Visualisation of probability <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \xi _{t}(i,j)}"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><msub><mi>ξ<!-- ξ --></mi><mrow class="MJX-TeXAtom-ORD"><mi>t</mi></mrow></msub><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo stretchy="false">)</mo></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle \xi _{t}(i,j)}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e4d94d72b11a8c2d37d311b389ee2e97c56917e" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:6.448ex; height:2.843ex;" alt="{\displaystyle \xi _{t}(i,j)}"></span> of change from state i to j in Hidden Markov Model: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{matrix}\xi _{t}(i,j)&amp;=&amp;P(q_{t}=S_{i},q_{t+1}=S_{j}|O,\lambda )\\&amp;=&{\frac {\alpha _{t}(i)a_{ij}b_{j}(O_{t+1})\beta _{t+1}(j)}{\sum \limits _{k=1}^{N}\sum \limits _{l=1}^{N}\alpha _{t}(k)a_{kl}b_{l}(O_{t+1})\beta _{t+1}(l)}}\end{matrix}}}"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mrow class="MJX-TeXAtom-ORD"><mtable rowspacing="4pt" columnspacing="1em"><mtr><mtd><msub><mi>ξ<!-- ξ --></mi><mrow class="MJX-TeXAtom-ORD"><mi>t</mi></mrow></msub><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo stretchy="false">)</mo></mtd><mtd><mo>=</mo></mtd><mtd><mi>P</mi><mo stretchy="false">(</mo><msub><mi>q</mi><mrow class="MJX-TeXAtom-ORD"><mi>t</mi></mrow></msub><mo>=</mo><msub><mi>S</mi><mrow class="MJX-TeXAtom-ORD"><mi>i</mi></mrow></msub><mo>,</mo><msub><mi>q</mi><mrow class="MJX-TeXAtom-ORD"><mi>t</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>=</mo><msub><mi>S</mi><mrow class="MJX-TeXAtom-ORD"><mi>j</mi></mrow></msub><mrow class="MJX-TeXAtom-ORD"><mo stretchy="false">|</mo></mrow><mi>O</mi><mo>,</mo><mi>λ<!-- λ --></mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd></mtd><mtd><mo>=</mo></mtd><mtd><mrow class="MJX-TeXAtom-ORD"><mfrac><mrow><msub><mi>α<!-- α --></mi><mrow class="MJX-TeXAtom-ORD"><mi>t</mi></mrow></msub><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo><msub><mi>a</mi><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mi>j</mi></mrow></msub><msub><mi>b</mi><mrow class="MJX-TeXAtom-ORD"><mi>j</mi></mrow></msub><mo stretchy="false">(</mo><msub><mi>O</mi><mrow class="MJX-TeXAtom-ORD"><mi>t</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo><msub><mi>β<!-- β --></mi><mrow class="MJX-TeXAtom-ORD"><mi>t</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo></mrow><mrow><munderover><mo movablelimits="false">∑<!-- ∑ --></mo><mrow class="MJX-TeXAtom-ORD"><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mrow class="MJX-TeXAtom-ORD"><mi>N</mi></mrow></munderover><munderover><mo movablelimits="false">∑<!-- ∑ --></mo><mrow class="MJX-TeXAtom-ORD"><mi>l</mi><mo>=</mo><mn>1</mn></mrow><mrow class="MJX-TeXAtom-ORD"><mi>N</mi></mrow></munderover><msub><mi>α<!-- α --></mi><mrow class="MJX-TeXAtom-ORD"><mi>t</mi></mrow></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><msub><mi>a</mi><mrow class="MJX-TeXAtom-ORD"><mi>k</mi><mi>l</mi></mrow></msub><msub><mi>b</mi><mrow class="MJX-TeXAtom-ORD"><mi>l</mi></mrow></msub><mo stretchy="false">(</mo><msub><mi>O</mi><mrow class="MJX-TeXAtom-ORD"><mi>t</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">)</mo><msub><mi>β<!-- β --></mi><mrow class="MJX-TeXAtom-ORD"><mi>t</mi><mo>+</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mi>l</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow></mtd></mtr></mtable></mrow></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}\xi _{t}(i,j)&amp;=&amp;P(q_{t}=S_{i},q_{t+1}=S_{j}|O,\lambda )\\&amp;=&amp;{\frac {\alpha _{t}(i)a_{ij}b_{j}(O_{t+1})\beta _{t+1}(j)}{\sum \limits _{k=1}^{N}\sum \limits _{l=1}^{N}\alpha _{t}(k)a_{kl}b_{l}(O_{t+1})\beta _{t+1}(l)}}\end{matrix}}}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2feac7c639b24f863a1b18c3a3663fb37f3a290f" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -4.736ex; margin-bottom: -0.269ex; width:39.634ex; height:11.176ex;" alt="{\displaystyle {\begin{matrix}\xi _{t}(i,j)&amp;=&amp;P(q_{t}=S_{i},q_{t+1}=S_{j}|O,\lambda )\\&amp;=&{\frac {\alpha _{t}(i)a_{ij}b_{j}(O_{t+1})\beta _{t+1}(j)}{\sum \limits _{k=1}^{N}\sum \limits _{l=1}^{N}\alpha _{t}(k)a_{kl}b_{l}(O_{t+1})\beta _{t+1}(l)}}\end{matrix}}}"></span>
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