ODIn/Website
9
0
Fork 1
Code Pull requests Releases Wiki Activity

ProbDB slides

Browse source
This commit is contained in:
Oliver Kennedy 2019-05-05 23:24:26 -04:00
parent 2b3ba60c6d
commit 8260cf4aa0
2 changed files with 334 additions and 14 deletions
Show stats Download patch file Download diff file Expand all files Collapse all files

15
src/teaching/cse-562/2019sp/slide/2019-05-01-IncompleteDBs.erb
View file

@ -1,6 +1,6 @@
---
template: templates/cse4562_2019_slides.erb
title: Incomplete and Probabilistic Databases
title: Querying Incomplete Databases
date: May 1, 2019
textbook: "<a href='https://github.com/UBOdin/mimir/wiki/Concepts-CTables'>PDB Concepts and C-Tables</a>"
dependencies:
@ -180,7 +180,7 @@ dependencies:
["Name", "ZipCode"],
[ ["Alice", "10003"],
["Bob","14260"],
["Bob","14290"],
["Bob","19260"],
["Carol","13201"],
["Carol","18201"]
],
@ -205,7 +205,7 @@ dependencies:
["Name", "ZipCode"],
[ ["Alice", "10003"],
["Bob","14260"],
["Bob","14290"],
["Bob","19260"],
["Carol","13201"],
["Carol","18201"]
],
@ -230,7 +230,7 @@ dependencies:
["Name", "ZipCode"],
[ ["Alice", "10003"],
["Bob","14260"],
["Bob","14290"],
["Bob","19260"],
["Carol","13201"],
["Carol","18201"]
],
@ -315,12 +315,7 @@ dependencies:
Not bigger than the aggregate input...
</p>
<p class="fragment">
...but at least it only reduces to bin-packing <br/>(or a similarly NP problem.)
...but at least it only reduces to bin-packing <br/>(or a similarly known NP problem.)
</p>
</section>
<section>
<p>In short, incomplete databases are limited, but have some uses.</p>
<p class="fragment">What about probabilities?</p>
</section>
</section>

333
src/teaching/cse-562/2019sp/slide/2019-05-06-ProbDBs.erb
View file

@ -1,6 +1,6 @@
---
template: templates/cse4562_2019_slides.erb
title: Incomplete and Probabilistic Databases
title: Probabilistic Databases
date: May 6, 2019
textbook: "<a href='https://github.com/UBOdin/mimir/wiki/Concepts-CTables'>PDB Concepts and C-Tables</a>"
dependencies:
@ -10,15 +10,74 @@ dependencies:
require "slide_utils.rb"
%>
<section>
<section>
<p><b>Idea: </b> Make $\texttt{bob}$ and $\texttt{carol}$ random variables.</p>
<img src="graphics/2019-04-31-NormalDB.svg" /><br/>
<hr class="fragment" data-fragment-index="1"/>
<svg data-src="graphics/2019-04-31-IncompleteDB.svg" class="fragment" data-fragment-index="1"/>
</section>
<section>
<h3>(One Form of) Incomplete Databases</h3>
<ul>
<li>Define each choice as a variable</li>
<li>Tag each row with a boolean formula over variables</li>
<li>Each possible world is one assignment of values to variables</li>
<li>The possible world has all rows tagged with formulas that evaluate to "true"</li>
</ul>
</section>
<section>
<dl>
<dt>Certain Tuple</dt>
<dd>A tuple that appears in all possible worlds</dd>
<dt>Possible Tuple</dt>
<dd>A tuple that appears in at least one possible world</dd>
</dl>
</section>
<section>
<p><b>Limitation: </b> Can't distinguish between possible-but unlikely and possible-but very likely.</p>
</section>
<section>
<p><b>Idea: </b> Make variables probabilistic</p>
</section>
<section>
<h3>Example</h3>
<p>$$\texttt{bob} = \begin{cases} 4 & p = 0.8 \\ 9 & p = 0.2\end{cases}$$</p>
<p>$$\texttt{carol} = \begin{cases} 3 & p = 0.4 \\ 8 & p = 0.6\end{cases}$$</p>
</section>
<section>
<%= data_table(
["Name", "ZipCode"],
[ ["Alice", "10003"],
["Bob","14260"],
["Bob","19260"],
["Carol","13201"],
["Carol","18201"]
],
name: "$\\mathcal R$",
rowids: true,
