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</head>

<body>




<h1 class="title toc-ignore">Efficient Computation of Ordinary and
Generalized Poisson Binomial Distributions</h1>


<div id="TOC">
<ul>
<li><a href="#introduction" id="toc-introduction">Introduction</a>
<ul>
<li><a href="#ordinary-poisson-binomial-distribution" id="toc-ordinary-poisson-binomial-distribution">Ordinary Poisson
Binomial Distribution</a></li>
<li><a href="#generalized-poisson-binomial-distribution" id="toc-generalized-poisson-binomial-distribution">Generalized Poisson
Binomial Distribution</a></li>
<li><a href="#existing-r-packages" id="toc-existing-r-packages">Existing
R Packages</a></li>
</ul></li>
<li><a href="#exact-procedures" id="toc-exact-procedures">Exact
Procedures</a>
<ul>
<li><a href="#ordinary-poisson-binomial-distribution-1" id="toc-ordinary-poisson-binomial-distribution-1">Ordinary Poisson
Binomial Distribution</a></li>
<li><a href="#generalized-poisson-binomial-distribution-1" id="toc-generalized-poisson-binomial-distribution-1">Generalized Poisson
Binomial Distribution</a></li>
<li><a href="#examples" id="toc-examples">Examples</a></li>
</ul></li>
<li><a href="#approximations" id="toc-approximations">Approximations</a>
<ul>
<li><a href="#ordinary-poisson-binomial-distribution-2" id="toc-ordinary-poisson-binomial-distribution-2">Ordinary Poisson
Binomial Distribution</a></li>
<li><a href="#generalized-poisson-binomial-distribution-2" id="toc-generalized-poisson-binomial-distribution-2">Generalized Poisson
Binomial Distribution</a></li>
<li><a href="#examples-1" id="toc-examples-1">Examples</a></li>
</ul></li>
<li><a href="#handling-special-cases-zeros-and-ones" id="toc-handling-special-cases-zeros-and-ones">Handling special cases,
zeros and ones</a>
<ul>
<li><a href="#ordinary-poisson-binomial-distributions" id="toc-ordinary-poisson-binomial-distributions">Ordinary Poisson
Binomial Distributions</a></li>
<li><a href="#generalized-poisson-binomial-distributions" id="toc-generalized-poisson-binomial-distributions">Generalized Poisson
Binomial Distributions</a></li>
</ul></li>
<li><a href="#usage-with-rcpp" id="toc-usage-with-rcpp">Usage with
Rcpp</a></li>
</ul>
</div>

<div id="introduction" class="section level2">
<h2>Introduction</h2>
<p>The Poisson binomial distribution (in the following abbreviated as
PBD) is becoming increasingly important, especially in the areas of
statistics, finance, insurance mathematics and quality management. This
package provides functions for two types of PBDs: ordinary and
generalized PBDs (henceforth referred to as O-PBDs and G-PBDs).</p>
<div id="ordinary-poisson-binomial-distribution" class="section level3">
<h3>Ordinary Poisson Binomial Distribution</h3>
<p>The O-PBD is the distribution of the sum of a number <span class="math inline">\(n\)</span> of independent Bernoulli-distributed
random indicators <span class="math inline">\(X_i \in \{0, 1\}\)</span>
<span class="math inline">\((i = 1, ..., n)\)</span>: <span class="math display">\[X := \sum_{i = 1}^{n}{X_i}.\]</span> Each of the
<span class="math inline">\(X_i\)</span> possesses a predefined
probability of success <span class="math inline">\(p_i := P(X_i =
1)\)</span> (subsequently <span class="math inline">\(P(X_i = 0) = 1 -
p_i =: q_i\)</span>). With this, mean, variance and skewness can be
expressed as <span class="math display">\[E(X) = \sum_{i = 1}^{n}{p_i}
\quad \quad Var(X) = \sum_{i = 1}^{n}{p_i q_i} \quad \quad Skew(X) =
\frac{\sum_{i = 1}^{n}{p_i q_i(q_i - p_i)}}{\sqrt{Var(X)}^3}.\]</span>
All possible observations are in <span class="math inline">\(\{0, ...,
n\}\)</span>.</p>
</div>
<div id="generalized-poisson-binomial-distribution" class="section level3">
<h3>Generalized Poisson Binomial Distribution</h3>
<p>The G-PBD is defined very similar. Again, it is the distribution of a
sum random variables, but here, each <span class="math inline">\(X_i \in
\{u_i, v_i\}\)</span> with <span class="math inline">\(P(X_i = u_i) =:
p_i\)</span> and <span class="math inline">\(P(X_i = v_i) = 1 - p_i =:
q_i\)</span>. Using ordinary Bernoulli-distributed random variables
<span class="math inline">\(Y_i\)</span>, <span class="math inline">\(X_i\)</span> can be expressed as <span class="math inline">\(X_i = u_i Y_i + v_i(1 - Y_i) = v_i + Y_i \cdot
(u_i - v_i)\)</span>. As a result, mean, variance and skewness are given
by <span class="math display">\[E(X) = \sum_{i = 1}^{n}{v_i} + \sum_{i =
1}^{n}{p_i (u_i - v_i)} \quad \quad Var(X) = \sum_{i = 1}^{n}{p_i
q_i(u_i - v_i)^2} \\Skew(X) = \frac{\sum_{i = 1}^{n}{p_i q_i(q_i -
p_i)(u_i - v_i)^3}}{\sqrt{Var(X)}^3}.\]</span> All possible observations
are in <span class="math inline">\(\{U, ..., V\}\)</span> with <span class="math inline">\(U := \sum_{i = 1}^{n}{\min\{u_i, v_i\}}\)</span>
and <span class="math inline">\(V := \sum_{i = 1}^{n}{\max\{u_i,
v_i\}}\)</span>. Note that the size <span class="math inline">\(m := V -
U\)</span> of the distribution does not generally equal <span class="math inline">\(n\)</span>!</p>
</div>
<div id="existing-r-packages" class="section level3">
<h3>Existing R Packages</h3>
<p>Computing these distributions exactly is computationally demanding,
but in the last few years, some efficient algorithms have been
developed. Particularly significant in this respect are the works of <a href="http://dx.doi.org/10.1016/j.csda.2012.10.006">Hong (2013)</a>, who
derived the DFT-CF procedure for O-PBDs, <a href="http://dx.doi.org/10.1016/j.csda.2018.01.007">Biscarri, Zhao &amp;
Brunner (2018)</a> who developed two immensely faster algorithms for
O-PBDs, namely the DC and DC-FFT procedures, and <a href="https://doi.org/10.1080/00949655.2018.1440294">Zhang, Hong and
Balakrishnan (2018)</a> who further developed <a href="http://dx.doi.org/10.1016/j.csda.2012.10.006">Hong’s (2013)</a>
DFT-CF algorithm for G-PBDs (in the following, this generalized
procedure is referred to as G-DFT-CF). Still, only a few R packages
exist for the calculation of either ordinary and generalized PBDs,
e.g. <a href="https://cran.r-project.org/package=poibin"><code>poibin</code></a>
and <a href="https://cran.r-project.org/package=poisbinom"><code>poisbinom</code></a>
for O-PBDs and <a href="https://cran.r-project.org/package=GPB"><code>GPB</code></a> for
G-PDBs. Before the release of this <code>PoissonBinomial</code> package,
there has been no R package that implemented the DC and DC-FFT
algorithms of <a href="http://dx.doi.org/10.1016/j.csda.2018.01.007">Biscarri, Zhao &amp;
Brunner (2018)</a>, as they only published a <a href="https://github.com/biscarri1/convpoibin">reference
implementation</a> for R, but refrained from releasing it as a package.
