.8. The detailed results are in the last columns of Tables 2, 3, 4, 5, and in the last two columns of Table 6. The results confirm the patterns found in our 1D examples.
8 Conclusions
This paper discussed an important topic of MIS estimators and the limited acceleration over equal count of samples balance heuristic. We have shown that in reality, this bound on acceleration does not hold, which justifies the search for better and more efficient estimators. We have obtained, both without and with taking into account the cost of sampling, new balance heuristic estimators that are provably better than balance heuristic with equal count of samples, and new heuristic estimators valid even when the independent techniques are biased. We have analyzed their behavior with 1D examples and with a 2D environment map example.
9 Appendix A: Inequality in Theorem 9.5 of Veach’s thesis does not hold for balance heuristic
as the last but two expression is clearly not the variance of a balance heuristic estimator with equal count of samples (Eq. 16). Inequality 61 thus holds only when the weights w_{ i }(x) do not depend on counts N_{ i }, and thus the proof of Theorem 9.5 is not valid in this case.
10 Appendix B: Section 5 proofs
We present the proof of Theorem 2. This proof is similar to the proof of Theorem 2 in [11].
Proof
and thus is positivedefinite. The variance function is then strictly convex in its convex domain \(\sum _{i=1}^{n} \alpha _{i} = 1\), where for all i, 0<α_{ i }<1, meaning that the critical point is unique and a minimum.
□
We present now the proof of Eq. 49.
Proof
and thus λ=C_{ T }μ^{2}. Substituting this value in Eq. 68 we find that the optimal sampling counts are the ones in Eq. 49. □
We present here the proof of Theorem 4.
Proof
Thus, we can guarantee that the general MIS estimator with weights w_{ i }(x) given by Eq. 14 and \(\alpha _{i} \propto {M_{i,eq}}/{\sqrt {c_{i}}}\) is more efficient than the onesample balance heuristic estimator with equal count of samples whenever the α_{ i } values are decreasing with c_{ i }. Considering now the onesample balance heuristic estimator with \(\alpha _{i} \propto {M_{i,{\text {eq}}}}/{\sqrt {c_{i}}}\), as the costs are the same and the variance is less than the general MIS estimator considered by virtue of Theorem 9.4 in Veach’s thesis, its efficiency is higher, and in turn, higher than the balance heuristic estimator with equal count of sampling. □
11 Appendix C: 1D examples
We compare the results for the functions and pdfs in Fig. 1 with the following estimators: equal count of samples, count inversely proportional to variances of independent techniques [9], the new heuristic defined in Section 6 of this paper, optimal count in [11], and the two new balance heuristic provably better estimators defined in Sections 4 and 5. These two new estimators are tried for both onesample and multisample.
11.1 Example 1
by MIS sampling on functions x, \(\left (x^{2}\frac {x}{\pi }\right)\), and sin(x), respectively. We first find the normalization constants: \(\int _{\frac {3}{2\pi }}^{\pi } x {\mathrm {d}}x = 4.82, \int _{\frac {3}{2\pi }}^{\pi } \left (x^{2}\frac {x}{\pi }\right) {\mathrm d}x = 8.76, \int _{\frac {3}{2\pi }}^{\pi } \sin (x) {\mathrm {d}}x=1.89\). We take into account the sampling costs given in [9], i.e., c_{1} = 1,c_{2} = 6.24,c_{3} = 3.28.
11.2 Example 2
using the same functions x, \(\left (x^{2}\frac {x}{\pi }\right)\), and sin(x) as before.
11.3 Example 3
using the same functions as before.
11.4 Example 4
In this case, we know the optimal (zero variance) α values: (0.3,0.3,0.4). This case should be most favorable to equal count of samples.
Declarations
Acknowledgements
The authors acknowledge the comments by anonymous reviewers that helped to improve a preliminary version of the paper.
Funding
The authors are funded in part by Czech Science Foundation research program GA1419213S by grant TIN201675866C33R from the Spanish Government and by grant OTKA K124124 and VKSZ14 PET/MRI 7T.
Authors’ contributions
Equal contribution from all authors. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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Authors’ Affiliations
References
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