annotations: [
"always",
"if $\\texttt{bob} = 4$",
"if $\\texttt{bob} = 9$",
"if $\\texttt{carol} = 3$",
"if $\\texttt{carol} = 8$"
]
) %>
<pre class="fragment"><code>
SELECT COUNT(*)
FROM R NATURAL JOIN ZipCodeLookup
WHERE State = 'NY'
</code></pre>
</section>
<section>
<p style="font-size: 70%">
$$Q(\mathcal D) = \begin{cases}
@ -52,12 +111,278 @@ dependencies:
</section>
<section>
<p>In general, computing probabilities exactly is <code>#P</code></p>
<p>In general, computing marginal probabilities for result tuples exactly is <code>#P</code></p>
<p style="margin-top: 50px;" class="fragment">... so we approximate</p>
</section>
</section>
<section>
<section>
<p><b>Idea 1</b>: Sample. Pick (e.g.) 10 random possible worlds and compute results for each.</p>
</section>
<%
bob = []
carol = []
counts = []
%>
<% (0...5).each do |i| %>
<section>
<%
bob[i] = if rand() < 0.8 then 4 else 9 end
carol[i] = if rand() < 0.4 then 3 else 8 end
counts[i] = [
1, # alice
(if bob[i] == 4 then 1 else 0 end),
(if carol[i] == 3 then 1 else 0 end),
].compact.sum
%>
<p>$$R_{<%=i+1%>} \Leftarrow \{\; \texttt{bob} \rightarrow <%= bob[i] %>, \; \texttt{carol} \rightarrow <%= carol[i] %>\}$$</p>
<%= data_table(
["Name", "ZipCode"],
[ ["Alice", "10003"],
["Bob","1#{bob[i]}260"],
["Carol","1#{carol[i]}201"]
],
name: "$\\mathcal R_{#{i+1}}$",
rowids: true,
) %>
<p>$$\mathcal Q = \{\;<%=counts.join(",\\;")%>\;\}$$</p>
<% if i == 10 %>
<p class="fragment">$$E[\mathcal Q] \approx <%=counts.avg.round(2)%>$$</p>
<p class="fragment">$$P[\mathcal Q \geq 2] \approx <%=(counts.select { |c| c >= 2 }.count / counts.size.to_f).round(2) %>$$</p>
<% else %>
<p>&nbsp;</p>
<p>&nbsp;</p>
<% end %>
</section>
<% end %>
</section>
<section>
<section>
<p><b>Problem</b>: Sloooooooooooow.</p>
<p class="fragment">Can we make it faster?</p>
</section>
<section>
<p><b>Idea 1</b>: Sample. Pick 10 random possible worlds and compute results for each.</p>
<p><b>Idea 1.A</b>: Combine all samples into one query.</p>
</section>
<section>
<div style="font-size: 80%">
<%= data_table(
[ "Name", "ZipCode", "$\\mathcal{ID}$"],
(0...5).map do |i|
[ ["Alice", "10003", "#{i+1}"],
["Bob", "1#{bob[i]}260", "#{i+1}"],
["Carol","1#{carol[i]}201","#{i+1}"] ]
end.flatten(1),
name: "$\\mathcal R$",
rowids: true,
) %>
</div>
</section>
<section>
<%= data_table(
[ "Count", "$\\mathcal{ID}$"],
(0...5).map do |i|
[counts[i], "#{i+1}"]
end,
name: "$\\mathcal Q$",
rowids: true,
) %>
</section>
<section>
<h3>Querying Joint Sample Tables</h3>
<p>$\pi_A(R) \rightarrow $ <span class="fragment">$\pi_{A, \mathcal{ID}}(R)$</span></p>
<p class="fragment">$\sigma_\phi(R) \rightarrow $ <span class="fragment">$\sigma_{\phi}(R)$</span></p>
<p class="fragment">$R \uplus S \rightarrow $ <span class="fragment">$R \uplus S$</span></p>
<p class="fragment">$R \times S \rightarrow $ <span class="fragment"><span class="fragment"><span class="fragment">$\pi_{R.*, S.*, R.\mathcal{ID}}\big($</span>$\sigma_{R.\mathcal{ID} = S.\mathcal{ID}}( $</span>$ R \times S)\big)$</span></p>
<p class="fragment">$\delta R \rightarrow $ <span class="fragment">$\delta R$</span></p>
<p class="fragment">$_A\gamma_{Agg(*)}(R) \rightarrow $ <span class="fragment">$_{A, \mathcal{ID}}\gamma_{Agg(*)}(R)$</span></p>
</section>
</section>
<section>
<section>
<p>Still sloooooow.</p>
<p class="fragment">There's a lot of repetition.</p>
</section>
<section>
<p><b>Idea 2.B</b> Use native array-types in DBs</p>
</section>