Additionally, there are no comparable approaches for G-PBDs to date.</p>
<p>The <code>poibin</code> package implements the DFT-CF algorithm along
with the exact recursive method of <a href="http://dx.doi.org/10.1109/TR.1984.5221843">Barlow &amp; Heidtmann
(1984)</a> and Normal and Poisson approximations. However, both exact
procedures of this package possess some disadvantages, i.e. they are
relatively slow at computing very large distributions, with the
recursive algorithm being also very memory consuming. The G-DFT-CF
procedure is implemented in the <code>GPB</code> package and inherits
this performance drawback. The <code>poisbinom</code> package provides a
more efficient and much faster DFT-CF implementation. The performance
improvement over the <code>poibin</code> package lies in the use of the
<a href="http://www.fftw.org">FFTW C library</a>. Unfortunately, it
sometimes yields some negative probabilities in the tail regions,
especially for large distributions. However, this numerical issue has
not been addressed to date. This <code>PoissonBinomial</code> also
utilizes FFTW for both DFT-CF and G-DFT-CF algorithms, but corrects that
issue. In addition to the disadvantages regarding computational speed
(<code>poibin</code> and <code>GPB</code>) or numerics
(<code>poisbinom</code>), especially for very large distributions, the
aforementioned packages do not provide headers for their internal C/C++
functions, so that they cannot be imported directly by C or C++ code of
other packages that use for example <code>Rcpp</code>.</p>
<p>In some situations, people might have to deal with Poisson binomial
distributions that include Bernoulli variables with <span class="math inline">\(p_i \in \{0, 1\}\)</span>. Calculation performance
can be further optimized by handling these indicators before the actual
computations. Approximations also benefit from this in terms of
accuracy. None of the aforementioned packages implements such
optimizations. Therefore, the advantages of this
<code>PoissonBinomial</code> package can be summarized as follows:</p>
<ul>
<li>Efficient computation of very large distributions with both exact
and approximate algorithms for O-PBDs and G-PBDs</li>
<li>Provides headers for the C++ functions so that other packages may
include them in their own C++ code</li>
<li>Handles (sometimes large numbers of) 0- and 1-probabilities to speed
up performance</li>
</ul>
<p>In total, this package includes 10 different algorithms of computing
ordinary Poisson binomial distributions, including optimized versions of
the Normal, Refined Normal and Poisson approaches, and 5 approaches for
generalized PBDs. In addition, the implementation of the exact recursive
procedure for O-PBDs was rewritten so that it is considerably less
memory intensive: the <code>poibin</code> implementation needs the
memory equivalent of <span class="math inline">\((n + 1)^2\)</span>
values of type <code>double</code>, while ours only needs <span class="math inline">\(3 \cdot (n + 1)\)</span>.</p>
<hr />
</div>
</div>
<div id="exact-procedures" class="section level2">
<h2>Exact Procedures</h2>
<div id="ordinary-poisson-binomial-distribution-1" class="section level3">
<h3>Ordinary Poisson Binomial Distribution</h3>
<p>In this package implements the following exact algorithms for
computing ordinary Poisson binomial distributions:</p>
<ul>
<li>the <em>Direct Convolution</em> approach of <a href="http://dx.doi.org/10.1016/j.csda.2018.01.007">Biscarri, Zhao &amp;
Brunner (2018)</a>,</li>
<li>the <em>Divide &amp; Conquer FFT Tree Convolution</em> procedure of
<a href="http://dx.doi.org/10.1016/j.csda.2018.01.007">Biscarri, Zhao
&amp; Brunner (2018)</a>,</li>
<li>the <em>Discrete Fourier Transformation of the Characteristic
Function</em> algorithm of <a href="http://dx.doi.org/10.1016/j.csda.2012.10.006">Hong (2013)</a>
and</li>
<li>the <em>Recursive Formula</em> of <a href="http://dx.doi.org/10.1109/TR.1984.5221843">Barlow &amp; Heidtmann
(1984)</a>.</li>
</ul>
</div>
<div id="generalized-poisson-binomial-distribution-1" class="section level3">
<h3>Generalized Poisson Binomial Distribution</h3>
<p>For generalized Poisson binomial distributions, this package
provides:</p>
<ul>
<li>a generalized adaptation of the <em>Direct Convolution</em> approach
of <a href="http://dx.doi.org/10.1016/j.csda.2018.01.007">Biscarri, Zhao
&amp; Brunner (2018)</a>,</li>
<li>a generalized <em>Divide &amp; Conquer FFT Tree Convolution</em>,
inspired by the respective procedure of <a href="http://dx.doi.org/10.1016/j.csda.2018.01.007">Biscarri, Zhao &amp;
Brunner (2018)</a> for O-PDBs,</li>
<li>the <em>Generalized Discrete Fourier Transformation of the
Characteristic Function</em> algorithm of <a href="https://doi.org/10.1080/00949655.2018.1440294">Zhang, Hong and
Balakrishnan (2018)</a>.</li>
</ul>
</div>
<div id="examples" class="section level3">
<h3>Examples</h3>
<p>Examples and performance comparisons of these procedures are
presented in a <a href="proc_exact.html">separate vignette</a>.</p>
<hr />
</div>
</div>
<div id="approximations" class="section level2">
<h2>Approximations</h2>
<div id="ordinary-poisson-binomial-distribution-2" class="section level3">
<h3>Ordinary Poisson Binomial Distribution</h3>
<p>In addition, the following O-PBD approximation methods are
included:</p>
<ul>
<li>the <em>Poisson Approximation</em> approach,</li>
<li>the <em>Arithmetic Mean Binomial Approximation</em> procedure,</li>
<li><em>Geometric Mean Binomial Approximation</em> algorithms,</li>
<li>the <em>Normal Approximation</em> and</li>
<li>the <em>Refined Normal Approximation</em>.</li>
</ul>
</div>
<div id="generalized-poisson-binomial-distribution-2" class="section level3">
<h3>Generalized Poisson Binomial Distribution</h3>
<p>For G-PBDs, there are</p>
<ul>
<li>the <em>Normal Approximation</em> and</li>
<li>the <em>Refined Normal Approximation</em>.</li>
</ul>
</div>
<div id="examples-1" class="section level3">
<h3>Examples</h3>
