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Fusion of probabilistic unreliable indirect information into estimation serving to decision making


Bayesian decision making (DM) quantifies information by the probability density (pd) of treated variables. Gradual accumulation of information during acting increases the DM quality reachable by an agent exploiting it. The inspected accumulation way uses a parametric model forecasting observable DM outcomes and updates the posterior pd of its unknown parameter. In the thought multi-agent case, a neighbouring agent, moreover, provides a privately-designed pd forecasting the same observation. This pd may notably enrich the information of the focal agent. Bayes’ rule is a unique deductive tool for a lossless compression of the information brought by the observations. It does not suit to processing of the forecasting pd. The paper extends solutions of this case. It: \(\triangleright\) refines the Bayes’-rule-like use of the neighbour’s forecasting pd \(\triangleright\) deductively complements former solutions so that the learnable neighbour’s reliability can be taken into account \(\triangleright\) specialises the result to the exponential family, which shows the high potential of this information processing \(\triangleright\) cares about exploiting population statistics.

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  1. The existence of regular probability densities of inspected probabilistic measures with respect to Lebesgue’s or counting measures is assumed [39].

  2. Our manipulations assume discrete-valued modelled variables. The uncertain pds acting on them are probabilistic vectors and their distributions are then modelled without technicalities of the measure theory. The found solution is valid without this assumption.

  3. The work [43] calls the same functional “cross-entropy”. The use of this term is often challenged so we stay with the name “Kulback–Leibler divergence”.

  4. The agreed implicit conditioning on the agent’s action, \(a_{\mathfrak {a}}\), and its regressor, \(r_{\mathfrak {a}}\), applies.

  5. It uses the implicit conditioning \(\mathsf {F}_{\mathfrak {n}}(o_{\mathfrak {a}})=\mathsf {F}_{\mathfrak {n}}(o_{\mathfrak {a}}|a_{\mathfrak {a}},r_{\mathfrak {a}})\), \(\mathsf {M}_{\mathfrak {a}}(o_{\mathfrak {a}}|p)=\mathsf {M}_{\mathfrak {a}}(o_{\mathfrak {a}}|p,a_{\mathfrak {a}},r_{\mathfrak {a}})\). The assumed neighbour, see Sect. 2, implies the relevance of \(a_{\mathfrak {a}},r_{\mathfrak {a}}\) in the forecasting pd \(\mathsf {F}_{\mathfrak {n}}\).

  6. Let us stress that the neighbour, \(\mathfrak {n}\), is generally unaware of the model, \(\mathsf {M}_{\mathfrak {a}}\), and its parameter, p.

  7. It uses the KLDs of posterior pds not the KLDs of joint pds.


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The paper was notably influenced by discussions with Dr. T.V. Guy.


The reported research has been supported by MŠMT ČR LTC18075 and EU-COST Action CA16228.

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Both authors tightly cooperated on the paper. MK dominated in writing the text and FH in experiments.

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Correspondence to Miroslav Kárný.

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Kárný, M., Hůla, F. Fusion of probabilistic unreliable indirect information into estimation serving to decision making. Int. J. Mach. Learn. & Cyber. 12, 3367–3378 (2021).

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  • Kullback–Leibler divergence
  • Distributed data fusion
  • Information fusion
  • Multi-agent
  • Parameter estimation
  • Decision making
  • Bayesian paradigm