<section>
<h3 class="fragment">Tuple Bundles</h3>
<div style="font-size: 80%">
<%= data_table(
[ "Name", "ZipCode" ],
[ ["Alice", "10003" ],
["Bob", "["+(0...5).map { |i| "1#{bob[i]}260" }.join(", ")+"]"],
["Carol", "["+(0...5).map { |i| "1#{carol[i]}201" }.join(", ")+"]"]
],
name: "$\\mathcal R$",
rowids: true,
) %>
</div>
<attribution><a href="https://dl.acm.org/citation.cfm?id=1376686">MCDB: a monte carlo approach to managing uncertain data</a> (Jampani et. al.)</attribution>
</section>
<section>
<h3>Querying Tuple Bundles</h3>
<p>$\pi_A(R) \rightarrow $ <span class="fragment">$\pi_{A}(R)$</span></p>
<p class="fragment">$\sigma_\phi(R) \rightarrow $ <span class="fragment">?</span></p>
<p>&nbsp;</p>
<p>&nbsp;</p>
<p>&nbsp;</p>
<p>&nbsp;</p>
</section>
<section>
<p><b>Idea 1.B'</b> Also mark which tuples are present in which samples</p>
</section>
<section>
<div style="font-size: 70%">
<%= data_table(
[ "Name", "ZipCode", "$\\mathcal W$" ],
[ ["Alice", "10003", "11111" ],
["Bob", "["+(0...5).map { |i| "1#{bob[i]}260" }.join(", ")+"]", "11111"],
["Carol", "["+(0...5).map { |i| "1#{carol[i]}201" }.join(", ")+"]", "11111"]
],
name: "$\\mathcal R$",
rowids: true,
) %>
</div>
<div class="fragment" style="margin: 50px;">↓ $\sigma_{InNYS(ZipCode)}(\mathcal R)$ ↓</div>
<div style="font-size: 70%" class="fragment">
<%= data_table(
[ "Name", "ZipCode", "$\\mathcal W$" ],
[ ["Alice", "10003", "11111" ],
["Bob",
"["+(0...5).map { |i| "1#{bob[i]}260" }.join(", ")+"]",
(0...5).map { |i| if bob[i] == 4 then 1 else 0 end }.join ],
["Carol",
"["+(0...5).map { |i| "1#{carol[i]}201" }.join(", ")+"]",
(0...5).map { |i| if carol[i] == 3 then 1 else 0 end }.join ]
],
name: "$\\mathcal R$",
rowids: true,
) %>
</div>
</section>
<section>
<h3>Querying Tuple Bundles</h3>
<p>$\pi_A(R) \rightarrow $ $\pi_{A}(R)$</p>
<p class="fragment">$\sigma_\phi(R) \rightarrow $ <span class="fragment"><span class="fragment">$\sigma_{\mathcal W = 0}($</span>$\pi_{\mathcal W \;\&\; \vec \phi}(R))$</span></p>
<p class="fragment">$R \uplus S \rightarrow $ <span class="fragment">$R \uplus S$</span></p>
<p class="fragment">$R \times S \rightarrow $ <span class="fragment"><span class="fragment"><span class="fragment">$\sigma_{\mathcal{W} = 0}\big($</span>$\pi_{R.*, S.*, R.\mathcal{W} \;\&\; S.\mathcal{W}}( $</span>$ R \times S)\big)$</span></p>
<p class="fragment">$_A\gamma_{Agg(B)}(R) \rightarrow $ <span class="fragment">$_A\gamma_{[ Agg\big(\textbf{if}(W[1])\{R.B[1]\}\big), Agg\big(\textbf{if}(W[2])\{R.B[2]\}\big), \ldots ]}(R)$</span></p>
</section>
<section>
<h3>Querying Joint Sample Tables</h3>
<p>$\pi_A(R) \rightarrow \pi_{A}(R)$</p>
<p>$\sigma_\phi(R) \rightarrow \sigma_{\mathcal W = 0}(\pi_{\mathcal W \;\&\; \vec \phi}(R))$</p>
<p>$R \uplus S \rightarrow R \uplus S$</p>
<p class="fragment highlight-blue" data-fragment-index="1">$R \times S \rightarrow \sigma_{\mathcal{W} = 0}\big(\pi_{R.*, S.*, R.\mathcal{W} \;\&\; S.\mathcal{W}}( R \times S)\big)$</p>
<p class="fragment highlight-blue" data-fragment-index="2">$_A\gamma_{Agg(B)}(R) \rightarrow $ $_A\gamma_{[ Agg\big(\textbf{if}(W[1])\{R.B[1]\}\big), Agg\big(\textbf{if}(W[2])\{R.B[2]\}\big), \ldots ]}(R)$</p>
<p>(Generate aggregates for each sample separately)</p>
<p class="fragment" data-fragment-index="1" style="margin-top: 60px; font-size: 60%">Good luck ever doing an equi-join.</p>
<p class="fragment" data-fragment-index="2" style="margin-top: 0px; font-size: 60%">Hope your group-by variables aren't uncertain.</p>