<p>Examples and performance comparisons of these approaches are provided
in a <a href="proc_approx.html">separate vignette</a> as well.</p>
<hr />
</div>
</div>
<div id="handling-special-cases-zeros-and-ones" class="section level2">
<h2>Handling special cases, zeros and ones</h2>
<p>Handling special cases, such as ordinary binomial distributions,
zeros and ones is useful to speed up performance. Unfortunately, some
approximations do not work well for Bernoulli trials with <span class="math inline">\(p_i \in \{0, 1\}\)</span>, e.g. the Geometric Mean
Binomial Approximations. This is why handling these values
<em>before</em> the actual computation of the distribution is not only a
performance tweak, but sometimes even a necessity. It is achieved by
some simple preliminary considerations.</p>
<div id="ordinary-poisson-binomial-distributions" class="section level3">
<h3>Ordinary Poisson Binomial Distributions</h3>
<ol style="list-style-type: decimal">
<li>All <span class="math inline">\(p_i = p\)</span> are equal?<br />
In this case, we have a usual binomial distribution. The specified
method of computation is then ignored. In particular, the following
applies:
<ol style="list-style-type: lower-alpha">
<li><span class="math inline">\(p = 0\)</span>: The only observable
value is <span class="math inline">\(0\)</span>, i.e. <span class="math inline">\(P(X = 0) = 1\)</span> and <span class="math inline">\(P(X \neq 0) = 0\)</span>.</li>
<li><span class="math inline">\(p = 1\)</span>: The only observable
value is <span class="math inline">\(n\)</span>, i.e. <span class="math inline">\(P(X = n) = 1\)</span> and <span class="math inline">\(P(X \neq n) = 0\)</span>.</li>
</ol></li>
<li>All <span class="math inline">\(p_i \in \{0, 1\} (i = 1, ...,
n)\)</span>?<br />
If one <span class="math inline">\(p_i\)</span> is 1, it is impossible
to measure 0 successes. Following the same logic, if two <span class="math inline">\(p_i\)</span> are 1, we cannot observe 0 and 1
successes and so on. In general, a number of <span class="math inline">\(n_1\)</span> values with <span class="math inline">\(p_i = 1\)</span> makes it impossible to measure
<span class="math inline">\(0, ..., n_1 - 1\)</span> successes.
Likewise, if there are <span class="math inline">\(n_0\)</span>
Bernoulli trials with <span class="math inline">\(p_i = 0\)</span>, we
cannot observe <span class="math inline">\(n - n_0 + 1, ..., n\)</span>
successes. If all <span class="math inline">\(p_i \in \{0, 1\}\)</span>,
it holds <span class="math inline">\(n = n_0 + n_1\)</span>. As a
result, the only observable value is <span class="math inline">\(n_1\)</span>, i.e. <span class="math inline">\(P(X
= n_1) = 1\)</span> and <span class="math inline">\(P(X \neq n_1) =
0\)</span>.</li>
<li>Are there <span class="math inline">\(p_i \notin \{0,
1\}\)</span>?<br />
Using the deductions from above, we can only observe an “inner”
distribution in the range of <span class="math inline">\(n_1, n_1 + 1,
..., n - n_0\)</span>, i.e. <span class="math inline">\(P(X \in \{n_1,
..., n - n_0\}) &gt; 0\)</span> and <span class="math inline">\(P(X &lt;
n_1) = P(X &gt; n - n_0) = 0\)</span>. As a result, <span class="math inline">\(X\)</span> can be expressed as <span class="math inline">\(X = n_1 + Y\)</span> with <span class="math inline">\(Y \sim PBin(\{p_i|0 &lt; p_i &lt; 1\})\)</span>
and <span class="math inline">\(|\{p_i|0 &lt; p_i &lt; 1\}| = n - n_0 -
n_1\)</span>. Subsequently, the Poisson binomial distribution must only
be computed for <span class="math inline">\(Y\)</span>. Especially, if
there is only one <span class="math inline">\(p_i \notin \{0,
1\}\)</span>, <span class="math inline">\(Y\)</span> follows a Bernoulli
distribution with parameter <span class="math inline">\(p_i\)</span>,
i.e. <span class="math inline">\(P(X = n_1) = P(Y = 0) = 1 -
p_i\)</span> and <span class="math inline">\(P(X = n_1 + 1) = P(Y = 1) =
p_i\)</span>.</li>
</ol>
<p>These cases are illustrated in the following example:</p>
<div class="sourceCode" id="cb1"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb1-1"><a href="#cb1-1" tabindex="-1"></a><span class="co"># Case 1</span></span>
<span id="cb1-2"><a href="#cb1-2" tabindex="-1"></a><span class="fu">dpbinom</span>(<span class="cn">NULL</span>, <span class="fu">rep</span>(<span class="fl">0.3</span>, <span class="dv">7</span>))</span>
<span id="cb1-3"><a href="#cb1-3" tabindex="-1"></a><span class="co">#&gt; [1] 0.0823543 0.2470629 0.3176523 0.2268945 0.0972405 0.0250047 0.0035721</span></span>
<span id="cb1-4"><a href="#cb1-4" tabindex="-1"></a><span class="co">#&gt; [8] 0.0002187</span></span>
<span id="cb1-5"><a href="#cb1-5" tabindex="-1"></a><span class="fu">dbinom</span>(<span class="dv">0</span><span class="sc">:</span><span class="dv">7</span>, <span class="dv">7</span>, <span class="fl">0.3</span>) <span class="co"># equal results</span></span>
<span id="cb1-6"><a href="#cb1-6" tabindex="-1"></a><span class="co">#&gt; [1] 0.0823543 0.2470629 0.3176523 0.2268945 0.0972405 0.0250047 0.0035721</span></span>
<span id="cb1-7"><a href="#cb1-7" tabindex="-1"></a><span class="co">#&gt; [8] 0.0002187</span></span>
<span id="cb1-8"><a href="#cb1-8" tabindex="-1"></a></span>
<span id="cb1-9"><a href="#cb1-9" tabindex="-1"></a><span class="fu">dpbinom</span>(<span class="cn">NULL</span>, <span class="fu">c</span>(<span class="dv">0</span>, <span class="dv">0</span>, <span class="dv">0</span>, <span class="dv">0</span>, <span class="dv">0</span>, <span class="dv">0</span>, <span class="dv">0</span>)) <span class="co"># only 0 is observable</span></span>
<span id="cb1-10"><a href="#cb1-10" tabindex="-1"></a><span class="co">#&gt; [1] 1 0 0 0 0 0 0 0</span></span>
<span id="cb1-11"><a href="#cb1-11" tabindex="-1"></a><span class="fu">dpbinom</span>(<span class="dv">0</span>, <span class="fu">c</span>(<span class="dv">0</span>, <span class="dv">0</span>, <span class="dv">0</span>, <span class="dv">0</span>, <span class="dv">0</span>, <span class="dv">0</span>, <span class="dv">0</span>)) <span class="co"># confirmation</span></span>
