simple typo fix, "distribution" -> "distributions"

index.html

@@ -371,7 +371,7 @@ <h3>MDN-RNN (M) Model</h3>
 <img src="assets/mdn_rnn_new.svg" style="display: block; margin: auto; width: 80%;"/>
 <figcaption>RNN with a Mixture Density Network output layer. The MDN outputs the parameters of a mixture of Gaussian distribution used to sample a prediction of the next latent vector <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>z</mi></mrow><annotation encoding="application/x-tex">z</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.43056em;"></span><span class="strut bottom" style="height:0.43056em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.04398em;">z</span></span></span></span>.</figcaption>
 </div>
-<p>In our approach, we approximate <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>p</mi><mo>(</mo><mi>z</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">p(z)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.75em;"></span><span class="strut bottom" style="height:1em;vertical-align:-0.25em;"></span><span class="base textstyle uncramped"><span class="mord mathit">p</span><span class="mopen">(</span><span class="mord mathit" style="margin-right:0.04398em;">z</span><span class="mclose">)</span></span></span></span> as a mixture of Gaussian distribution, and train the RNN to output the probability distribution of the next latent vector <span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>z</mi><mrow><mi>t</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">z_{t+1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.43056em;"></span><span class="strut bottom" style="height:0.638891em;vertical-align:-0.208331em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit" style="margin-right:0.04398em;">z</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.04398em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit">t</span><span class="mbin">+</span><span class="mord mathrm">1</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span> given the current and past information made available to it.</p>
+<p>In our approach, we approximate <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>p</mi><mo>(</mo><mi>z</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">p(z)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.75em;"></span><span class="strut bottom" style="height:1em;vertical-align:-0.25em;"></span><span class="base textstyle uncramped"><span class="mord mathit">p</span><span class="mopen">(</span><span class="mord mathit" style="margin-right:0.04398em;">z</span><span class="mclose">)</span></span></span></span> as a mixture of Gaussian distributions, and train the RNN to output the probability distribution of the next latent vector <span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>z</mi><mrow><mi>t</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">z_{t+1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.43056em;"></span><span class="strut bottom" style="height:0.638891em;vertical-align:-0.208331em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit" style="margin-right:0.04398em;">z</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.04398em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit">t</span><span class="mbin">+</span><span class="mord mathrm">1</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span> given the current and past information made available to it.</p>
 <p>More specifically, the RNN will model <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>P</mi><mo>(</mo><msub><mi>z</mi><mrow><mi>t</mi><mo>+</mo><mn>1</mn></mrow></msub><mspace width="0.277778em"></mspace><mi mathvariant="normal">∣</mi><mspace width="0.277778em"></mspace><msub><mi>a</mi><mi>t</mi></msub><mo separator="true">,</mo><msub><mi>z</mi><mi>t</mi></msub><mo separator="true">,</mo><msub><mi>h</mi><mi>t</mi></msub><mo>)</mo></mrow><annotation encoding="application/x-tex">P(z_{t+1} \; | \; a_t, z_t, h_t)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.75em;"></span><span class="strut bottom" style="height:1em;vertical-align:-0.25em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.13889em;">P</span><span class="mopen">(</span><span class="mord"><span class="mord mathit" style="margin-right:0.04398em;">z</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.04398em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord scriptstyle cramped"><span class="mord mathit">t</span><span class="mbin">+</span><span class="mord mathrm">1</span></span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mord mspace thickspace"></span><span class="mord mathrm">∣</span><span class="mord mspace thickspace"></span><span class="mord"><span class="mord mathit">a</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord mathit">t</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mpunct">,</span><span class="mord"><span class="mord mathit" style="margin-right:0.04398em;">z</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:-0.04398em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord mathit">t</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mpunct">,</span><span class="mord"><span class="mord mathit">h</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord mathit">t</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span><span class="mclose">)</span></span></span></span>, where <span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>a</mi><mi>t</mi></msub></mrow><annotation encoding="application/x-tex">a_t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.43056em;"></span><span class="strut bottom" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit">a</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord mathit">t</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span> is the action taken at time <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>t</mi></mrow><annotation encoding="application/x-tex">t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.61508em;"></span><span class="strut bottom" style="height:0.61508em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit">t</span></span></span></span> and <span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>h</mi><mi>t</mi></msub></mrow><annotation encoding="application/x-tex">h_t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.69444em;"></span><span class="strut bottom" style="height:0.84444em;vertical-align:-0.15em;"></span><span class="base textstyle uncramped"><span class="mord"><span class="mord mathit">h</span><span class="vlist"><span style="top:0.15em;margin-right:0.05em;margin-left:0em;"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span><span class="reset-textstyle scriptstyle cramped"><span class="mord mathit">t</span></span></span><span class="baseline-fix"><span class="fontsize-ensurer reset-size5 size5"><span style="font-size:0em;">​</span></span>​</span></span></span></span></span></span> is the <em>hidden state</em> of the RNN at time <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>t</mi></mrow><annotation encoding="application/x-tex">t</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.61508em;"></span><span class="strut bottom" style="height:0.61508em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit">t</span></span></span></span>. During sampling, we can adjust a <em>temperature</em> parameter <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>τ</mi></mrow><annotation encoding="application/x-tex">\tau</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.43056em;"></span><span class="strut bottom" style="height:0.43056em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.1132em;">τ</span></span></span></span> to control model uncertainty, as done in <dt-cite key="sketchrnn"></dt-cite> -- we will find adjusting <span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>τ</mi></mrow><annotation encoding="application/x-tex">\tau</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.43056em;"></span><span class="strut bottom" style="height:0.43056em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit" style="margin-right:0.1132em;">τ</span></span></span></span> to be useful for training our controller later on.</p>
 <div style="text-align: center;">
 <video autoplay muted playsinline loop style="display: block; margin: auto; width: 100%;"><source src="https://storage.googleapis.com/quickdraw-models/sketchRNN/world_models/assets/mp4/sketch_rnn_insect.mp4" type="video/mp4"/></video>
@@ -2120,4 +2120,4 @@ <h3>DoomRNN</h3>
 <script src="demo/start_demo.js"></script>
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