</section>
</section>
<section>
<section>
<p>Inefficient equi-joins on uncertain variables.</p>
<p>Inefficient aggregates with uncertain variables.</p>
<p class="fragment">How many samples necessary to get desired precision?</p>
</section>
<section>
<p><b>Idea 2</b>: Symbolic Execution (Provenance)</p>
</section>
<section>
<p>$\sigma_{count \geq 2}(Q) =$</p>
<p class="fragment">$\texttt{bob} = 4 \wedge \texttt{carol} = 8 $<br/>
$\vee\; \texttt{bob} = 9 \wedge \texttt{carol} = 3 $<br/>
$\vee\; \texttt{bob} = 4 \wedge \texttt{carol} = 3$</p>
<p class="fragment">$P[\sigma_{count \geq 2}(Q)] = ?$ <span class="fragment">$\approx$ #SAT</span></p>
</section>
<section>
<h3>Computing Probabilities</h3>
<p class="fragment">$P[\texttt{x} \wedge \texttt{y}] = P[\texttt{x}] \cdot P[\texttt{y}]$<br/>(iff $\texttt{x}$ and $\texttt{y}$ are independent)</p>
<p class="fragment">$P[\texttt{x} \wedge \texttt{y}] = 0$<br/>(iff $\texttt{x}$ and $\texttt{y}$ are mutually exclusive)</p>
<p class="fragment">$P[\texttt{x} \vee \texttt{y}] = 1- (1-P[\texttt{x}]) \cdot (1-P[\texttt{y}])$<br/>(iff $\texttt{x}$ and $\texttt{y}$ are independent)</p>
<p class="fragment">$P[\texttt{x} \vee \texttt{y}] = P[\texttt{x}] + P[\texttt{y}]$<br/>(iff $\texttt{x}$ and $\texttt{y}$ are mutually exclusive)</p>
<p class="fragment" style="font-size: 70%; font-weight: bold;">Good enough to get us the probability of any boolean formula over mutually exclusive or independent variables</p>
<p class="fragment" style="font-size: 70%; font-weight: bold;">... and otherwise?</p>
</section>
<section>
<h3>Shannon Expansion</h3>
<p>For a boolean formula $f$ and variable $\texttt{x}$:</p>
<p>$$f = (\texttt{x} \wedge f[\texttt{x}\backslash T]) \vee (\neg \texttt{x} \wedge f[\texttt{x}\backslash F])$$</p>
<p class="fragment" style="margin-top: 50px;margin-bottom:0px;">Disjunction of mutually-exclusive terms!</p>
<p class="fragment" style="margin-top: 0px;margin-bottom:0px;">... each a conjunction of independent terms.</p>
<p class="fragment" style="margin-top: 0px;margin-bottom:0px;">... and $\texttt{x}$ removed from $f$</p>
<p class="fragment">Ok... just keep applying Shannon!</p>
<p class="fragment">Each application creates 2 new formulas (ExpTime!)</p>
</section>
<section>
<p><b>Idea 2.A</b>: Combine the two. Use Shanon expansion as long as time/resources permit, then use a #SAT approximation.</p>
<attribution class="fragment"><a href="https://ieeexplore.ieee.org/abstract/document/4812442/">Sprout: Lazy vs. eager query plans for tuple-independent probabilistic databases</a> (Olteanu et. al.)</attribution>
</section>
</section>
<section>
<section>
<h3>More Resources</h3>
<dl style="font-size: 80%">
<dt><a href="https://dl.acm.org/citation.cfm?id=1376686">MCDB</a></dt>
<dd>Sampling-based probabilistic databases</dd>
<dt><a href="https://ieeexplore.ieee.org/abstract/document/4812442/">Sprout</a></dt>
<dd>"Any-time" Approximation.</dd>
<dt><a href="https://dl.acm.org/citation.cfm?id=2824055">Mimir</a></dt>
<dd>PL tricks to make ProbDBs faster</dd>
<dt><a href="https://dl.acm.org/citation.cfm?id=2809991">DeepDive</a></dt>
<dd>ProbDBs used in practice to populate Knowledge Bases.</dd>
<dt><a href="https://ieeexplore.ieee.org/abstract/document/4812509/">Integrating and Ranking Uncertain Scientific Data</a></dt>
<dd>ProbDBs used in practice to predict gene expressions / propose experiments.</dd>
</dl>
</section>
</section>