<span id="cb1-12"><a href="#cb1-12" tabindex="-1"></a><span class="co">#&gt; [1] 1</span></span>
<span id="cb1-13"><a href="#cb1-13" tabindex="-1"></a></span>
<span id="cb1-14"><a href="#cb1-14" tabindex="-1"></a><span class="fu">dpbinom</span>(<span class="cn">NULL</span>, <span class="fu">c</span>(<span class="dv">1</span>, <span class="dv">1</span>, <span class="dv">1</span>, <span class="dv">1</span>, <span class="dv">1</span>, <span class="dv">1</span>, <span class="dv">1</span>)) <span class="co"># only 7 is observable</span></span>
<span id="cb1-15"><a href="#cb1-15" tabindex="-1"></a><span class="co">#&gt; [1] 0 0 0 0 0 0 0 1</span></span>
<span id="cb1-16"><a href="#cb1-16" tabindex="-1"></a><span class="fu">dpbinom</span>(<span class="dv">7</span>, <span class="fu">c</span>(<span class="dv">1</span>, <span class="dv">1</span>, <span class="dv">1</span>, <span class="dv">1</span>, <span class="dv">1</span>, <span class="dv">1</span>, <span class="dv">1</span>)) <span class="co"># confirmation</span></span>
<span id="cb1-17"><a href="#cb1-17" tabindex="-1"></a><span class="co">#&gt; [1] 1</span></span>
<span id="cb1-18"><a href="#cb1-18" tabindex="-1"></a></span>
<span id="cb1-19"><a href="#cb1-19" tabindex="-1"></a><span class="co"># Case 2</span></span>
<span id="cb1-20"><a href="#cb1-20" tabindex="-1"></a><span class="fu">dpbinom</span>(<span class="cn">NULL</span>, <span class="fu">c</span>(<span class="dv">0</span>, <span class="dv">0</span>, <span class="dv">0</span>, <span class="dv">0</span>, <span class="dv">1</span>, <span class="dv">1</span>, <span class="dv">1</span>)) <span class="co"># only 3 is observable</span></span>
<span id="cb1-21"><a href="#cb1-21" tabindex="-1"></a><span class="co">#&gt; [1] 0 0 0 1 0 0 0 0</span></span>
<span id="cb1-22"><a href="#cb1-22" tabindex="-1"></a><span class="fu">dpbinom</span>(<span class="dv">3</span>, <span class="fu">c</span>(<span class="dv">0</span>, <span class="dv">0</span>, <span class="dv">0</span>, <span class="dv">0</span>, <span class="dv">1</span>, <span class="dv">1</span>, <span class="dv">1</span>)) <span class="co"># confirmation</span></span>
<span id="cb1-23"><a href="#cb1-23" tabindex="-1"></a><span class="co">#&gt; [1] 1</span></span>
<span id="cb1-24"><a href="#cb1-24" tabindex="-1"></a></span>
<span id="cb1-25"><a href="#cb1-25" tabindex="-1"></a><span class="co"># Case 3</span></span>
<span id="cb1-26"><a href="#cb1-26" tabindex="-1"></a><span class="fu">dpbinom</span>(<span class="cn">NULL</span>, <span class="fu">c</span>(<span class="dv">0</span>, <span class="dv">0</span>, <span class="fl">0.1</span>, <span class="fl">0.2</span>, <span class="fl">0.4</span>, <span class="fl">0.8</span>, <span class="dv">1</span>)) <span class="co"># only 1-5 are observable</span></span>
<span id="cb1-27"><a href="#cb1-27" tabindex="-1"></a><span class="co">#&gt; [1] 0.0000 0.0864 0.4344 0.3784 0.0944 0.0064 0.0000 0.0000</span></span>
<span id="cb1-28"><a href="#cb1-28" tabindex="-1"></a><span class="fu">dpbinom</span>(<span class="dv">1</span><span class="sc">:</span><span class="dv">5</span>, <span class="fu">c</span>(<span class="dv">0</span>, <span class="dv">0</span>, <span class="fl">0.1</span>, <span class="fl">0.2</span>, <span class="fl">0.4</span>, <span class="fl">0.8</span>, <span class="dv">1</span>)) <span class="co"># confirmation</span></span>
<span id="cb1-29"><a href="#cb1-29" tabindex="-1"></a><span class="co">#&gt; [1] 0.0864 0.4344 0.3784 0.0944 0.0064</span></span>
<span id="cb1-30"><a href="#cb1-30" tabindex="-1"></a></span>
<span id="cb1-31"><a href="#cb1-31" tabindex="-1"></a><span class="fu">dpbinom</span>(<span class="cn">NULL</span>, <span class="fu">c</span>(<span class="dv">0</span>, <span class="dv">0</span>, <span class="fl">0.4</span>, <span class="dv">1</span>)) <span class="co"># only 1 and 2 are observable</span></span>
<span id="cb1-32"><a href="#cb1-32" tabindex="-1"></a><span class="co">#&gt; [1] 0.0 0.6 0.4 0.0 0.0</span></span>
<span id="cb1-33"><a href="#cb1-33" tabindex="-1"></a><span class="fu">dpbinom</span>(<span class="dv">1</span><span class="sc">:</span><span class="dv">2</span>, <span class="fu">c</span>(<span class="dv">0</span>, <span class="dv">0</span>, <span class="fl">0.4</span>, <span class="dv">1</span>)) <span class="co"># confirmation</span></span>
<span id="cb1-34"><a href="#cb1-34" tabindex="-1"></a><span class="co">#&gt; [1] 0.6 0.4</span></span></code></pre></div>
</div>
<div id="generalized-poisson-binomial-distributions" class="section level3">
<h3>Generalized Poisson Binomial Distributions</h3>
<ol style="list-style-type: decimal">
<li>All <span class="math inline">\(u_i \in \{0, 1\}\)</span> and all
<span class="math inline">\(v_i = 1 - u_i\)</span>?<br />
Then, it is an ordinary Poisson binomial distribution with parameters
<span class="math inline">\(p_i&#39; = p_i\)</span> for all <span class="math inline">\(i\)</span> for which <span class="math inline">\(u_i = 1\)</span> and <span class="math inline">\(p_i&#39; = 1 - p_i\)</span> otherwise. This
includes all the special cases described above.</li>
<li>All <span class="math inline">\(u_i = u\)</span> are equal and all
<span class="math inline">\(v_i = v\)</span> are equal?<br />
In this case, we have a linearly transformed ordinary Poisson binomial
distribution, i.e. <span class="math inline">\(X\)</span> can be
expressed as <span class="math inline">\(X = uY + v(n - Y)\)</span> with
<span class="math inline">\(Y \sim PBin(p_1, ..., p_n)\)</span>. In
particular, if all <span class="math inline">\(p_i = p\)</span> are also
the same, we have a linear transformation of the usual binomial
distribution, i.e. <span class="math inline">\(X = uZ + v(n -
Z)\)</span> with <span class="math inline">\(Z \sim Bin(n, p)\)</span>.
Summarizing this, the following applies:
<ol style="list-style-type: lower-alpha">
<li>All <span class="math inline">\(p_i = 0\)</span>: The only
observable value is <span class="math inline">\(n \cdot v\)</span>,
i.e. <span class="math inline">\(P(X = n \cdot v) = 1\)</span> and <span class="math inline">\(P(X \neq n \cdot v) = 0\)</span>.</li>
<li>All <span class="math inline">\(p_i = 1\)</span>: The only
observable value is <span class="math inline">\(n \cdot u\)</span>,
i.e. <span class="math inline">\(P(X = n \cdot u) = 1\)</span> and <span class="math inline">\(P(X \neq n \cdot u) = 0\)</span>.</li>
<li>All <span class="math inline">\(p_i = p\)</span>: Observable values
are in <span class="math inline">\(\{u \cdot k + v \cdot (n - k) | k =
0, ..., n\}\)</span> and <span class="math inline">\(P(X = u \cdot k + v
\cdot (n - k)) = P(Z = k)\)</span>.</li>
<li>Otherwise: Observable values are in <span class="math inline">\(\{u
\cdot k + v \cdot (n - k) | k = 0, ..., n\})\)</span> and <span class="math inline">\(P(X = u \cdot k + v(n - k)) = P(Y =
k)\)</span></li>
</ol></li>
<li>All <span class="math inline">\(p_i \in \{0, 1\}\)</span>?<br />
Let <span class="math inline">\(I = \{i\, |\, p_i = 1\} \subseteq \{1,
..., n\}\)</span> and <span class="math inline">\(J = \{i\, |\, p_i =
0\} \subseteq \{1, ..., n\}\)</span>. Then, we have:
<ol style="list-style-type: lower-alpha">
<li>All <span class="math inline">\(p_i = 0\)</span>: The only
observable value is <span class="math inline">\(v^* := \sum_{i =
1}^{n}{v_i}\)</span>, i.e. <span class="math inline">\(P(X = v^*) =
1\)</span> and <span class="math inline">\(P(X \neq v^*) =
0\)</span>.</li>
<li>All <span class="math inline">\(p_i = 1\)</span>: The only
observable value is <span class="math inline">\(u^* := \sum_{i =
1}^{n}{u_i}\)</span>, i.e. <span class="math inline">\(P(X = u^*) =
1\)</span> and <span class="math inline">\(P(X \neq u^*) =
0\)</span>.</li>
<li>Otherwise, The only observable value is <span class="math inline">\(w^* := \sum_{i \in I}{u_i} + \sum_{i \in
J}{v_i}\)</span>, i.e. <span class="math inline">\(P(X = w^*) =
1\)</span> and <span class="math inline">\(P(X \neq w^*) = 0\)</span>.
Note that the case that any <span class="math inline">\(u_i =
v_i\)</span> is equivalent to <span class="math inline">\(p_i =
1\)</span>, because the corresponding random variable <span class="math inline">\(X_i\)</span> has always the same (non-random)
value.<br />
</li>
</ol></li>
<li>Are there <span class="math inline">\(p_i \notin \{0,
1\}\)</span>?<br />
Let <span class="math inline">\(I\)</span>, <span class="math inline">\(J\)</span> and <span class="math inline">\(w^*\)</span> as above and <span class="math inline">\(K = \{i\, |\, p_i &gt; 0 \, \wedge p_i &lt; 1\}
\subseteq \{1, ..., n\}\)</span>. Then, <span class="math inline">\(X\)</span> can be expressed as <span class="math inline">\(X = w^* + Z\)</span> with <span class="math inline">\(Z = \sum_{i \in K}{X_i}\)</span> following a
(reduced) generalized Poisson Bernoulli distribution. In particular, if
only one <span class="math inline">\(p_i \notin \{0, 1\}\)</span>, Z
follows a linearly transformed Bernoulli distribution.</li>
</ol>
<p>These cases are illustrated in the following example:</p>
<div class="sourceCode" id="cb2"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb2-1"><a href="#cb2-1" tabindex="-1"></a><span class="fu">set.seed</span>(<span class="dv">1</span>)</span>
<span id="cb2-2"><a href="#cb2-2" tabindex="-1"></a>pp <span class="ot">&lt;-</span> <span class="fu">runif</span>(<span class="dv">7</span>)</span>
<span id="cb2-3"><a href="#cb2-3" tabindex="-1"></a>va <span class="ot">&lt;-</span> <span class="fu">sample</span>(<span class="dv">0</span><span class="sc">:</span><span class="dv">6</span>, <span class="dv">7</span>, <span class="cn">TRUE</span>)</span>
<span id="cb2-4"><a href="#cb2-4" tabindex="-1"></a>vb <span class="ot">&lt;-</span> <span class="fu">sample</span>(<span class="dv">0</span><span class="sc">:</span><span class="dv">6</span>, <span class="dv">7</span>, <span class="cn">TRUE</span>)</span>
<span id="cb2-5"><a href="#cb2-5" tabindex="-1"></a></span>
<span id="cb2-6"><a href="#cb2-6" tabindex="-1"></a><span class="co"># Case 1</span></span>
<span id="cb2-7"><a href="#cb2-7" tabindex="-1"></a><span class="fu">dgpbinom</span>(<span class="cn">NULL</span>, pp, <span class="fu">rep</span>(<span class="dv">1</span>, <span class="dv">7</span>), <span class="fu">rep</span>(<span class="dv">0</span>, <span class="dv">7</span>))</span>
<span id="cb2-8"><a href="#cb2-8" tabindex="-1"></a><span class="co">#&gt; [1] 8.114776e-05 3.112722e-03 4.063146e-02 2.115237e-01 3.793308e-01</span></span>
<span id="cb2-9"><a href="#cb2-9" tabindex="-1"></a><span class="co">#&gt; [6] 2.735489e-01 8.297278e-02 8.798512e-03</span></span>
<span id="cb2-10"><a href="#cb2-10" tabindex="-1"></a><span class="fu">dpbinom</span>(<span class="cn">NULL</span>, pp) <span class="co"># equal results</span></span>
<span id="cb2-11"><a href="#cb2-11" tabindex="-1"></a><span class="co">#&gt; [1] 8.114776e-05 3.112722e-03 4.063146e-02 2.115237e-01 3.793308e-01</span></span>
<span id="cb2-12"><a href="#cb2-12" tabindex="-1"></a><span class="co">#&gt; [6] 2.735489e-01 8.297278e-02 8.798512e-03</span></span>
<span id="cb2-13"><a href="#cb2-13" tabindex="-1"></a></span>
<span id="cb2-14"><a href="#cb2-14" tabindex="-1"></a><span class="fu">dgpbinom</span>(<span class="cn">NULL</span>, pp, <span class="fu">rep</span>(<span class="dv">0</span>, <span class="dv">7</span>), <span class="fu">rep</span>(<span class="dv">1</span>, <span class="dv">7</span>))</span>
<span id="cb2-15"><a href="#cb2-15" tabindex="-1"></a><span class="co">#&gt; [1] 8.798512e-03 8.297278e-02 2.735489e-01 3.793308e-01 2.115237e-01</span></span>
<span id="cb2-16"><a href="#cb2-16" tabindex="-1"></a><span class="co">#&gt; [6] 4.063146e-02 3.112722e-03 8.114776e-05</span></span>
<span id="cb2-17"><a href="#cb2-17" tabindex="-1"></a><span class="fu">dpbinom</span>(<span class="cn">NULL</span>, <span class="dv">1</span> <span class="sc">-</span> pp) <span class="co"># equal results</span></span>
<span id="cb2-18"><a href="#cb2-18" tabindex="-1"></a><span class="co">#&gt; [1] 8.798512e-03 8.297278e-02 2.735489e-01 3.793308e-01 2.115237e-01</span></span>
<span id="cb2-19"><a href="#cb2-19" tabindex="-1"></a><span class="co">#&gt; [6] 4.063146e-02 3.112722e-03 8.114776e-05</span></span>
<span id="cb2-20"><a href="#cb2-20" tabindex="-1"></a></span>
<span id="cb2-21"><a href="#cb2-21" tabindex="-1"></a><span class="fu">dgpbinom</span>(<span class="cn">NULL</span>, pp, <span class="fu">c</span>(<span class="fu">rep</span>(<span class="dv">1</span>, <span class="dv">3</span>), <span class="fu">rep</span>(<span class="dv">0</span>, <span class="dv">4</span>)), <span class="fu">c</span>(<span class="fu">rep</span>(<span class="dv">0</span>, <span class="dv">3</span>), <span class="fu">rep</span>(<span class="dv">1</span>, <span class="dv">4</span>)))</span>
<span id="cb2-22"><a href="#cb2-22" tabindex="-1"></a><span class="co">#&gt; [1] 3.062225e-02 1.998504e-01 3.769239e-01 2.828424e-01 9.450797e-02</span></span>
<span id="cb2-23"><a href="#cb2-23" tabindex="-1"></a><span class="co">#&gt; [6] 1.426764e-02 9.620692e-04 2.331571e-05</span></span>
<span id="cb2-24"><a href="#cb2-24" tabindex="-1"></a><span class="fu">dpbinom</span>(<span class="cn">NULL</span>, <span class="fu">c</span>(pp[<span class="dv">1</span><span class="sc">:</span><span class="dv">3</span>], <span class="dv">1</span> <span class="sc">-</span> pp[<span class="dv">4</span><span class="sc">:</span><span class="dv">7</span>])) <span class="co"># reorder for 0 and 1; equal results</span></span>
<span id="cb2-25"><a href="#cb2-25" tabindex="-1"></a><span class="co">#&gt; [1] 3.062225e-02 1.998504e-01 3.769239e-01 2.828424e-01 9.450797e-02</span></span>
<span id="cb2-26"><a href="#cb2-26" tabindex="-1"></a><span class="co">#&gt; [6] 1.426764e-02 9.620692e-04 2.331571e-05</span></span>
<span id="cb2-27"><a href="#cb2-27" tabindex="-1"></a></span>
<span id="cb2-28"><a href="#cb2-28" tabindex="-1"></a><span class="co"># Case 2 a)</span></span>
<span id="cb2-29"><a href="#cb2-29" tabindex="-1"></a><span class="fu">dgpbinom</span>(<span class="cn">NULL</span>, <span class="fu">rep</span>(<span class="dv">0</span>, <span class="dv">7</span>), <span class="fu">rep</span>(<span class="dv">4</span>, <span class="dv">7</span>), <span class="fu">rep</span>(<span class="dv">2</span>, <span class="dv">7</span>)) <span class="co"># only 14 is observable</span></span>
<span id="cb2-30"><a href="#cb2-30" tabindex="-1"></a><span class="co">#&gt;  [1] 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0</span></span>
<span id="cb2-31"><a href="#cb2-31" tabindex="-1"></a><span class="fu">dgpbinom</span>(<span class="dv">7</span> <span class="sc">*</span> <span class="dv">2</span>, <span class="fu">rep</span>(<span class="dv">0</span>, <span class="dv">7</span>), <span class="fu">rep</span>(<span class="dv">4</span>, <span class="dv">7</span>), <span class="fu">rep</span>(<span class="dv">2</span>, <span class="dv">7</span>)) <span class="co"># confirmation</span></span>
<span id="cb2-32"><a href="#cb2-32" tabindex="-1"></a><span class="co">#&gt; [1] 1</span></span>
<span id="cb2-33"><a href="#cb2-33" tabindex="-1"></a></span>
<span id="cb2-34"><a href="#cb2-34" tabindex="-1"></a><span class="co"># Case 2 b)</span></span>
<span id="cb2-35"><a href="#cb2-35" tabindex="-1"></a><span class="fu">dgpbinom</span>(<span class="cn">NULL</span>, <span class="fu">rep</span>(<span class="dv">1</span>, <span class="dv">7</span>), <span class="fu">rep</span>(<span class="dv">4</span>, <span class="dv">7</span>), <span class="fu">rep</span>(<span class="dv">2</span>, <span class="dv">7</span>)) <span class="co"># only 28 is observable</span></span>
<span id="cb2-36"><a href="#cb2-36" tabindex="-1"></a><span class="co">#&gt;  [1] 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1</span></span>
<span id="cb2-37"><a href="#cb2-37" tabindex="-1"></a><span class="fu">dgpbinom</span>(<span class="dv">7</span> <span class="sc">*</span> <span class="dv">4</span>, <span class="fu">rep</span>(<span class="dv">1</span>, <span class="dv">7</span>), <span class="fu">rep</span>(<span class="dv">4</span>, <span class="dv">7</span>), <span class="fu">rep</span>(<span class="dv">2</span>, <span class="dv">7</span>)) <span class="co"># confirmation</span></span>
<span id="cb2-38"><a href="#cb2-38" tabindex="-1"></a><span class="co">#&gt; [1] 1</span></span>
<span id="cb2-39"><a href="#cb2-39" tabindex="-1"></a></span>
<span id="cb2-40"><a href="#cb2-40" tabindex="-1"></a><span class="co"># Case 2 c)</span></span>
<span id="cb2-41"><a href="#cb2-41" tabindex="-1"></a><span class="fu">dgpbinom</span>(<span class="cn">NULL</span>, <span class="fu">rep</span>(<span class="fl">0.3</span>, <span class="dv">7</span>), <span class="fu">rep</span>(<span class="dv">4</span>, <span class="dv">7</span>), <span class="fu">rep</span>(<span class="dv">2</span>, <span class="dv">7</span>))</span>
<span id="cb2-42"><a href="#cb2-42" tabindex="-1"></a><span class="co">#&gt;  [1] 0.0823543 0.0000000 0.2470629 0.0000000 0.3176523 0.0000000 0.2268945</span></span>
<span id="cb2-43"><a href="#cb2-43" tabindex="-1"></a><span class="co">#&gt;  [8] 0.0000000 0.0972405 0.0000000 0.0250047 0.0000000 0.0035721 0.0000000</span></span>
<span id="cb2-44"><a href="#cb2-44" tabindex="-1"></a><span class="co">#&gt; [15] 0.0002187</span></span>
<span id="cb2-45"><a href="#cb2-45" tabindex="-1"></a><span class="fu">dbinom</span>(<span class="dv">0</span><span class="sc">:</span><span class="dv">7</span>, <span class="dv">7</span>, <span class="fl">0.3</span>) <span class="co"># equal results, but on different support set</span></span>
<span id="cb2-46"><a href="#cb2-46" tabindex="-1"></a><span class="co">#&gt; [1] 0.0823543 0.2470629 0.3176523 0.2268945 0.0972405 0.0250047 0.0035721</span></span>
<span id="cb2-47"><a href="#cb2-47" tabindex="-1"></a><span class="co">#&gt; [8] 0.0002187</span></span>
<span id="cb2-48"><a href="#cb2-48" tabindex="-1"></a></span>
<span id="cb2-49"><a href="#cb2-49" tabindex="-1"></a><span class="co"># Case 2 d)</span></span>
<span id="cb2-50"><a href="#cb2-50" tabindex="-1"></a><span class="fu">dgpbinom</span>(<span class="cn">NULL</span>, pp, <span class="fu">rep</span>(<span class="dv">4</span>, <span class="dv">7</span>), <span class="fu">rep</span>(<span class="dv">2</span>, <span class="dv">7</span>))</span>
<span id="cb2-51"><a href="#cb2-51" tabindex="-1"></a><span class="co">#&gt;  [1] 8.114776e-05 0.000000e+00 3.112722e-03 0.000000e+00 4.063146e-02</span></span>
<span id="cb2-52"><a href="#cb2-52" tabindex="-1"></a><span class="co">#&gt;  [6] 0.000000e+00 2.115237e-01 0.000000e+00 3.793308e-01 0.000000e+00</span></span>
<span id="cb2-53"><a href="#cb2-53" tabindex="-1"></a><span class="co">#&gt; [11] 2.735489e-01 0.000000e+00 8.297278e-02 0.000000e+00 8.798512e-03</span></span>
<span id="cb2-54"><a href="#cb2-54" tabindex="-1"></a><span class="fu">dpbinom</span>(<span class="cn">NULL</span>, pp) <span class="co"># equal results, but on different support set</span></span>
<span id="cb2-55"><a href="#cb2-55" tabindex="-1"></a><span class="co">#&gt; [1] 8.114776e-05 3.112722e-03 4.063146e-02 2.115237e-01 3.793308e-01</span></span>
<span id="cb2-56"><a href="#cb2-56" tabindex="-1"></a><span class="co">#&gt; [6] 2.735489e-01 8.297278e-02 8.798512e-03</span></span>
<span id="cb2-57"><a href="#cb2-57" tabindex="-1"></a></span>
<span id="cb2-58"><a href="#cb2-58" tabindex="-1"></a><span class="co"># Case 3 a)</span></span>
<span id="cb2-59"><a href="#cb2-59" tabindex="-1"></a><span class="fu">dgpbinom</span>(<span class="cn">NULL</span>, <span class="fu">c</span>(<span class="dv">0</span>, <span class="dv">0</span>, <span class="dv">0</span>, <span class="dv">0</span>, <span class="dv">0</span>, <span class="dv">0</span>, <span class="dv">0</span>), va, vb) <span class="co"># only sum(vb) is observable</span></span>
<span id="cb2-60"><a href="#cb2-60" tabindex="-1"></a><span class="co">#&gt;  [1] 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0</span></span>
<span id="cb2-61"><a href="#cb2-61" tabindex="-1"></a><span class="fu">dgpbinom</span>(<span class="fu">sum</span>(vb), <span class="fu">rep</span>(<span class="dv">0</span>, <span class="dv">7</span>), va, vb) <span class="co"># confirmation</span></span>
<span id="cb2-62"><a href="#cb2-62" tabindex="-1"></a><span class="co">#&gt; [1] 1</span></span>
<span id="cb2-63"><a href="#cb2-63" tabindex="-1"></a></span>
<span id="cb2-64"><a href="#cb2-64" tabindex="-1"></a><span class="co"># Case 3 b)</span></span>
<span id="cb2-65"><a href="#cb2-65" tabindex="-1"></a><span class="fu">dgpbinom</span>(<span class="cn">NULL</span>, <span class="fu">c</span>(<span class="dv">1</span>, <span class="dv">1</span>, <span class="dv">1</span>, <span class="dv">1</span>, <span class="dv">1</span>, <span class="dv">1</span>, <span class="dv">1</span>), va, vb) <span class="co"># only sum(va) is observable</span></span>
<span id="cb2-66"><a href="#cb2-66" tabindex="-1"></a><span class="co">#&gt;  [1] 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0</span></span>
<span id="cb2-67"><a href="#cb2-67" tabindex="-1"></a><span class="fu">dgpbinom</span>(<span class="fu">sum</span>(va), <span class="fu">rep</span>(<span class="dv">1</span>, <span class="dv">7</span>), va, vb) <span class="co"># confirmation</span></span>
<span id="cb2-68"><a href="#cb2-68" tabindex="-1"></a><span class="co">#&gt; [1] 1</span></span>
<span id="cb2-69"><a href="#cb2-69" tabindex="-1"></a></span>
<span id="cb2-70"><a href="#cb2-70" tabindex="-1"></a><span class="co"># Case 3 c)</span></span>
<span id="cb2-71"><a href="#cb2-71" tabindex="-1"></a><span class="fu">dgpbinom</span>(<span class="cn">NULL</span>, <span class="fu">c</span>(<span class="dv">0</span>, <span class="dv">0</span>, <span class="dv">0</span>, <span class="dv">1</span>, <span class="dv">1</span>, <span class="dv">1</span>, <span class="dv">1</span>), va, vb) <span class="co"># only sum(va[4:7], vb[1:3]) is observable</span></span>
<span id="cb2-72"><a href="#cb2-72" tabindex="-1"></a><span class="co">#&gt;  [1] 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0</span></span>
<span id="cb2-73"><a href="#cb2-73" tabindex="-1"></a><span class="fu">dgpbinom</span>(<span class="fu">sum</span>(va[<span class="dv">4</span><span class="sc">:</span><span class="dv">7</span>], vb[<span class="dv">1</span><span class="sc">:</span><span class="dv">3</span>]), <span class="fu">c</span>(<span class="dv">0</span>, <span class="dv">0</span>, <span class="dv">0</span>, <span class="dv">1</span>, <span class="dv">1</span>, <span class="dv">1</span>, <span class="dv">1</span>), va, vb) <span class="co"># confirmation</span></span>
<span id="cb2-74"><a href="#cb2-74" tabindex="-1"></a><span class="co">#&gt; [1] 1</span></span>
<span id="cb2-75"><a href="#cb2-75" tabindex="-1"></a></span>
<span id="cb2-76"><a href="#cb2-76" tabindex="-1"></a><span class="co"># Case 4</span></span>
<span id="cb2-77"><a href="#cb2-77" tabindex="-1"></a><span class="fu">dgpbinom</span>(<span class="cn">NULL</span>, <span class="fu">c</span>(<span class="dv">0</span>, <span class="dv">0</span>, <span class="fl">0.3</span>, <span class="fl">0.6</span>, <span class="dv">1</span>, <span class="dv">1</span>, <span class="dv">1</span>), va, vb)</span>
<span id="cb2-78"><a href="#cb2-78" tabindex="-1"></a><span class="co">#&gt;  [1] 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.18 0.00 0.00 0.12 0.42 0.00 0.00 0.28</span></span>
<span id="cb2-79"><a href="#cb2-79" tabindex="-1"></a><span class="co">#&gt; [16] 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00</span></span>
<span id="cb2-80"><a href="#cb2-80" tabindex="-1"></a>sure <span class="ot">&lt;-</span> <span class="fu">sum</span>(va[<span class="dv">5</span><span class="sc">:</span><span class="dv">7</span>], vb[<span class="dv">1</span><span class="sc">:</span><span class="dv">2</span>])</span>
<span id="cb2-81"><a href="#cb2-81" tabindex="-1"></a>x.transf <span class="ot">&lt;-</span> <span class="fu">sum</span>(<span class="fu">pmin</span>(va[<span class="dv">3</span><span class="sc">:</span><span class="dv">4</span>], vb[<span class="dv">3</span><span class="sc">:</span><span class="dv">4</span>]))<span class="sc">:</span><span class="fu">sum</span>(<span class="fu">pmax</span>(va[<span class="dv">3</span><span class="sc">:</span><span class="dv">4</span>], vb[<span class="dv">3</span><span class="sc">:</span><span class="dv">4</span>]))</span>
<span id="cb2-82"><a href="#cb2-82" tabindex="-1"></a><span class="fu">dgpbinom</span>(sure <span class="sc">+</span> x.transf, <span class="fu">c</span>(<span class="dv">0</span>, <span class="dv">0</span>, <span class="fl">0.3</span>, <span class="fl">0.6</span>, <span class="dv">1</span>, <span class="dv">1</span>, <span class="dv">1</span>), va, vb)</span>
<span id="cb2-83"><a href="#cb2-83" tabindex="-1"></a><span class="co">#&gt; [1] 0.18 0.00 0.00 0.12 0.42 0.00 0.00 0.28</span></span>
<span id="cb2-84"><a href="#cb2-84" tabindex="-1"></a><span class="fu">dgpbinom</span>(x.transf, <span class="fu">c</span>(<span class="fl">0.3</span>, <span class="fl">0.6</span>), va[<span class="dv">3</span><span class="sc">:</span><span class="dv">4</span>], vb[<span class="dv">3</span><span class="sc">:</span><span class="dv">4</span>]) <span class="co"># equal results</span></span>
<span id="cb2-85"><a href="#cb2-85" tabindex="-1"></a><span class="co">#&gt; [1] 0.18 0.00 0.00 0.12 0.42 0.00 0.00 0.28</span></span>
<span id="cb2-86"><a href="#cb2-86" tabindex="-1"></a></span>
<span id="cb2-87"><a href="#cb2-87" tabindex="-1"></a><span class="fu">dgpbinom</span>(<span class="cn">NULL</span>, <span class="fu">c</span>(<span class="dv">0</span>, <span class="dv">0</span>, <span class="dv">0</span>, <span class="fl">0.6</span>, <span class="dv">1</span>, <span class="dv">1</span>, <span class="dv">1</span>), va, vb)</span>
<span id="cb2-88"><a href="#cb2-88" tabindex="-1"></a><span class="co">#&gt;  [1] 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.6 0.0 0.0 0.4 0.0 0.0 0.0 0.0</span></span>
<span id="cb2-89"><a href="#cb2-89" tabindex="-1"></a><span class="co">#&gt; [20] 0.0 0.0 0.0 0.0 0.0 0.0</span></span>
<span id="cb2-90"><a href="#cb2-90" tabindex="-1"></a>sure <span class="ot">&lt;-</span> <span class="fu">sum</span>(va[<span class="dv">5</span><span class="sc">:</span><span class="dv">7</span>], vb[<span class="dv">1</span><span class="sc">:</span><span class="dv">3</span>])</span>
<span id="cb2-91"><a href="#cb2-91" tabindex="-1"></a>x.transf <span class="ot">&lt;-</span> va[<span class="dv">4</span>]<span class="sc">:</span>vb[<span class="dv">4</span>]</span>
<span id="cb2-92"><a href="#cb2-92" tabindex="-1"></a><span class="fu">dgpbinom</span>(sure <span class="sc">+</span> x.transf, <span class="fu">c</span>(<span class="dv">0</span>, <span class="dv">0</span>, <span class="dv">0</span>, <span class="fl">0.6</span>, <span class="dv">1</span>, <span class="dv">1</span>, <span class="dv">1</span>), va, vb)</span>
<span id="cb2-93"><a href="#cb2-93" tabindex="-1"></a><span class="co">#&gt; [1] 0.6 0.0 0.0 0.4</span></span>
<span id="cb2-94"><a href="#cb2-94" tabindex="-1"></a><span class="fu">dgpbinom</span>(x.transf, <span class="fl">0.6</span>, va[<span class="dv">4</span>], vb[<span class="dv">4</span>]) <span class="co"># equal results; essentially transformed Bernoulli</span></span>
<span id="cb2-95"><a href="#cb2-95" tabindex="-1"></a><span class="co">#&gt; [1] 0.6 0.0 0.0 0.4</span></span></code></pre></div>
<hr />
</div>
</div>
<div id="usage-with-rcpp" class="section level2">
<h2>Usage with Rcpp</h2>
<p>How to import and use the internal C++ functions in <code>Rcpp</code>
based packages is described in a <a href="use_with_rcpp.html">separate
vignette</a>.</p>
</div>



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